Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators Chapter Outline 6.1 General 471 6.2 Three-phase voltage-sou...

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Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators Chapter Outline 6.1 General 471 6.2 Three-phase voltage-source converters

472

6.2.1 Basic operation of voltage-source converters 472 6.2.2 Basic control of voltage-source converters 477 6.2.3 Current capability of converter insulated gate bipolar transistor (IGBT) switches 477

6.3 Types of wind turbine generator technologies 477 6.4 Type 1 fixed-speed wind turbine induction generators 478 6.4.1 Modelling and analysis of short-circuit current contribution 478 6.4.2 Example 479

6.5 Type 2 variable slip wind turbine wound rotor induction generators 480 6.5.1 Modelling and analysis of short-circuit current contribution 480 6.5.2 Example 483

6.6 Type 3 variable-speed wind turbine doubly fed induction generators (DFIG) 484 6.6.1 6.6.2 6.6.3 6.6.4 6.6.5 6.6.6 6.6.7 6.6.8 6.6.9 6.6.10 6.6.11 6.6.12

Background 484 Basic operation principle 485 Rotor protection 486 Passive and active crowbar protection 487 dc chopper protection 489 General model of doubly fed induction generators (DFIG) 490 DFIG steady-state equivalent circuit 490 DFIG natural stator and rotor short-circuit currents under constant ac excitation 491 DFIG stator and rotor short-circuit currents under crowbar action 497 DFIG short-circuit currents with dc chopper control 505 DFIG short-circuit currents with rotor converter control 506 Examples 507

6.7 Type 4 variable-speed inverter-interfaced wind turbine generators 511 6.8 Type 5 variable-speed wind turbine synchronous generators 6.9 Solar photovoltaic (PV) generators and connection to the ac grid 512 6.9.1 Solar PV applications 512

Power Systems Modelling and Fault Analysis. DOI: https://doi.org/10.1016/B978-0-12-815117-4.00006-0 Copyright © 2019 Dr. Abdul Nasser Dib Tleis. Published by Elsevier Ltd. All rights reserved.

511

6

470

Power Systems Modelling and Fault Analysis

6.9.2 6.9.3 6.9.4 6.9.5

Solar PV generator components 512 Solar PV voltage-source inverters 513 Grid connection of solar PV power plant 514 Short-circuit current contribution of solar PV inverters 516

6.10 Technologies interfaced to the ac grid through voltage-source inverters 516 6.11 Modelling and analysis of grid-connected voltage-source inverters 517 6.11.1 6.11.2 6.11.3 6.11.4 6.11.5

General inverter model 517 Phase-locked loop 520 Inverter inner current control loop 521 Inverter outer control loops 525 Short-circuit current contribution of voltage-source inverters with frozen controls 527

6.12 Grid code requirements for dynamic reactive current injection from inverters 530 6.13 Short-circuit current contribution of voltage-source inverters during balanced three-phase voltage dips 533 6.13.1 Dynamic reactive current control 533 6.13.2 Examples of inverter positive-sequence short-circuit current contribution 534

6.14 Short-circuit current contribution of voltage-source inverters during unbalanced voltage dips 537 6.14.1 Inverter model in positive- and negative-sequence synchronous reference frames 537 6.14.2 Examples of inverter positive- and negative-sequence short-circuit current contributions

539

6.15 Sequence network representation of voltage-source inverters during balanced and unbalanced short-circuit faults 542 6.16 Short-circuit currents in ac grids containing mixed synchronous generators and inverters 543 6.16.1 6.16.2 6.16.3 6.16.4 6.16.5 6.16.6

General 543 Steady-state and transient characteristics of inverter short-circuit reactive currents 544 Analysis of three-phase short-circuit fault currents 545 Analysis of single-phase short-circuit fault currents 548 Analysis of two-phase short-circuit fault currents 554 Examples 559

6.17 Grid-forming voltage-source inverters

568

6.17.1 Background to grid-following and grid-forming inverters 568 6.17.2 Emerging challenges in power systems dominated by grid-following voltage-source inverters 568 6.17.3 Control structure of grid-forming inverters 569

6.18 Grid-forming virtual synchronous machine (VSM) inverters

570

6.18.1 Possible VSM inverter features of real synchronous machines 570 6.18.2 A model of grid-forming VSM inverters 571

6.19 Grid-forming voltage-source inverters using droop control 575 6.20 Natural short-circuit current contribution of grid-forming inverters 6.20.1 Natural three-phase short-circuit current of grid-forming inverters 579 6.20.2 Natural two-phase short-circuit current of grid-forming inverters 580

579

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

6.21 Over-current limitation strategies of grid-forming inverters

471

581

6.21.1 General 581 6.21.2 Current limitation in inverters using single voltage controller structure 582 6.21.3 Current limitation strategies in inverters using cascaded control structure 586

6.22 Symmetrical components sequence equivalent circuits of ‘grid-forming’ inverters 589 6.23 Examples 591 Further reading 595

6.1

General

Until the end of the 20th century, almost all of the world’s electric power supply was produced by synchronous generators in thermal power and hydropower generation plant. With mounting concerns over climate change and the need for reduced green house gas such as CO2 emissions, renewable energy, particularly wind and solar photovoltaic (PV) generators, have now become a mainstream source of electric energy supply. Wind turbine generators utilising a range of nonsynchronous generators are used as a major new energy source in many countries around the world that have good wind resources. Medium- to large-scale onshore and offshore wind parks consist of tens to hundreds of wind turbine generators. At the time of writing, individual turbines have reached a rating of 4.8 MW onshore and 8 MW offshore. Wind turbine sizes of 1012 MW offshore are nearing full-scale deployment. The most installed wind turbine generator technologies are the doubly fed and full-converter generators and the latter is likely to dominate most future installations. The largest wind turbine generators offshore are direct drive, that is, without a gearbox, employing permanent magnet synchronous generators which are interfaced with the ac grid through a full-size ac/dc/ac converter. Solar PV generators utilising full-size dc to ac voltage-source inverters are also used as a major new energy source in many countries around the world that have a good or even average solar irradiance resource. Solar PV generators range from the very small with a few kW in size installed on domestic rooftops to very large, utility-scale, ground-mounted, solar parks of hundreds of MW in size. An individual solar PV generator interfaced to the ac grid through a dc/ac voltage-source inverter can be as small as 1 or 2 kVA single-phase or as large as 4.4 MVA three-phase in a large solar park containing tens to hundreds of such individual generators. High-voltage direct-current (HVDC) transmission systems employing voltagesource converter technology represent a system that is interfaced to the ac grid through a dc/ac voltage-source converter. Typical uses are as a transmission link connecting a remote, large, offshore wind farm to the onshore ac grid, a long submarine cable HVDC link and an interconnection between two asynchronous ac systems.

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Power Systems Modelling and Fault Analysis

Battery energy storage systems (BESS) are interfaced to the ac grid through a dc/ac voltage-source inverter technology. They are used to provide energy storage or ancillary services to the ac grid such as fast frequency response, reserves, reactive power and voltage control, and other ancillary services. Currently, installations in the few MW to a few tens of MW capacity are emerging but with further advances in battery technologies and reduction in installed costs, large BESS in the hundreds of MW capacity may become cost-effective and an indispensable solution to the integration of very large amounts of variable and nondispatchable wind and solar PV generation systems. Voltage-source inverters that use alternative control strategies to become gridforming inverters are an exciting alternative in grid-connected applications to conventional current-controlled grid-following converters. The latter follow the already available grid voltage and control their current output. However, grid-forming inverters can form their voltage and act as a true voltage source. Grid-forming inverters can either be implemented to replicate certain aspects of the behaviour of real synchronous machines known as virtual synchronous machines (VSM or VISMA) or be droop controlled. The latter have been used in small isolated grids, microgrids, and special applications such as marine and aeronautical power systems but connections to large-scale grids are also possible. Grid-forming inverters are expected to be adopted by inverter manufacturers in large-scale grid-connected applications.

6.2

Three-phase voltage-source converters

6.2.1 Basic operation of voltage-source converters There are many types and configurations of power electronics converters in use. However, within the scope of this book, we will briefly introduce some of the basic characteristics of three-phase voltage-source converters used in wind turbine generators, solar PV generators, battery energy storage, HVDC transmission and other applications. A three-phase voltage-source converter can be viewed as a three-phase voltage source whose magnitude, frequency and phase can be controlled simultaneously. A dc to ac converter is called an inverter and an ac to dc converter is called a rectifier. Fig. 6.1 shows a three-phase two-level six-switch voltage-source converter using insulated gate bipolar transistor (IGBT) switches predominantly used in low-voltage systems. Three-level converters and the more recent modular multilevel converters typically used in higher voltage systems are outside the scope of this book but the operation principle is similar. In this topology, a relatively large capacitor feeds the three-phase bridge inverter circuit with a dc voltage, hence the term voltage-source inverter. The bridge consists of six switches; each is a power transistor IGBT with a free-wheeling diode to provide bidirectional current flow and a unidirectional voltage-blocking capability. With proper control algorithms, this enables the converter to operate either as a

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

473

+C 2

–

,

( ) ( )

+ C 2 –

( )

Figure 6.1 Six-switch two-level voltage-source converter.

rectifier or an inverter without any change in topology. The dc capacitor is charged to a certain voltage level which ensures the basic function of the converter which is to control the ac current through the switching strategy. The LC, or in many installations LCL filter on the ac side, is used as a low-pass filter as well as to smooth the ripples in the ac current and dc voltage waveforms. The two-level refers to the number of voltage levels that can be produced at the output of each leg of the bridge. The converter switches are switched on and off with a fixed frequency but with a pulse width that is varied in order to control the output voltage. Fig. 6.2A shows one leg or one phase, phase r, of the converter with two IGBT switches S1 and S2, each with its antiparallel diode. When switch S1 is closed, the ac output voltage ero with respect to the zero-voltage node is Vdc/2 but when switch S2 is on, ero is 2 Vdc/2. The output voltage ero can be controlled between 2 Vdc/2 and Vdc/2. In a three-level inverter, the inverter is controlled so that in the positive half cycle of reference waveform, the output voltage is controlled between Vdc/2 and zero and in the negative half cycle between 2 Vdc/2 and zero, hence the three levels 2 Vdc/2, zero and Vdc/2. In the positive half cycle, switch S1 would be on and off but switch S2 would be off. The converse applies in the negative half cycle. The ac output voltage can be changed by modulating the width of the Vdc pulse. Fig. 6.2B illustrates a sinusoidal pulse-width modulated waveform and shows how the width of the pulse can be adjusted by comparing a modulating or reference waveform with a carrier waveform of a triangular shape. The reference waveform is the desired waveform at the output and the triangular waveform determines the frequency with which the IGBT switches are switched. When the reference waveform is higher than the triangular waveform, a switching pulse is generated and switch S1 is turned on. The carrier frequency should always be an odd multiple of 3 as this provides halfand quarter-wave symmetry which eliminates even harmonics and generates symmetrical three-phase voltages. The frequency modulation index fm is defined as fm 5

fsw fo

(6.1)

where fsw is the switching frequency in Hz and fo is the fundamental power frequency in Hz.

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Power Systems Modelling and Fault Analysis

(A)

+

2

– 0

+

2

(B)

–

Reference waveform

Carrier waveform

0

2 0

− 2

ΔT1, on

ΔT2, on

Fundamental er0

Figure 6.2 (A) One leg of six-switch two-level voltage-source inverter and (B) generation of an ac waveform using carrier-based pulse width modulation.

The duty cycle or amplitude modulation index is given by mo 5

E^ out Vdc =2

(6.2a)

0 , mo # 1

(6.2b)

where E^ out is the peak of the fundamental component of the ac output voltage ero . Eq. (6.2a) can also be expressed as mo 5 ΔT1;on 3 fsw 5 Tsw 5

1 fsw

ΔT1;on Tsw

(6.3a) (6.3b)

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

475

and ΔT1;on is the on-time of the switch S1 and Tsw is the period of the carrier waveform. mo lies between 0 (with S2 on continuously) and 1 (with S1 on continuously). However, mo can be varied in time and therefore any desired voltage and frequency can be generated for a given switching frequency and dc voltage. In Fig. 6.2B, the following parameters have been used: fo 5 50Hz; mo 5 0:9; fm 5 15 or fsw 5 750 Hz. A low value for fsw has been deliberately chosen to improve the clarity of the illustration although typical values in practice are much higher (a few kHz). Sinusoidal pulse-width modulation of two-level converters generates odd harmonics at the switching frequency fsw and its multiples as well as the sidebands around them. The high-frequency harmonics are filtered using passive tuned filters. An alternative to modelling every switching state of the converter is to use an average converter model which assumes that mo is constant during one switching cycle as implied in Eq. (6.2a). This is generally sufficiently accurate because the carrier signal switching frequency is at least 1540 times higher than the frequency of the modulating signal. Therefore in the three-phase converter as shown in Fig. 6.1, there are three modulation indices mr ðtÞ, my ðtÞ and mb ðtÞ; and if these are designed to vary sinusoidally, the converter average output voltages with respect to the zero-voltage node of the dc capacitors, ignoring high-order harmonics, are given by er0 ðtÞ 5 mr ðtÞ

Vdc 2

(6.4a)

ey0 ðtÞ 5 my ðtÞ

Vdc 2

(6.4b)

eb0 ðtÞ 5 mb ðtÞ

Vdc 2

(6.4c)

where mr ðtÞ 5 mo sinðωo t 1 ϕÞ 2π my ðtÞ 5 mo sin ωo t 1 ϕ 2 3 2π mb ðtÞ 5 mo sin ωo t 1 ϕ 1 3

(6.5a) (6.5b) (6.5c)

and ωo 5 2πfo , fo is the power frequency. Therefore Eqs. (6.4a)(6.4c) can be written as Vdc sinðωo t 1 ϕÞ 2 Vdc 2π sin ωo t 1 ϕ 2 ey0 ðtÞ 5 mo 3 2 er0 ðtÞ 5 mo

(6.6a) (6.6b)

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Power Systems Modelling and Fault Analysis

Vdc 2π sin ωo t 1 ϕ 1 eb0 ðtÞ 5 mo 3 2

(6.6c)

Also, the converter output line-to-line voltages are given by ery 5 er0 2 ey0 5

pﬃﬃﬃ Vdc π sin ωo t 1 ϕ 1 3mo 6 2

pﬃﬃﬃ Vdc π sin ωo t 1 ϕ 2 3m o 2 2 pﬃﬃﬃ Vdc 5π sin ωo t 1 ϕ 1 ebr 5 eb0 2 er0 5 3mo 6 2 eyb 5 ey0 2 eb0 5

(6.7a)

(6.7b)

(6.7c)

and the converter output rms line or phase-to-phase voltage is related to the dc voltage by pﬃﬃﬃ 3 Ery rms 5 Eyb rms 5 Ebrrms 5 pﬃﬃﬃ mo Vdc 2 2

(6.8)

Eq. (6.8) represents balanced three-phase output line voltages whose amplitude is controlled by mo and output frequency and phase are controlled by the frequency of the modulating or reference waveform. Fig. 6.3 shows the three-phase pulse-width modulated output waveforms of the three-phase two-level converter.

er0 ey0 eb0 ery eyb ebr Figure 6.3 Three-phase line output voltages of six-switch two-level voltage-source inverter.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

477

6.2.2 Basic control of voltage-source converters The currently dominant technology of grid-connected voltage-source converters operates as controlled current sources. This means that the pulse width modulation (PWM) switching of the converter is controlled so that its output current is forced to follow a reference value in its inner current control loop. Even during large changes in grid voltage, the converter output current continues to follow the reference values for both positive- and negative-sequence currents because, as ‘seen’ from the grid, the converter appears to have a very high impedance. In this method of control, the current-controlled voltage-source converter controls the output current indirectly. The converter uses the grid voltage feedback and the voltage drop across the output filter inductance to synthesise the converter voltage feedback that will produce the required converter output currents. This is presented in detail in Section 6.11.

6.2.3 Current capability of converter insulated gate bipolar transistor (IGBT) switches The transient output current of the converter that can be sustained for a short period of a few seconds is generally limited to no more than 1.1 pu of rated continuous current. This protects the IGBT switches from damage, avoid oversizing the converter and help to select an economic converter rating that is consistent with the rated active power of the feeding energy source, be it a solar field, a wind turbine generator or a HVDC transmission link, etc. Typically, therefore, the IGBT switches can carry up to 1.55 pu instantaneous current and up to 2 pu current for a maximum of 1 ms. Internal protection in the power stack IGBT unit samples voltages and currents at typically 1 μs and can block the IGBT current, that is, reduce it to zero in a few microseconds. External voltage dips cause an outrush current from the filter capacitor to flow out of the converter terminals. This is a high-frequency, oscillatory transient current with an initial peak of typically 2.55 pu and a frequency of several kHz. This damped current decays completely to zero within a few milliseconds. Obviously, this physical current does not flow through the converter IGBTs and hence cannot be controlled by the converter.

6.3

Types of wind turbine generator technologies

Several types of wind turbine generators are used in onshore and offshore wind parks. Wind turbine generators that are directly connected to the ac grid employ induction, rather than synchronous, generators. Synchronous generators are also widely used but, currently, these are not directly connected to the ac network. Rather, they are connected through a back-to-back power electronics voltage-source converter. Directly connected synchronous generators may be used in future if the development of variable-speed gearbox technology proves sufficiently reliable and

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Power Systems Modelling and Fault Analysis

commercially attractive. The variable-speed gearbox has a variable input speed on the turbine side but produces constant output synchronous speed. At the time of writing, their large-scale commercial use is yet to materialise. Wind turbine generator technologies have been classified for convenience into five types based on their specific technology. These are discussed in the following sections.

6.4

Type 1 fixed-speed wind turbine induction generators

6.4.1 Modelling and analysis of short-circuit current contribution Type 1 fixed-speed induction generators are essentially similar to squirrel-cage induction motors. These are driven by a wind turbine prime mover at a speed just above synchronous speed, normally up to 1% rated slip for today’s large wind turbines. Because the speed variation from no load to full load is very small, the term ‘fixed’ speed is widely used. The generator is coupled to the wind turbine rotor via a gearbox as shown in Fig. 6.4. The design and construction of the stator and rotor of an induction generator are similar to those of a large induction motor with a squirrel-cage rotor. In Chapter 5, Modelling of rotating ac synchronous and induction machines, we presented the modelling of induction motors and analysed their short-circuit current contribution. The positive- and negative-sequence short-circuit contribution of a fixed-speed induction generator can be represented in a similar way to that of an induction motor. The equations of transient reactances and time constants derived for induction motors can also be used for induction generators. The stator windings of these generators are usually connected in delta or star with an isolated neutral. Thus their zero-sequence impedance to the flow of zero-sequence currents is infinite.

Wind turbine

Squirrel-cage Gearbox induction generator

Generator- MV transformer Grid

Power factor correction capacitors

Figure 6.4 Type 1 fixed-speed wind turbine induction generator.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

479

Using Eq. (5.112b), to represent a single-cage fixed-speed induction generator, the instantaneous short-circuit currents delivered by the machine for a three-phase solid fault at its terminals are given by ii ðtÞ 5

pﬃﬃﬃ 2Vrms

1 1 2t=T 0 1 2t=Ta 2 sin ð ω t 1 θ Þ 2 e sinθ e o i i X0 X X0

(6.9)

where θr 5 θo

i 5 r; y; b

θy 5 θo 2 2π=3

θb 5 θo 1 2π=3

and θo is the instant of fault occurrence on machine stator phase r voltage waveform and the machine parameters are as given in Chapter 5, Modelling of rotating ac synchronous and induction machines.

6.4.2 Example Example 6.1 Consider a type 1 fixed-speed squirrel-cage wind turbine induction generator having the following parameters:

Machine voltages (pu)

Rated voltage 0.69 kV Stator resistance 0.0014 Ω Stator leakage reactance 0.018 Ω

Apparent power 2.3 MVA, 50 Hz Rotor resistance (referred to stator) 0.0023 Ω Rotor leakage reactance (referred to stator) 0.014 Ω, magnetising reactance 0.7 Ω

1.5

Phase R Phase Y Phase B

1 0.5

0 –40 –0.5

0

40

80

120

160

200 Time (ms)

40

80

120

160

200 Time (ms)

–1

Machine currents (pu)

–1.5 15

R

10

5 0 –40 –5 –10

0 Y

B

–15

Figure 6.5 Three-phase short-circuit currents of type 1 fixed-speed induction generator.

480

Power Systems Modelling and Fault Analysis

15

R

Components of phase r fault current (pu)

10 5 0 –40

0

40

80

120

–5

160

200 Time (ms)

–10

10 dc component of fault current

5 0 –40

0

–5

40

80

120

160 200 Time (ms)

ac component of fault current

–10

Figure 6.6 Components of phase R short-circuit current of type 1 fixed-speed induction generator.

Using the rated voltage and apparent power as base quantities, the machine parameters in per unit are: stator resistance 0.00676 pu, rotor resistance 0.01111 pu, magnetising reactance 3.3816 pu and stator leakage reactance 0.086957 pu and rotor leakage reactance 0.07198 pu. Using Eqs. (5.93b), (5.94c) and (5.108), we obtain: 0

0

X 5 0:1574 pu, T 5 44:9ms and Ta 5 74:1ms. We assume that θo 5 0 to obtain maximum dc current offset on phase r. The fault currents are shown in Fig. 6.5. Fig. 6.6 shows the ac and dc components of phase r which has the maximum dc current offset.

6.5

Type 2 variable slip wind turbine wound rotor induction generators

6.5.1 Modelling and analysis of short-circuit current contribution This type of induction generator historically uses a three-phase wound rotor winding that is accessible via brushes and slip rings. The rotor windings are connected to an external resistor circuit through an ac/dc power electronics converter that modifies the rotor circuit resistance by injecting a variable external resistance on the rotor circuit. The variation in the effective rotor resistance enables control of

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

481

the magnitude of rotor currents and hence generator electromagnetic torque. The rated slip of large fixed-speed squirrel-cage induction generators is typically around 2 1% but the speed of the wound rotor induction generator can be varied over a small range; typically up to 10% giving a variable slip range of up to 2 10%. Typical values of effective external resistance, as a multiple of rotor winding resistance, required to drive the machine at slips of 2 2%, 2 5% and 2 10% are 1, 4 and 8, respectively. Fig. 6.7A illustrates this type of wind turbine generator with a typical ac/dc power converter for external resistance control. The converter typically consists of a three-phase diode bridge rectifier, three-phase resistors and an IGBT chopper. The rectifier consists of six diodes and converts the three-phase ac voltages at the rotor terminals to a dc voltage and the IGBT is inserted on the dc side of the rectifier. When the IGBT is on, it short-circuits the rotor winding and this reduces the external resistance to zero. However, when the IGBT is off, the entire external resistance appears in series with the rotor circuit. The average value of the external resistor is controlled by the duty cycle mo of the IGBT switch using a classical PWM technique. Eq. (6.3a) gives the IGBT duty cycle and if ideal diodes and IGBT chopper are assumed, then the effective value of the external resistor appearing in series with the generator rotor winding resistor would be Rext ð1 2 mo Þ where 0 , mo # 1. An alternative arrangement is the use of the diode bridge with a single external resistor on the dc side in parallel with the IGBT chopper instead of the three resistances on the rotor winding ac side. This arrangement, shown in Fig. 6.7B,

(A) Wind turbine

Wound rotor Gearbox induction generator

Generatortransformer MV Grid Power factor correction capacitors

Rext Three-phase diode bridge and IGBT chopper (B)

To rotor Rext

IGBT

Three-phase diode bridge with single resistor and IGBT chopper

Figure 6.7 Type 2 variable slip wind turbine induction generator: (A) General diagram and (B) alternative variable external rotor resistor circuit.

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Power Systems Modelling and Fault Analysis

requires only one resistor but tends to cause mechanical vibrations in the generator drive train which result from current rectification and hence harmonics in rotor currents. Modern designs have also been built that avoid slip rings and brushes, the socalled brushless designs. The most widely used approach is to mount the resistors, rectifier, current sensors and controller on the rotating rotor of the generator with communication signals transmitted over fibre optic cables. When a three-phase short-circuit fault occurs at the terminals of an induction generator, the generator will inherently supply a large stator short-circuit current. As we have already seen for synchronous generators and induction motors, due to the theorem of constant flux linkages, a corresponding increase in the generator’s rotor currents occurs. For a variable slip wound rotor induction generator, the rotor converter used to control the rotor currents is suddenly subjected to large overcurrents. However, the short-term over-current capability of these switches is extremely limited and if their junction temperature is exceeded, they would be damaged. When the instantaneous rotor current reaches the IGBT current threshold limit, the IGBT is immediately blocked and this effectively inserts the entire external resistor circuit in series with the rotor circuit. The blocking of the IGBT switch is extremely fast and typically occurs in less than a few microseconds for the higher rated IGBTs. This causes the rotor winding of the generator to become effectively similar to that of a conventional wound rotor induction motor but with an additional external resistance. Fig. 6.8 shows the induction generator’s equivalent circuit. Therefore the generator’s initial short-circuit current contribution to a fault on the ac grid can be calculated with an effective rotor resistance equal to the sum of the rotor winding resistance and the external resistance Rext . The effect of Rext on the time constant of the ac short-circuit current component is given as 0

0

Te 5

Xr ðRr 1 Rext Þ

Figure 6.8 Steady state equivalent circuit of type 2 variable slip induction generator.

(6.10a)

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

483

! 0

Xr 5 Xσr 1

1 Xσs

1 1

(6.10b)

1 Xm

The additional rotor resistance Rext reduces the generator ac transient time constant and hence increases the rate of decay of the ac current component of the short-circuit current. To the best of the author’s knowledge, the effect of Rext on the dc time constant of the stator dc short-circuit current component of a type 2 induction generator has never been addressed in the published literature. The implicit assumption has been that Rext does not affect the dc time constant. However, for the first time, we show in Section 6.6.9 that this is only valid for small values of Rext but not moderate and large values. The effect can be calculated using Eq. 6.33b. As the stator short-circuit current decays, and the rotor current drops below the protection or the converter-controlled reference value, some generator manufacturers design their converters to quickly unblock the IGBT switch and regain control of the rotor currents back to some specified value. This means that the generator starts to supply a constant, predefined, low-value stator short-circuit current.

6.5.2 Example Example 6.2 Consider a type 2 wind turbine wound rotor induction generator having the following parameters: Rated voltage 0.69 kV Stator resistance 0.0054 pu Stator leakage reactance 0.102 pu

Apparent power 2.5 MVA, 60 Hz Rotor resistance (referred to stator) 0.00608 pu Rotor leakage reactance (referred to stator) 0.11 pu, magnetising reactance 4.36 pu

Calculate and plot the phase r fault current for a three-phase solid fault at the machine terminals assuming that θo 5 0 to obtain maximum dc current offset on phase r for the following cases: a. The external resistance is short-circuited. b. The external resistance is chosen by design to be equal to four times the rotor winding resistance. c. The external resistance is chosen by design to be equal to nine times the rotor winding resistance. Neglect the effect of the external resistance on the dc time constant Using Eqs. (5.93b), (5.94c) and (5.108), and Eqs. (6.10a) and (6.10b), we obtain 0 Xr 5 0:20929 pu, Ta 5 102:81ms and Case (a): Case (b): Case (c):

0

T 5 91:475 ms 0 Te 5 18:295ms 0 Te 5 9:147ms

Fig. 6.9 illustrates the effect of external rotor resistance on the stator ac rms short-circuit current and asymmetric current.

Power Systems Modelling and Fault Analysis

Machine currents (pu)

Machine currents (pu)

484

14

Asymmetric phase R fault currents

= 0 = 4 = 9

10 6 2 –2 –50

8

0

50

100

150

200

250 300 Time (ms)

=0 =4 =9

dc components of fault currents

4 0 –50

0

50

100

150

–4

200

250 300 Time (ms)

ac components of fault currents

–8

Figure 6.9 Effect of external resistor on short-circuit current of type 2 variable slip induction generator. Case (b) shows that a moderate value of external resistance is sufficient to cause a substantial reduction in the rotor transient time constant and a reduction in the first peak of the shortcircuit current which is caused by the reduction in the ac component of the short-circuit. Within four cycles, the ac current component is almost completely damped out. In case (c), the ac current is damped in around two cycles.

6.6

Type 3 variable-speed wind turbine doubly fed induction generators (DFIG)

6.6.1 Background These are variable-speed wound rotor induction generators with two bidirectional back-to-back voltage-source converters, as shown in Fig. 6.10. The three-phase stator winding is directly connected to the grid. The three-phase rotor windings are connected to a voltage-source inverter via slip rings. This rotor-side inverter injects three-phase rotor voltages of variable frequency, magnitude and phase. The ac terminals of the grid-connected converter are connected to the grid either at the generator stator terminals or the tertiary winding of a three-winding generator step-up transformer. The grid-connected converter controls the dc voltage and sometimes some reactive power output, if it was sized and specified to do so. The rotor-side converter injects a rotor voltage and controls the rotor currents almost instantaneously. This control provides two important functions. The first is variation of generator

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

Wind turbine

Wound rotor doubly-fed Gearbox induction generator

485

Generatortransformer Grid

ac-rotor frequency

PWM converters Grid frequency

Crowbar circuit

dc Rotor-side Grid-side converter chopper converter

Figure 6.10 Type 3 variable-speed doubly fed wind turbine induction generator.

electromagnetic torque and hence rotor speed. The second is usually stator terminal voltage control or, unusually, constant stator reactive power control or stator power factor control. The typical speed control range for a modern MW class wind turbine doubly fed induction generator (DFIG) may be between 70% and 130% of nominal synchronous speed, with the 130% usually referred to as the rated speed at which rated MW output is produced. The rating of the back-to-back converter corresponds to the speed variation so that, for a speed range of 30%, the rating of the converter is 30% of the generator rating. This reduces the cost of the converter compared to type 4 full-size converter-interfaced generators discussed in Section 6.7.

6.6.2 Basic operation principle In a synchronous machine, dc excitation produces a static magnetic field and this rotates at the speed of the rotor when the latter is driven by a prime mover. As a result, this rotating magnetic field produces ac voltages in the stator windings. For a conventional or singly fed induction generator, electrical power is produced when the generator’s rotor speed is driven above synchronous speed, resulting in a negative slip. The frequency of the rotor speed corresponds to the frequency of the stator plus the electrical frequency of the rotor current. This principle can be applied to DFIG with ac current excitation fed into the three-phase rotor windings. The speed of the rotating stator magnetic field is given by the rotational speed of the rotor and the frequency of the ac currents supplied to the rotor windings. In other words, we can write fstator 5

Nrotor 3 npp 6 frotor 60

(6.11)

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Power Systems Modelling and Fault Analysis

where fstator is the frequency of the induced stator voltage in Hz; Nrotor is the mechanical speed of the rotor in rpm; npp is the number of pole pairs of the rotor; andfrotor is the frequency of rotor currents in Hz. The positive sign in Eq. (6.11) applies when the generator rotor rotates below synchronous speed and the rotor current excitation creates a magnetic field that rotates in the same direction as the generator rotor. However, the negative sign applies when the generator rotor rotates above synchronous speed and the ac current excitation applied to the rotor windings is negative sequence so that it creates a magnetic field that rotates in the opposite direction to the generator rotor. It is noted that due to the ac rotor current excitation, a laminated rotor design is used which results in short rotor flux time constants. The phase angle of the ac rotor current supplied to the rotor windings sets up the phase angle of the internal stator source voltage and hence affects the flow of active power. The magnitude of the ac rotor currents determines the magnitude of the internal stator source voltage and hence the flow of reactive power. Eq. (6.11) shows that in order to produce a constant stator frequency fstator of say, 50 or 60 Hz, the frequency of the rotor ac currents frotor supplied to the rotor must be continuously adjusted to counteract variations in rotor speed Nrotor caused by wind turbine mechanical power variations. Also, when the generator is connected to a network having a fixed power frequency equals to fnetwork , the adjustment of the rotor current frequency allows control of the rotor speed in such a way as to maintain a constant stator frequency equal to the network frequency. Example 6.3 Consider a wind turbine DFIG with four magnetic poles. The generator supplies power to a 50 Hz ac grid. The wind turbine drives the generator rotor at a speed of 1650 rpm. Calculate the frequency of the rotor ac currents that need to be fed into the rotor windings to produce a constant stator frequency of 50 Hz. The number of pole pairs is npp 5 2. The rotor synchronous speed is given by Nrotor 5 ð50 3 60Þ=2 5 1500 rpm. At 1650 rpm, the generator rotor is running above synchronous speed. Using Eq. (6.11), the frequency of the rotor ac currents is calculated as frotor 5 ð1650 3 2Þ=60 2 50 5 5Hz. If the rotor were running at a subsynchronous speed of 1410 rpm, then the frequency of the rotor ac currents is calculated as frotor 5 50 2 ð1410 3 2Þ=60 5 3Hz.

6.6.3 Rotor protection A simultaneous three-phase short-circuit fault in the network causes a symmetrical voltage dip at the generator terminals and large oscillatory currents in the stator windings and rotor windings connected to the rotor-side converter. Controlling large rotor currents requires a large and uneconomic rotor voltage rating. The large rotor currents flow through the converter switches and the dc link capacitor causing a steep rise in the dc link voltage. However, to protect the converter switches and dc capacitor from damage, a variety of methods and protection circuits have been used. These are generally called crowbar and dc chopper circuits.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

487

6.6.4 Passive and active crowbar protection Many DFIGs use a rotor converter protective circuit called a ‘crowbar’. The crowbar circuit is connected between the rotor winding and the rotor-side converter and is activated when either an instantaneous rotor current in any phase exceeds the allowable converter limit, or an overvoltage on the dc link capacitor exceeds its voltage limit. The rotor current limit is typically in the order of 1.1 pu of rated, since converter IGBTs can carry a very limited overcurrent for a very short time. The dc link voltage limit is typically set at 1.11.2 pu of rated dc link voltage. When these limits are exceeded, the converter switches are immediately blocked by blocking the firing pulses and the controllable switch of the crowbar circuit is simultaneously fired. This crowbar action effectively short-circuits the rotor winding, directly or through a small resistor. An earlier generation of DFIG used the so-called passive crowbar circuit where, following a large voltage dip on the network and a corresponding large dip on the generator terminals, the crowbar short-circuits the rotor winding then the main circuit breaker of the generator eventually disconnects the generator from the network. In these passive crowbars, the rotor winding remains short-circuited by the crowbar circuit until the generator stator is disconnected from the grid and the rotor currents decay to zero. Fig. 6.11 shows three typical passive crowbar circuits. The thyristor switches used in passive crowbar circuits cannot turn off the current until it decays to zero. However, following a grid voltage dip, the rotor current will invariably contain a dc current component that can typically take a few hundred milliseconds to decay. This precludes the ability to resume normal operation and rotor-side converter control during the voltage dip or disturbance. (A)

Thyristor crowbar

(B)

To rotor

Rotor-side converter

To rotor

Diode bridge thyristor crowbar

Rotor-side converter

To rotor

(C)

R Diode bridge thyristor crowbar

Rotor-side converter

Figure 6.11 DFIG passive crowbar circuits: (A) phase-to-phase antiparallel thyristor crowbar circuit, (B) diode bridge and a controlled thyristor crowbar circuit and (C) diode bridge and a controlled thyristor crowbar circuit with a crowbar resistor.

488

Power Systems Modelling and Fault Analysis

With increased penetration of wind turbine generation in ac power grids, network operators imposed a requirement called ‘low-voltage ride-through’ capability which generally dictates that the generator must not be disconnected and must quickly regain normal active and reactive power operation and control during or immediately following the low-voltage disturbance. In response, wind turbine generator manufacturers introduced the so-called active crowbar circuit with earlier gate turn-off (GTO) switches, but nowadays IGBT switches can switch off a current several times their rated value. Again, when rotor current or, mostly, dc link voltage limit is exceeded, the control system simultaneously disconnects the rotor-side converter by blocking IGBTs and connects the active crowbar circuit. This results in rerouting much of the high rotor currents through the external crowbar resistance which causes the high rotor currents to decay. Importantly, with an active crowbar, the crowbar resistor voltage and the dc link voltage are monitored during crowbar operation. When both voltages are low enough, the crowbar is turned off. After a short delay, to allow for the rotor currents decay, the rotor-side inverter is restarted, and the reactive current component of the generator is ramped up in order to support the grid. The crowbar may be reactivated after typically 4060 ms of the onset of the disturbance. Fig. 6.12 shows typical active crowbar circuits. During the period when the crowbar is activated, the rotor winding of the generator appears effectively similar to that of a conventional singly fed wound rotor induction generator with an external rotor resistance Rcb or a rectified equivalent resistance of Rcb , and no control over active or reactive power. The generator goes from generating to absorbing reactive power from the network which depresses the grid voltage and may delay voltage recovery. This is controlled by minimising the operation time of the crowbar by increasing the value of the crowbar resistance and hence increasing the rate of decay of rotor currents. However, too high a value of crowbar resistance can cause high voltage at the rotor-side converter terminals which drives a higher current through the antiparallel diodes of the IGBT switches and charges the dc link capacitor. This increases the dc link voltage and, if the limit is exceeded, the crowbar circuit could be triggered again after its deactivation, leading to an entire generator shutdown. Therefore the selection of the optimum value of resistor in an active crowbar circuit is very important.

To rotor IGBT R Active diode bridge crowbar with IGBT chopper

Figure 6.12 An active crowbar circuit of a DFIG.

Rotor-side converter

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

489

6.6.5 dc chopper protection A recent alternative to the active crowbar circuit is the use of a chopper circuit in the intermediate dc bus of the ac/dc/ac converter, as shown in Fig. 6.13A. The chopper circuit is connected in parallel with the dc link capacitor and consists of a controlled IGBT switch and a chopper resistor sized to handle a specific amount of energy as dictated by the specified grid code low-voltage ride-through requirement. In Fig. 6.13A, the dc chopper is used alone without a crowbar and therefore the rotor winding itself is not short-circuited. In the event of a short-circuit, the rotorside converter IGBT switches are blocked, causing the rotor current to be diverted through the antiparallel diodes to the dc link capacitor. The dc chopper resistor is switched by the chopper control in order to regulate and maintain the dc link voltage within specified limits, typically using a hysteresis controller that is generally set to around 1.11.2 pu. The chopper resistor quickly decreases the rotor current and the converter is restarted when the rotor current and dc link voltage have dropped below certain thresholds. However, the disadvantage of this method is that it requires the antiparallel diodes of the converter switches to be oversized to carry and withstand the highest rotor transient currents during grid voltage dips which may be economically unattractive. An alternative operation in this configuration is that the converter remains in operation, maintains control of the generator and supplies reactive current during grid faults or dips as required by most grid codes. However, in this case, the rotor-side converter IGBTs have to be oversized which, again, is generally economically unattractive. In some recent designs, both dc chopper and crowbar circuits are used as shown in Fig. 6.13B. The active crowbar offers the overcurrent protection of the converter, whereas the dc chopper controls the dc bus voltage to within specified limits. (A)

To rotor Rch

Rotor-side converter dc chopper

(B)

To rotor Rch

IGBT R Active diode bridge crowbar with IGBT chopper

Rotor-side converter

dc chopper

Figure 6.13 (A) dc chopper in parallel with dc link capacitor and (B) dc chopper with an active crowbar.

490

Power Systems Modelling and Fault Analysis

Overall, the rotor protection philosophy, implementation method, sizing of converter components, and settings of limits of rotor-side converter in a type 3 wind turbine DFIG vary among manufacturers. Actual equipment data, control parameters and models for specific project installations need to be based on actual information provided by manufacturers.

6.6.6 General model of doubly fed induction generators (DFIG) In Appendix A.2, we present modelling of induction machines in various reference frames including reference frame transformations. Because of the symmetry of the magnetic system, the analysis of stator and rotor short-circuit currents of induction generators is simplified using a complex space vector induction machine model in a rotor reference frame. Using the generator convention for stator current and all rotor quantities referred to the stator, the DFIG equations in per unit are v s ðtÞ 5 2 Rs i s ðtÞ 1 jωr ψ s ðtÞ 1 v r ðtÞ 5 Rr i r ðtÞ 1

dψ s ðtÞ dt

dψ r ðtÞ dt

(6.12a)

(6.12b)

ψ s ðtÞ 5 2 Ls i s ðtÞ 1 Lm i r ðtÞ

(6.13a)

ψ r ðtÞ 5 2 Lm i s ðtÞ 1 Lr i r ðtÞ

(6.13b)

where Ls 5 Lσs 1 Lm

Lr 5 Lσr 1 Lm

ωr 5 ð1 2 sÞωs

(6.13c)

ωr and ωs are rotor and synchronous angular frequencies, respectively, and s is the rotor slip. The variables and parameters in Eqs. (6.12) and (6.13) are described in Appendix A.2.

6.6.7 DFIG steady-state equivalent circuit In the steady state, voltages and currents are represented as complex phasors and in the rotor reference frame, the equivalent circuit is derived by substituting d=dt 5 jsωs in Eq. (6.12). Using Eqs. (6.13a)(6.13c), Eq. (6.12a) can be written as Vs 5 2 Rs Is 1 ½jð1 2 sÞωs 1 jsωs ð 2Ls Is 1 Lm Ir Þ 5 2 ðRs 1 jωs Ls ÞIs 1 jωs Lm Ir or Vs 5 2 ðRs 1 jXs ÞIs 1 jXm Ir

(6.14a)

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

491

/

(A)

+ / −

(B)

/ + / −

Figure 6.14 (A) DFIG steady-state equivalent circuit and (B) DFIG equivalent circuit with crowbar activated.

where Xs 5 ωs Ls and Xm 5 ωs Lm . Now, using Xs 5 Xσs 1 Xm from Eq. (6.13c), we have Vs 5 2 ½Rs 1 jðXσs 1 Xm ÞIs 1 jXm Ir

(6.14b)

Similarly, Eq. (6.12b) can be written as Vr 5 Rr Ir 1 jsωs ð 2 Lm Is 1 Lr Ir Þ or Vr 5 ðRr 1 jsXr ÞIr 2 jsXm Is

(6.14c)

where Xr 5 ωs Lr . Dividing by s throughout and using Xr 5 Xσr 1 Xm from Eq. (6.13c), we have Vr Rr 5 1 jðXσr 1 Xm Þ Ir 2 jXm Is s s

(6.14d)

Eqs. (6.14b)and (6.14d) are represented in the DFIG steady-state equivalent circuit shown in Fig. 6.14A.

6.6.8 DFIG natural stator and rotor short-circuit currents under constant ac excitation In this section, we derive the equations that model the natural responses of the DFIG stator and rotor currents to a solid three-phase short-circuit fault at the DFIG stator terminals or to a symmetrical voltage dip at the DFIG stator terminals which may be caused by remote three-phase short-circuit faults on the ac grid. The natural

492

Power Systems Modelling and Fault Analysis

stator and rotor current responses under a symmetrical voltage dip at the DFIG terminals assume constant ac rotor excitation and no crowbar action. In practice, the rotor currents can be too high under electrically close faults, that is, large voltage dips, and could destroy the rotor-side converter if allowed to flow freely as discussed in Section 6.6.3. However, our objective in this section is to provide an insight into the natural stator and rotor short-circuit current responses of the DFIG that would result if allowed to flow. The currents due to converter crowbar protection action are presented in Section 6.6.9. We assume that the rotor speed does not change during these disturbances and neglect saturation of leakage and magnetising inductances. The large rotor currents would flow unhindered because the rotor-side converter appears as a zero impedance to the flow of these currents. We maintain the use of s as the machine slip and use the complex frequency p 5 σ 1 jω as the Laplace operator. We choose a solution method that, although demanding, provides an insight into the various components of stator and rotor short-circuit currents. Therefore we take the Laplace transforms of Eqs. (6.12) and (6.13) then substitute Eq. (6.13b) into Eq. (6.12b) and, after some algebra, we obtain the following general expression that relates stator and rotor voltages to their corresponding currents

2 ½Rs 1 Ls ðp 1 jωr Þ V s ðpÞ 5 2 Lm p V r ðpÞ

Lm ðp 1 jωr Þ Rr 1 Lr p

Ls I s ðpÞ 1 L I r ðpÞ m

2 Lm 2 Lr

is ir

(6.15a)

where is and ir represent the initial stator and rotor currents, respectively. Dropping the complex frequency operator ðpÞ for convenience, the stator and rotor currents can be expressed as " # " # 2 Lm ðp 1 jωr Þ Is Vs 21 Rr 1 Lr p 5 D Lm p 2 ½Rs 1 Ls ðp 1 jωr Þ V r Ir " #" #

2 Lm ðRr 2 jωr Lr Þ is 1 Rr Ls 1 Lr Ls 2 L2m p 2 jωr L2m

1 2 D 2 Lm ðRs 1 jωr Ls Þ Lr Ls 2 Lm p 1 Lr ðRs 1 jωr Ls Þ ir (6.15b) where

D 5 Lr Ls 2 L2m p2 1 Rr Ls 1 Rs Lr 1 jωr Lr Ls 2 L2m p 1 Rr ðRs 1 jωr Ls Þ (6.15c)

Our research experience shows that the analytical solutions of the stator and rotor current equations can be greatly simplified by making use of a few useful equations that use the basic transient parameters of the machine as follows: L0s 5 Ls 2

L2m Lr

T dc 5

L0s Rs

(6.16a)

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

L0r 5 Lr 2

L2m Ls

T 0r 5

L0r Rr

T 0o 5

493

Lr Rr

(6.16b)

Ls Lr T0 5 0 5 o0 0 Ls Lr Tr

Lr Ls 2 L2m 5 Lr L0s 5 Ls L0r

(6.16c)

where L0s is stator transient inductance; L0r is rotor transient inductance; Tdc is stator short-circuit dc time constant; T 0r is rotor transient short-circuit time constant; and T 0o is rotor transient open circuit time constant Therefore using Eq. (6.16) in Eqs. (6.15b) and (6.15c), the stator and rotor currents are obtained as

Is 5

21 L0s

p1

1 T 0o

L2 V s 1 LL0 mLr ðp 1 jωr ÞV r 1 p 1 T10 2 jωr L0 mLr is 2 s r s 1 1 1 Rs 2 p 1 T 0 1 Tdc 1 jωr p 1 T 0 Ls 1 jωr r

Lm L0s

1 T 0o

2 jωr ir

r

(6.17a)

Ir 5

2Lm L0s Lr

pV s 1

1 L0s Lr

½Rs 1 ðp 1 jωr ÞLs V r 2 LL0 mLr ðRs 1 jωr Ls Þis 1 p 1 s p2 1 T10 1 T1dc 1 jωr p 1 T10 RLss 1 jωr r

1 Tdc

1 jωr LLs0 ir s

r

(6.17b) The stator and rotor currents have the same denominator and therefore the same eigenfrequencies and damping time constants. The two roots can be determined from 2 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 1 1 1 1 4 Rs p1;2 5 4 2 1 1 jωr 6 1 1jωr 2 0 1 jωr 5 2 T 0r Tdc T 0r Tdc T r Ls (6.18) For practical DFIG parameters and rotor speed range of typically 0:7 pu # ωr # 1:3 pu, the small rotor resistance allows us to make simplifications that enable us to derive a closed form solution of the two roots with negligible errors in practical applications. As shown in Eqs. (6.17a) and (6.17b), the Laplace transforms of the initial stator and rotor voltages are required. Since the frequency of the original steady-state stator voltage and current in the rotor reference frame is sωs , the predisturbance instantaneous stator voltage can be written as o pﬃﬃﬃ vs ðtÞ 5 Real V s ðtÞ 5 2Vs cosðsωs t 1 θo Þ

(6.19a)

494

Power Systems Modelling and Fault Analysis o

where V s ðtÞ is the initial complex instantaneous stator voltage given by pﬃﬃﬃ o V s ðtÞ 5 2Vos ejsωs t

Vs 5 Vs ejθo

(6.19b) (6.19c)

where Vos is the initial stator voltage phasor with an angle θo . Similarly, the DFIG original steady-state instantaneous rotor voltage is given by o pﬃﬃﬃ vr ðtÞ 5 Real V r ðtÞ 5 2Vr cosðsωs t 1 δo Þ (6.20a) o

where V r ðtÞ is the initial complex instantaneous rotor voltage given by

V r ðtÞ 5

pﬃﬃﬃ jsω t 2V r e s

Vr 5 Vr ejδo

(6.20b) (6.20c)

where Vor is the initial rotor voltage phasor with an angle δo . When a symmetrical voltage dip occurs at the DFIG terminals, the retained complex instantaneous stator voltage at the dip instant can be expressed as pﬃﬃﬃ V s ðtÞ 5 k 2Vs ejθo ejsωs t (6.21a) where 0 # k , 1 and the case of k 5 0 it represents a solid three-phase fault at the DFIG terminals. Taking the Laplace transform of Eq. (6.21a), we have pﬃﬃﬃ k 2Vs ejθo V s ðpÞ 5 (6.21b) p 2 jsωs The natural DFIG stator and rotor currents are calculated with a constant rotor voltage. Therefore taking the Laplace transform of Eq. (6.20b), we have pﬃﬃﬃ jδ 2 Vr e o V r ðpÞ 5 (6.21c) p 2 jsωs Substituting Eqs. (6.21b) and (6.21c) into Eqs. (6.18b) and (6.18c) and taking their inverse Laplace transforms, we obtain the complex instantaneous currents I s ðtÞ and I r ðtÞ. Since our equations are in the rotor reference frame, the real instantaneous stator current in the stationary reference frame is obtained as follows (6.22a) is ðtÞ 5 Real I s ðtÞejωr t and the real instantaneous rotor current is obtained using ir ðtÞ 5 Real I r ðtÞ

(6.22b)

Following extensive mathematical analysis, it can be shown that the stator shortcircuit fault current is given by

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

495

pﬃﬃﬃ pﬃﬃﬃ

Xm 2Vr

2 k 2Vs Zr 0 0 is ðtÞ 5 cos ωs t 2 α01 1 α04 1 δo 0 cos ω s t 1 θo 2 α1 1 α2 1 0 Xs Zr Xs Zr ( 0 1 0 1 pﬃﬃﬃ k 2 Vs @ 1 1 A Rr π 2 2 cos@ωr t 1 θo 2 α01 2 A Xs Zr0 2 1 2 s Xs0 pﬃﬃﬃ

Xm 2 V r 1 cos ωr t 2 α01 1 α05 1 δo 0 Xs Zr ) t 20 1 3 2 0 Tr 1 1 X m 1 4@ 0 2 AXs is 2 0 ir 5cosðωr tÞ e Xs Xs Xs

(

1

pﬃﬃﬃ pﬃﬃﬃ

Xm 2Vr

2 k 2Vs 1 0 cos α04 1 δo cos θo 1 α3 1 0 0 Xs Xs Xr ½ð1 2 sÞωs Tdc

)

1

1 0 Xs i s 2 Xm i r Xs

t 2T

dc

e

(6.23a) The phase y and b currents are obtained by replacing θo by θo 2 2π=3 and θo 1 2π=3, respectively, and replacing δo by δo 2 2π=3 and δo 1 2π=3, respectively, The rotor short-circuit fault current is given by pﬃﬃﬃ

2 Vr cos sωs t 2 α01 1 α04 1 δo 0 Zr ( 0 1 pﬃﬃﬃ pﬃﬃﬃ

2 V Xm k 2 Vs 1 1 r 1 2 cos 2 ωr t 1 θo 1 α03 1 Rs @ 0 2 A Xs Xs Xs Xr0 ð1 2 sÞXr0 ) t 2 0 1 3 2T

dc X 1 1 m 3 cos 2 ωr t 1 α04 1 δo 1 4 0 is 2 @ 0 2 AXr ir 5cosð 2 ωr tÞ e Xr Xr Xr

ir ðt Þ 5 2

Xm pﬃﬃﬃ s k 2Vs 0 cos sωs t 1 θo 2 α01 1 α04 1 Zr Xs

(

2

pﬃﬃﬃ pﬃﬃﬃ

2V r Xm k 2 V s Rr 0 cos 2 α01 1 α05 1 δo 0 0 sin θo 2 α1 1 0 Xs ð1 2 sÞXr Zr Zr

)

2

1

1 0 Xm is 2 Xr ir e Xr

t Tr0

(6.23b)

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Power Systems Modelling and Fault Analysis

where Zr 5

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R2r 1 ðsXr Þ2

0 0

α1 5 tan21 sωs Tr 2

13 0 0 sT X α2 5 Argument4sωs Tr 1 j@ r 2 s A5 Tdc Xs 2 3 1 1 0 2 1 j5 α3 5 Argument4 ð1 2 sÞωs Tr0 ωs Tdc 0

0

0

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ﬃ R2r 1 sXr0 0 1 1 0 A α4 5 tan21 @ ωs Tdc 2 3 1 0 5 α5 5 tan21 4 ð1 2 sÞωs Tr0 0

Zr 5

0

0

Xs 5 ωs Ls

0

0

Xr 5 ωs Lr (6.23c)

Eq. (6.23a) shows that the stator current consists of three current components: 1. a steady-state current component due to the retained stator terminal voltage and constant rotor voltage and has a frequency equal to ωs ; 2. a transient ac current component that has a frequency equal to rotor speed ωr and decays with a time constant T 0r ; and 3. a transient dc or zero frequency current component that decays with a stator dc time constant Tdc .

Eq. (6.23b) shows that the rotor current consists of three current components: 1. a steady-state current component due to the retained stator terminal voltage and constant rotor voltage and has a frequency equal to slip frequency sωs ; 2. a transient ac current component that has a frequency equal to 2ωr and decays with a time constant Tdc ; and 3. a transient dc or zero frequency current component that decays with a time constant T 0r .

The predisturbance stator and rotor currents and rotor voltage are calculated from the DFIG’s known initial operating conditions of stator voltage, active power, reactive power and rotor slip. The initial rms stator current phasor is given by ðPs 1jQs Þ Is 5 jθ 5 Is ejðθo 2ϕo Þ Vs e o

(6.24a)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

where Is 5 ð P2s 1 Q2s Þ=Vs , ϕo 5 tan21 Qs =Ps and Ps and Qs are initial stator active and reactive power outputs, respectively. The initial, real, instantaneous stator current is given by

is ðtÞ 5

pﬃﬃﬃ

2Is cos ωs t 1 θo 2 ϕo

(6.24b)

The initial rms rotor current phasor is calculated from Eq. (6.14a) as follows

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

Ir 5

497

Vs 1 ðRs 1 jXs ÞIs 5 Ir ejβo jX m

(6.25a)

and the initial, real, instantaneous rotor current can be written as pﬃﬃﬃ ior ðtÞ 5 2Iro cosðsωs t 1 β o Þ

(6.25b)

The predisturbance rotor voltage is calculated using Eq. (6.14c) as follows

Vr 5 ðRr 1 jsXr ÞIr 2 jsXm Is 5 Vr ejδo and the initial, real, instantaneous rotor voltage can be written as pﬃﬃﬃ vr ðtÞ 5 2Vr cosðsωs t 1 δo Þ

(6.26a)

(6.26b)

The stator and rotor short-circuit currents are illustrated in the examples given in Section 6.6.12.

6.6.9 DFIG stator and rotor short-circuit currents under crowbar action As discussed in Sections 6.6.3 and 6.6.4, crowbar protection operates to protect the converter from the high and damaging rotor currents calculated in Section 6.6.8. In this section, we limit our attention to the derivation of the DFIG’s short-circuit currents during the period of crowbar operation. The short-circuit current in the period after the crowbar is removed and the converter regaining control of rotor currents is discussed in Section 6.6.11. As shown in Fig. 6.12, the blocking of the converter switches and firing of the crowbar inserts an effective crowbar resistance Rcb that appears in series with the rotor winding. Rcb is the effective resistance seen on the ac side of the threephase diode bridge and is therefore the ‘rectified’ value of R. The crowbar action is equivalent to short-circuiting the rotor winding through an equivalent resistance Rcb as illustrated in the steady-state equivalent circuit in Fig. 6.14B. Therefore the rotor voltage can be written as V r 5 2 Rcb I r

(6.27a)

The subscript cb is used throughout to denote crowbar. Substituting Eq. (6.27a) in Eq. (6.15a) and rearranging, we obtain " # " # 2Lm ðp1jωr Þ Is Vs 21 Rr 1Rcb 1Lr p 5 D Lm p 2 ½Rs 1Ls ðp1jωr Þ V r Ir " #" #

2Lm ½ðRr 1Rcb Þ2jωr Lr is 1 ðRr 1Rcb ÞLs 1 Lr Ls 2L2m p2jωr L2m

1 2 D 2Lm ðRs 1jωr Ls Þ Lr Ls 2Lm p1Lr ðRs 1jωr Ls Þ ir (6.27b)

498

Power Systems Modelling and Fault Analysis

where

D 5 Lr Ls 2 L2m p2 1 ðRr 1 Rcb ÞLs 1 Rs Lr 1 jωr Lr Ls 2 L2m p 1 ðRr 1 Rcb ÞðRs 1 jωr Ls Þ

(6.27c)

The crowbar action is mathematically equivalent to setting the rotor voltage to zero and replacing the rotor winding resistance Rr with the total resistance Rr 1 Rcb . Similar steps to those used in Section 6.6.8 are followed making use of the following equations that greatly simplify the analysis 0

L2m Lr

0

Ls 5 Ls 2

T dc 5

Ls Rs

0

Lr 5 Lr 2

L2m Ls

(6.28a)

0

Lr Ls 2 L2m

0

0

5 Lr Ls 5 Ls Lr

Lr T cb 5 Rr 1 Rcb 0

0

T o;cb 5

Lr Rr 1 Rcb

(6.28b)

0

T Ls Lr 5 0 5 o;cb 0 L0s Lr T cb

(6.28c)

In defining the above equations, we note that we have intentionally reused L0s and Tdc that do not include the effect of crowbar resistance as defined in Eq. (6.16a) since this simplifies the mathematical solutions greatly but, as we show later, the effective stator transient reactance and stator dc time constant are affected and modified by the high rotor resistance. We also used L0r as defined in Eq. (6.16b) and, as we show later, this is not affected by the high rotor resistance. 0 0 Also, we included the crowbar resistance Rcb in the time constants Tcb and To;cb in Eq. (6.28b) which, again, facilitate the derivation of a closed form solution of stator and rotor currents and will be shown to be a valid practical representation of the DFIG short-circuit and open-circuit time constants. Now, substituting Eqs. (6.28a)(6.28c) in Eqs. (6.27b) and (6.27c) and rearranging, it can be shown that the stator and rotor currents are given by Is 5

21 L0s

p1

L2 V s 1 p 1 T10 2 jωr L0 mLr is 2 LLm0 T 01 2 jωr ir s s cb o;cb p2 1 T10 1 T1dc 1 jωr p 1 T10 RLss 1 jωr 1

0 To;cb

cb

cb

2Lm Lm Lr 1 pV 2 ð R 1 jω L Þi 1 p 1 1 jω s s r s s r L0 i r Tdc L0 Lr L0 Lr s r Ir 5 s p2 1 T10 1 T1dc 1 jωr p 1 T10 RLss 1 jωr cb

(6.29a)

(6.29b)

cb

The denominator of Eqs. (6.29a) and (6.29b) shows that the stator and rotor currents have the same frequencies and decay time constants as those of Eq. (6.17) and

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

499

0

that T 0r is now replaced by Tcb . However, since the crowbar resistance Rcb can be tens of times greater than the rotor winding resistance Rr , the roots of the denominator of Eq. (6.29) cannot be obtained using the simplification of negligibly small rotor resistance as was used in obtaining the roots of Eq. (6.18a). Physically, the high rotor resistance Rr 1 Rcb substantially increases the magnetic coupling between the stator and rotor fluxes. Fortunately, our extensive research has succeeded in obtaining an accurate closed-form solution for these roots and hence the formulation of a closed form solution for stator and rotor current transients. We start next by analysing the effect of the high rotor resistance on the rotor and stator transient inductances and stator time constant. In the stationary reference frame presented in Appendix A.2 and with a generator sign convention for stator current, substituting ψs of Eq. (A.2.19a) and V s 5 0 in Eq. (A.2.20a), we obtain I s 5 pLm =ðRs 1 pLs ÞI r . Now, substituting this into ψr of Eq. (A.2.19b), we obtain ψ r 5 Lr 2

pL2m I r 5 2 Lr ðpÞI r Rs 1 pLs

(6.30a)

Lr ðpÞ 5 Lr 2

pL2m Rs 1 pLs

(6.30b)

where Lr ðpÞ is the DFIG operator inductance, as seen from the rotor with the stator short-circuited. Since Rs is very small compared to Ls , Lr ðpÞ reduces to L2

Lr ðpÞ 5 Lr 2 Lms 5 L0r . This means that L0r is not affected by the rotor resistance. Also, in the rotor reference frame presented in Appendix A.2, substituting ψr of Eq. (A.2.19b) and Vr 50 in Eq. (A.2.22b), we obtain I r 5 pLm =½ðRr 1 Rcb Þ 1 pLr I s . Now, substituting this into ψs of Eq. (A.2.19a), we obtain ψ s 5 2 Ls 2

Ls ðpÞ 5 Ls 2

pL2m I s 5 2 Ls ðpÞI s ðRr 1 Rcb Þ 1 pLr

(6.31a)

pL2m ðRr 1 Rcb Þ 1 pLr

(6.31b)

where Ls ðpÞ is the DFIG operator inductance, as seen from the stator with the rotor short-circuited through a high crowbar resistance Rcb . With Rcb 5 0, we obtain the L2

familiar equation of stator transient inductance L0s 5 Ls 2 Lmr since Rr is very small compared to Lr which means that L0s is not affected by the small rotor resistance Rr . However, with practical values of crowbar resistance Rcb , Rr 1 Rcb can be comparable to or even greater than Lr and therefore cannot be neglected in the denominator of Eq. (6.31b).

500

Power Systems Modelling and Fault Analysis

In our chosen rotor reference frame, a stationary stator flux with respect to the stator, induces a voltage in the rotor whose angular frequency is ωr and, as implied in Eq. (6.22b) and shown in Eq. (6.23b), the frequency of the resultant rotor current is 2ωr . Therefore the effective stator transient inductance that includes the effect of the high crowbar resistance Rcb can be derived from the operator inductance given in Eq. (6.31b) by setting p 5 2 jωr . Therefore the resultant complex stator transient inductance is given by 0

L s;cb 5 Ls ð 2 jωr Þ 5 Ls 2

2 jωr L2m Rr 1 Rcb 2 jωr Lr

(6.32a)

Dividing by stator resistance Rs , the complex stator time constant is given by 0 L s;cb 1 2 jωr L2m T dc;cb 5 5 Ls 2 (6.32b) Rs Rs Rr 1 Rcb 2 jωr Lr Following further extensive analysis, it can be shown that the effective stator transient inductance and stator dc time constant, in the presence of a high rotor resistance, are given by Rr 1Rcb 2 1 1 L10 2 r 0 0 1s Ls;cb 5 Ls 3 (6.33a) 2 R r 1Rcb 1 1 1 L L0 r r 1s

Rr 1Rcb 2 1s Tdc;cb 5 Tdc 3 Rr 1Rcb 2 1 1 L 1L0 r r 1s 11

1 L0r2

(6.33b)

The complex stator time constant given in Eq. (6.32b) means that the stator flux is no longer stationary with respect to the stator but it slowly rotates at a certain angular frequency. It can be shown that the angular frequency of the slowly rotating component of stator flux is given by 1 1 Rr 1 Rcb 2 1 L0r Lr 1s ωdc 5 3 (6.33c) Tdc 1 Rr 1Rcb 2 1 1 02 Lr 1s Eqs. (6.33a) and (6.33b) show the effect of high rotor resistance as a multiplier applied to L0s and Tdc which apply in the case the machine has a negligibly small

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

501

rotor resistance. In fact, Eqs. (6.33a) and (6.33b) are general and apply for any 0 value of rotor resistance. Ls;cb and Tdc;cb show strong dependence on crowbar or total rotor resistance and rotor slip. To gain a feel of the effect of high rotor resistance, consider a DFIG machine having the following parameters in per unit: Rs 5 0:0069 pu, Xs 5 3:3158 pu, Rr 5 0:00867 pu, Xr 5 3:3239 pu, 0 0 Xs 5 0:2351 pu, Xr 5 0:2356 pu, Tdc 5 34:07 pu. Network frequency is 50Hz. With Rcb 5 0, the factor that multiplies L0s and Tdc in Eqs. (6.33a) and (6.33b) ranges 0 from 1.0007 to 1.0026 for 0:7 pu # ωr # 1:3 pu thus Ls;cb 5 L0s and Tdc;cb 5 Tdc . However, for a moderately high rotor resistance, for example, Rcb 5 20Rr , this factor increases significantly and now ranges from 1.32 to 2.04 for 0:7 pu # ωr # 1:3 pu. Also, Eq. (6.34) shows that with Rcb 5 0, the angular frequency of the rotating stator flux ranges from 0:038 to 0:07Hz for 0:7 pu # ωr # 1:3 pu and hence it is practically negligible. However, for Rcb 5 20Rr , this angular frequency ranges from 0:6 to 0:68Hz. pﬃﬃ k 2Vs ejθo Now, we return to Eqs. (6.29a) and (6.29b), use Eq. (6.21b) or V s ðpÞ 5 p 2 jsω , s take the inverse Laplace transforms and obtain I s ðtÞ and I r ðtÞ. It can be shown that following extensive mathematical analysis, the stator phase r short-circuit current response of the DFIG with crowbar action is given by pﬃﬃﬃ 2 k 2Vs Zr1 0 0

is ðtÞ 5 cos ωs t 1 θo 2 β 1 1 β 2 Xs Zr2 9 8 0 pﬃﬃﬃ 1 1 Rr 1 Rcb Xr 0 0 > > > > > k 2 Vs cos ðωr 2 ωdc Þt 1 θo 2 β 1 1 β 3 > > > 0 2 > > X X Z Z > > s r3 r2 s > > > > > > 0 = t < 1 1 ð1 2 sÞXr 0 π 2 Xs is cos ðωr 2 ωdc Þt 1 β 3 2 e2Tac 1 0 2 Xs Xs 2 Zr3 > > > > > > > > > > Xm Zr4 0 0 > > > > > > 1 ir cos ðωr 2 ωdc Þt 1 β 3 1 β 5 > > ; : Xs Zr3 9 8 pﬃﬃﬃ 0 > 0 > 2 k 2Vs ð1 2 sÞXr > > > cos ðωr 2 ωac Þt 1 θo 1 β 4 > > > = < Xs0 Zr3 2 t e Tdc;cb 1 0 0 > > Z X r4 m > > > > > ; : 1 Zr3 is 2 Xs ir cos ðωr 2 ωac Þt 1 β 3 1 β 5 > and the rotor phase r short-circuit fault current is given by

(6.34a)

502

Power Systems Modelling and Fault Analysis

pﬃﬃﬃ Xm k 2 Vs s 0

ir ðtÞ 5 2 cos sωs t 1 θo 2 β 1 Xs Zr2 9 pﬃﬃﬃ 0 Xm k 2Vs Rr 1 Rcb Xr 0 0

> > > cos 2 ωdc t 1 θo 2 β 1 1 β 3 0 > = t Xr X s Zr3 Zr2 e2Tac 1 0 > 0 0 0 > X ð 1 2 s ÞX π Z m r4 > > r > > > ; : 1 Xr Zr3 is cos 2 ωdc t 1 β 3 1 2 1 Zr3 ir cos 2 ωdc t 1 β 3 1 β 5 > 8 > > > > <

9 pﬃﬃﬃ 0 > 0

Xm k 2Vs ð1 2 sÞXr > > cos 2 ωac t 1 θo 1 β 3 > 0 = Xr Xs Zr3 2 t e Tdc;cb 2 0 > > ð 1 2 s Þ X π 0 0 m > > > > X i 2 ðXr 2 Xr Þir cos 2 ωac t 1 β 3 1 > ; : 1 Zr3 2 > Xr r s 8 > > > > <

(6.34b) where Zr1 5

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðRr 1Rcb Þ2 1 s2 Xr2

Zr3 5

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðRr 1Rcb Þ2 1 ð12sÞ2 Xr0 2

Zr2 5

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðRr 1Rcb Þ2 1 s2 Xr0 2

Zr4 5

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðRr 1Rcb Þ2 1 ð12sÞ2 Xr2

0

21

β 1 5 tan

0

sωs Tcb

"

#

T β 2 5 Argument sωs Tcb 2 j 0 cb To;cb 0

0 β 3 5 tan ð1 2 sÞωs Tcb 0

0

(6.35)

21

0

" 0

21

β 4 5 tan

h i 0 0 β 5 5 2 tan21 ð1 2 sÞωs To;cb

1R 2 # 0 cb Xr Xr 1 Rr12s

ðXr 2 Xr0 Þ Rr112Rscb (6.36)

and Tac 5

1 1 0 Tcb

1

1 Tdc

Tdc;cb 5 Tdc 3

2

1 Tdc;cb

11 11

1 Rr 1Rcb 2 12s L0r2 R 1R 2 1 r cb 12s Lr L0r

(6.37a)

(6.37b)

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

0 ωr @ 12 ωdc 5 2 0 ωr @ ωac 5 11 2

1 0 Tcb 1 0 Tcb

1 1 0 Tcb

1 0 Tcb

1

2 1 Tdc

2 1 Tdc

1

1 Tdc

2

2 Tdc;cb

A

(6.38a)

1

1 Tdc

2

503

2 Tdc;cb

A

(6.38b)

ωr 5 ωac 1 ωdc

(6.38c)

Eq. (6.38a) is presented as a practical equivalent but more intuitive and arguably more elegant alternative to Eq. (6.33c). It is interesting to observe that for practical DFIG parameters and 0:7 pu # ωr # 1:3 pu, the incremental angular frequency term in Eqs. (6.38a) and (6.38b)

1 0 Tcb

2

1 Tdc

=

1 0 Tcb

1

1 Tdc

2

2 Tdc;cb

is found to range from

0.94 to 0.99 indicating that ωdc is generally less than 3% of ωr and ωac is generally greater than 97% of ωr . In other words, the high DFIG rotor resistance causes the stator flux to rotate slowly at near-dc or near-zero frequency ωdc instead of being purely stationary, and the ac flux to rotate at near-rotor speed frequency ωac instead of rotor speed ωr . In addition, Eqs. (6.37a), (6.37b), (6.38a), and (6.38b) show that for Rcb 5 0, 0 Tdc;cb DTdc , ωdc 5 0, ωac 5 ωr and Tac 5 Tcb . This results in a stationary or dc stator flux and an ac flux that rotates at rotor speed. Eqs. (6.38a) and (6.38b) show that the incremental term affects equally the angular frequencies of the stator flux ωdc and ac rotating flux ωac as seen from the stationary stator. This is confirmed in Eq. (6.38c). Physically, this is expected since the presence of this small incremental term may be attributed to the magnetic drag effect that acts equally on both stator and rotor fluxes. Eq. (6.34a) shows that the stator current consists of three components: 1. A steady-state current component that exists if the retained stator terminal voltage is not zero and has a frequency equal to ωs . 2. A transient ac current component that has a frequency of ωr 2 ωdc , that is, near-rotor speed ωr and decays with an effective time constant Tac which, for practical DFIG para0 0 meters, is equal to Tcb 5 Rr 1LrRcb given in Eq. (6.28b). 3. A transient near-dc current component that has a frequency of ωr 2 ωac , that is, near-dc or zero frequency and decays with an effective time constant Tdc;cb .

Eq. (6.34b) shows that the rotor current consists of three components: 1. A steady-state current component that exists if the retained stator terminal voltage or rotor slip is not zero and has a frequency equal to slip frequency sωs . 2. A transient ac component that has a frequency equal to 2ωac , that is, near-rotor speed 2ωr and decays with a time constant Tdc;cb . 3. A transient near-dc component that has a frequency of 2 ωdc , that is, near-dc or zero frequency and decays with a time constant Tac .

504

Power Systems Modelling and Fault Analysis

It is interesting to note that the frequency of the stator transient fault current of Eq. (6.34a), ðωr 2ωdc Þ=2π5ð12sÞfs 2 fdc , is different from the nominal frequency fs because of primarily the slip and to a small extent the magnetic drag effect that acts on the stator and rotor fluxes. This difference is significant when the DFIG slip is large, for example, 6 0:2 pu or 6 0:3 pu and affects the time instant that corresponds to the first and subsequent peaks of the short-circuit current. For example, whereas the frequency of the short-circuit current in the first few cycles is practically 50 Hz for a synchronous and fixed speed induction generators in a 50 Hz power system, it would be 1:2 3 50 Hz 2 0:5 Hz 5 59:5 Hz for a DFIG if it is initially operating at a high-active power output, for example, at a slip s 5 2 0:2 pu and fac 5 0:5 Hz. The effect of this is that the instant of the first peak on the current waveform will occur before the usual half cycle of nominal frequency. The converse occurs if the machine is initially operating at subsynchronous speed when the instant of the first current peak will occur after the first half cycle peak of nominal frequency. The second and subsequent current peaks are similarly affected. Although unnecessary, if the approximations ωdc D0 and ωac Dωr are made, Eqs. (6.34a) and (6.34b) can be written as i s ðt Þ 5

pﬃﬃﬃ 2 k 2Vs Zr1 0 0

cos ωs t 1 θo 2 β 1 1 β 2 Xs Zr2

9 8 0 pﬃﬃﬃ 1 0 0 > 1 Rr 1 Rcb Xr > > > > k 2 Vs 2 cos ωr t 1 θo 2 β 1 1 β 3 > > > > > Xs0 Xs Zr3 Zr2 > > > > > > > > 0 = t < 0 1 1 ð1 2 sÞXr π 2 2 X i cos ω t 1 β 2 e2Tac 1 s s r 0 3 X X 2 Z > > s r3 s > > > > > > > > 0 0

Xm Zr4 > > > > > > 1 i cos ω t 1 β 1 β r > > r 3 5 ; : Xs Zr3 9 8 pﬃﬃﬃ 0 > 2 k 2Vs ð1 2 sÞXr 0 > > > > cos θo 1 β 4 > > > = < Xs0 Zr3 2 t e Tdc;cb 1 0 Zr4 Xm 0 > > > > > > > ; : 1 Zr3 is 2 Xs ir cos β 3 1 β 5 > and

(6.39a)

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

505

pﬃﬃﬃ Xm k 2 Vs s 0

i r ðt Þ 5 2 cos sωs t 1 θo 2 β 1 Xs Zr2 9 8 pﬃﬃﬃ 0 > Xm k 2Vs Rr 1 Rcb Xr 0 0 > > > > cos θo 2 β 1 1 β 3 > > > 0 > > X X Z Z > > r r3 r2 s > > > > > > 0 = t < 0 Xm ð1 2 sÞXr π 1 is cos β 3 1 e2Tac 1 > > 2 Xr Zr3 > > > > > > > > Zr4 0 0

> > > > > > 1 i cos β 1 β > > r 3 5 ; : Zr3 9 pﬃﬃﬃ 0 > Xm k 2Vs ð1 2 sÞXr 0

> > cos 2 ωr t 1 θo 1 β 3 > = Xr Xs0 Zr3 2 t e Tdc;cb 2 0 > ð 1 2 s Þ Xm 0 π > 0 > > > > Xr is 2 ðXr 2 Xr Þir cos 2 ωr t 1 β 3 1 > ; : 1 Zr3 2 > Xr 8 > > > > <

(6.39b)

6.6.10 DFIG short-circuit currents with dc chopper control The stator and rotor transient current analysis presented in Section 6.6.9 is also generally applicable to a DFIG where a dc chopper circuit is used instead of a crowbar circuit. Where the dc chopper inserts a fixed resistance, then the effective value of the rectified resistance appears in series with the rotor winding resistance and the analysis presented for the crowbar action in Section 6.6.9 is applicable as shown in Fig. 6.15. However, where chopper control action occurs in response to rotor currents or dc link voltage, the effective value of the rectified chopper resistance is not fixed and becomes time-dependent with the resistance value increasing with time as the rotor current decreases with time. The analysis of the stator and rotor transient currents with the nonlinear behaviour of the chopper resistance is not amenable to closed form solutions and requires digital computer simulations. However, an average fixed value of the rectified chopper resistance can still be derived and our crowbar analysis applied if a small degree of inaccuracy is deemed acceptable.

506

Power Systems Modelling and Fault Analysis

To rotor

(A)

Rch

Rotor-side converter dc chopper (B)

/ +

+ C

−

−

Figure 6.15 dc chopper action: (A) rotor current flow through antiparallel diodes with IGBT switches blocked and (B) equivalent circuit.

6.6.11 DFIG short-circuit currents with rotor converter control In the grid-friendly DFIG design, the crowbar circuit is quickly switched out and the converter switches unblocked after some time delay from the instant of shortcircuit current once the instantaneous rotor current has decayed to a sufficiently low value. The switching out of the crowbar and unblocking of the converter switches allows the converter to regain control of the rotor currents. This enables the generator to supply a predefined and constant value of stator short-circuit current with a magnitude that is dependent on the retained stator voltage and the grid code requirement for reactive current injection (discussed in Section 6.12). In other words, the machine acts as a constant current source whose magnitude is determined by the rotor-side converter control strategy rather than the physics of the induction generator. If the fault location is on the network and is sufficiently remote from the generator, the rotor-side converter current limit may not be exceeded and hence the protective crowbar circuit would not operate. This is typically the case for a symmetrical voltage dip at the DFIG terminals of up to 10%. Thus the magnitude of the stator current supplied is affected and modified by the converter’s voltage/ reactive power control strategy. For example, consider a rotor-side converter operating in stator terminal voltage control mode using an automatic voltage regulator (AVR) acting through the converter to deliver the required change in rotor voltage ΔVr . The extent to which the stator short-circuit current supplied by the generator is affected is influenced by the converter proportional-integral (PI) controller gains

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

507

as well as the generator short-circuit transient time constant T0 . Eq. (6.18b)and (6.18c) can be used to calculate the stator and rotor transient currents as a result of rotor converter control and change in rotor voltage ΔVr . However, this analysis is left for the interested researcher/reader. The short-circuit current behaviour of the doubly fed generator depends on whether the fault is balanced or unbalanced. During unbalanced faults on the network that produce a large negative-phase sequence voltage at the generator terminals and a corresponding large negative-phase sequence rotor current, permanent crowbar operation may result until the fault is cleared.

6.6.12 Examples Example 6.4 This example illustrates the theoretical or natural DFIG stator and rotor short-circuit currents as derived in Section 6.6.8. Consider a DFIG with the following machine data: Rated apparent power: 2.222 MVA Rated voltage: 690 V Stator resistance: 0.0064 pu Stator leakage reactance: 0.1674 pu Magnetising reactance: 3.80 pu

Rated power: 2 MW Rated frequency: 50 Hz Rotor resistance: 0.00842 pu Rotor leakage reactance: 0.1808 pu

All impedances are in per unit on 690 V and 2.222 MVA base. Calculate and plot the phase r stator and rotor short-circuit currents for a three-phase shortcircuit fault on the ac grid that results in a symmetrical voltage dip at the DFIG’s stator terminals of 70%. Consider the following initial DFIG operating conditions: a. supersynchronous operation: Ps 5 1:66MW, Qs 5 0:2Mvar and ωr 5 1:3 pu and b. synchronous operation: Ps 5 0:73MW, Qs 5 0:1Mvar and ωr 5 1 pu.

Initial stator voltage Vs 5 1ej90 pu. Assume maximum dc offset of stator current phase r. Solution The symmetrical voltage dip of 70% results in a retained voltage of k 5 0:3. From the machine 0 data, the following parameters are calculated: Xs 5 3:9674 pu, Xr 5 3:9808 pu, Xs 5 0:340, 0 Xr 5 0:3411, T 0o 5 472:78 pu or 1.5 s, T 0r 5 40:5 pu or 0:129s, Tdc 5 53:12 pu or 0:169s. Case (a), S 5 1:66 1 j0:2 MVA, ωr 5 1:3 pu. The initial phasor values of stator current, rotor current and rotor voltage are calculated using Eqs. (6.24a), (6.25a) and (6.26a) and are Ios 5 0:75ej83:1 pu, Ior 5 0:858ej65:3 pu and Vro 5 0:329e2j75:5 pu. Also, using Eq. (6.23c), the following parameters are calculated: Zr 5 1:194 pu, 0 0 0 0 0 Zr 5 0:103 pu, α1 5 2 85:3degrees, α2 5 178:5degrees, α3 5 90degrees, α4 5 1:08degrees, 0 α5 5 1:09degrees. Fig. 6.16 shows the phase r stator and rotor short-circuit currents. Case (b), ωr 5 1 pu. The initial phasor values of stator current, rotor current and rotor voltage are Ios 5 0:332ej83:2 pu, Ior 5 0:463ej47:8 pu and Vro 5 0:004ej47:8 pu. Zr 5 0:00842 pu,

508

Power Systems Modelling and Fault Analysis 7 6 5 4 3 2 1 0 –1 –2 –3

(A)

–100

–50

(B)

Pre-dip ac trans.

Pre-dip ac trans.

8

Steady state dc trans.

4 2 0 –2 –4

ms 0

–50

10 6

50

100

150

200

250

300

350

400

Pre-dip

ms

–6 –100

–50

(D)

7 6 5 4 3 2 1 0 –1 –2 –3

–100

12

(C)

Steady state dc trans.

0

50

100

150

200

250

300

350

400

12

Fault current

10

Pre-dip

8

Fault current

6 4 2

0 –2 –4

ms 0

50

100

150

200

250

300

350

400

ms

–6 –100

–50

0

50

100

150

200

250

300

350

400

Figure 6.16 DFIG fault currents for Example 6.4 with constant ac excitation and 70% symmetrical voltage dip, Ps 5 1:66 MW, Qs 5 0:2Mvar and ωr 5 1:3 pu: (A) stator current components, (B) stator current, (C) rotor current components and (D) rotor current.

7

(A)

Pre-dip ac trans.

5

9

(C)

Steady state dc trans.

Pre-dip ac trans.

7

5 3

Steady state dc trans.

3 1

1

–1

–1

–3

ms

–3 –100

(B)

–50

0

50

100

150

200

250

300

350

400

7

(D) Pre-dip

5

ms

–5 –100

Fault current

–50

0

50

100

150

200

250

300

350

400

9

7

Pre-dip

Fault current

5 3 3 1

–100

–50 –1 0

1

50

100

150

200

250

300

–3

350

400

ms

–100

–50 –1 0

50

100

150

200

250

300

350

–3

400

ms

–5

Figure 6.17 DFIG fault currents for Example 6.4 with constant ac excitation and 70% symmetrical voltage dip, Ps 5 0:73 MW, Qs 5 0:1Mvar and ωr 5 1 pu: (A) stator current components, (B) stator current, (C) rotor current components and (D) rotor current. 0

0

0

0

0

Zr 5 0:00842 pu, α1 5 0degrees, α2 5 88:9degrees, α3 5 89:7degrees, α4 5 1:08degrees, 0 α5 5 1:4degrees. Fig. 6.17 shows the stator and rotor short-circuit currents. It is instructive for the reader to calculate and plot the stator and rotor fault currents for a DFIG in a subsynchronous operating condition with Ps 5 0:35MW, Qs 5 0Mvar and ωr 5 0:7 pu.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

509

Example 6.5 In this example, we illustrate the effect of crowbar action on the DFIG stator and rotor currents as derived in Section 6.6.9. Consider the DFIG of Example 6.4 and the following three cases for a symmetrical voltage dip at the stator terminals of 70%: Case (a), ωr 5 1:3 pu, crowbar resistance Rcb 5 10Rr Case (b), ωr 5 1:3 pu, crowbar resistance Rcb 5 30Rr Case (c), ωr 5 0:8 pu, Ps 5 0:465 pu and three cases of crowbar resistance values namely Rcb 5 0; 10Rr and 30Rr . Solution Case (a), ωr 5 1:3 pu. Crowbar resistance Rcb 5 10Rr . The initial phasor values of stator current, rotor current and rotor voltage are as given in Example 6.4 Case (a). Using Eq. (6.38c), the following parameters are calculated: Zr1 5 0 1:198 pu, Zr2 5 0:138 pu, Zr3 5 0:453 pu, Zr4 5 5:176 pu, β 1 5 2 47:9 degrees, 0 0 0 0 β 2 5 2 175:6 degrees, β 3 5 78:2 degrees, β 4 5 79:2 degrees, β 5 5 2 89 degrees, 0 Tcb 5 3:68 pu, Tdc;cb 5 55:23 pu, Tac 5 3:67 pu, ωdc 5 0:00368 pu, ωac 5 1:29632 pu. In Hz, the rotor speed is fr 5 1:3 3 50 5 65Hz, near-dc frequency is fdc 5 0:184 Hz and near-rotor speed frequency is fac 5 64:816Hz. (A)

(C)

5

Pre-dip dc trans.

3

7

ac trans. Steady state

Pre-dip ac trans.

5

dc trans. Steady state

3

1

1 –1

–1

–3

ms

–3 –100

–50

0

50

100

150

200

250

300

350

400

5

(B)

–50

(D) Fault current

ms

–5 –100

0

50

100

150

200

250

300

350

400

7

Pre-dip

5

3

Fault current

Pre-dip

3 1 –100

–50

–1

1 0

50

100

150

200

250

300

350

–1

400

ms

–3 –3

ms

–5 –100

–50

0

50

100

150

200

250

300

350

400

Figure 6.18 DFIG fault currents for Example 6.5 with crowbar action, Rcb 5 10Rr , Ps 5 1:66 MW, Qs 5 0:2Mvar and ωr 5 1:3 pu and 70% symmetrical voltage dip: (A) stator current components, (B) stator current, (C) rotor current components and (D) rotor current. Fig. 6.18 shows the stator and rotor short-circuit currents. Case (b), ωr 5 1:3 pu. Crowbar resistor Rcb 5 30Rr. Using Eq. (6.38c), the following parameters are calculated: Zr1 5 1:222 pu, Zr2 5 0:2804 pu, 0 0 0 Zr3 5 0:5146 pu, Zr4 5 5:182 pu, β 1 5 2 21:4 degrees, β 2 5 2 167:7 degrees, β 3 5 0 0 0 59:5 degrees, β 4 5 62:4 degrees, β 5 5 2 87:1 degrees, Tcb 5 1:31 pu, Tdc;cb 5 69:46 pu, Tac 5 1:3 pu, ωdc 5 0:00762 pu, ωac 5 1:29238 pu. In Hz, the rotor speed is fr 5 65 Hz, near-dc frequency is fdc 5 0:381 Hz and near-rotor speed frequency is fac 5 64:619 Hz. Fig. 6.19 shows the stator and rotor short-circuit currents.

5

(A)

(C) Pre-dip dc trans.

3

5

ac trans. Steady state

Pre-dip ac trans.

3

dc trans. Steady state

1 1 –1 –1 –3

ms

–3 –100

–50

0

50

100

150

200

250

300

350

(B)

–100

–50

0

(D) Pre-dip

ms

–5

400

5

50

100

150

200

250

300

350

400

5

Fault current

Fault current

Pre-dip

3

3

1 1 –1 –100

–50

–1

0

50

100

150

200

250

300

350

400

ms

–3

ms

–5

–3 –100

–50

0

50

100

150

200

250

300

350

400

Figure 6.19 DFIG fault currents for Example 6.5 with crowbar action, Rcb 5 30Rr , Ps 5 1:66 MW, Qs 5 0:2Mvar and ωr 5 1:3 pu and 70% symmetrical voltage dip: (A) stator current components, (B) stator current, (C) rotor current components and (D) rotor current. (A)

7 5

Pre-dip current

Rcb = 0

Rcb = 10 Rr

Rcb = 30 Rr

3 1

–1 ms

–3 –50

0

(B)

50

100

150

200

250

300

350

400

7 5

Pre-dip

Rcb = 0

Rcb = 10 Rr

Rcb = 30 Rr

3

1 –1 –3 –100

–50

0

50

100

150

200

250

300

350

ms 400

Figure 6.20 DFIG fault currents for Example 6.5 with crowbar action, Ps 5 0:465 MW, Qs 5 0Mvar, ωr 5 0:8 pu, symmetrical voltage dip of 70% and three cases of crowbar resistance of Rcb 5 0, Rcb 5 10Rr and Rcb 5 30Rr : (A) stator currents and (B) rotor currents.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

511

Case (c), ωr 5 0:8 pu, Ps 5 0:465 pu and three cases of crowbar resistance values of Rcb 5 0; 10 and 30Rr . The initial conditions and parameters can be calculated as in the above cases. We give below the parameters for the case of Rcb 5 30Rr as follows: Zr1 5 0:8378 pu, Zr2 5 0:2698 pu, 0 0 Zr3 5 0:3776 pu, Zr4 5 3:195 pu, β 1 5 14:65 degrees, β 2 5 2 18:2 degrees, 0 0 0 0 β 3 5 46:28 degrees, β 4 5 50:96 degrees, β 5 5 2 85:3 degrees, Tcb 5 1:3 pu, Tdc;cb 5 94:3 pu, Tac 5 1:29 pu, ωdc 5 0:00862 pu, ωac 5 0:7914 pu. In Hz, the rotor speed is fr 5 40 Hz, near-dc frequency is fdc 5 0:43 Hz and near-rotor speed frequency is fac 5 39:569 Hz. It is worth noting the substantial increase in Tdc;cb as predicted in Eq. (6.33b). Fig. 6.20 shows the stator and rotor short-circuit currents for the three cases of crowbar resistance values.

6.7

Type 4 variable-speed inverter-interfaced wind turbine generators

These are ac generators that are connected to the ac network through a full-size ac/ dc/ac back-to-back converter with a rating that matches that of the generator. Many such electrical generators presently in use are low-speed multipole ‘ring’-type permanent magnet synchronous generators. Conventional ac excitation is also used. These generators are directly driven by the wind turbine rotor, that is, there is no gearbox between the turbine rotor and generator. Other types of electrical generators used are squirrel-cage induction generators but these are generally connected to the turbine rotor through a gearbox. Fig. 6.21 shows typical type 4 wind turbine generators. The modelling and analysis of the short-circuit contribution of such converterinterfaced generators is presented in Section 6.11.

6.8

Type 5 variable-speed wind turbine synchronous generators

A type 5 variable-speed wind turbine generator consists of a mechanical speed converter or a variable-speed gearbox to achieve the speed variation, and a standard fixed-speed synchronous generator directly connected to the ac grid. This is Wind turbine

Induction generator

dc

Generatortransformer Grid

Gearbox Converter-side Grid-side voltage-source voltage-source converter converter

Figure 6.21 Type 4 variable-speed wind turbine generator.

512

Power Systems Modelling and Fault Analysis

Wind turbine Mechanical variable - speed converter

Synchronous Generatorgenerator transformer MV Grid

Figure 6.22 Type V variable-speed mechanical converter directly connected synchronous generator.

illustrated in Fig. 6.22. At the time of writing, this type of wind turbine generator has not seen any large-scale use in the commercial wind turbines market. The short-circuit contribution of type 5 wind turbine generators is the same as standard conventional synchronous generators covered in Chapter 5, Modelling of rotating ac synchronous and induction machines.

6.9

Solar photovoltaic (PV) generators and connection to the ac grid

6.9.1 Solar PV applications Solar PV generators have become a mainstream cost-effective source of electric energy supply. The physics of solar energy conversion to dc electricity is beyond the scope of this book. In brief, however, solar PV panels convert directly and instantaneously incoming solar energy into electrical energy using the photoelectric effect where semiconductors, suitably doped, generate dc electric current when exposed to solar irradiance. The main applications of solar PV generators are: 1. Stand-alone off-grid PV systems feeding a load and some applications include batteries for energy storage. 2. Hybrid PV systems that consist of a combination of solar PV modules and a complementary method of electricity generation such as a wind turbine, gas or diesel generator, usually with batteries to provide energy storage. 3. Grid-connected PV systems. These can be divided into categories based on installation size. Small-scale systems are installed on domestic rooftops, sheds and small buildings, have a rating of typically a few kW and connected to the low-voltage grid. Large-scale systems of a few to hundreds of MWs are connected to the utility grids at medium, high or even extra high voltage through appropriate electrical networks.

6.9.2 Solar PV generator components A solar PV cell typically produces an open-circuit dc voltage of about 0.5 V and a short-circuit current of several Amps, typically 45 A depending on cell area. A typical PV module may consist of 60 series-connected cells (10 3 6). Modern PV modules may also contain 72 (12 3 6) or 96 (12 3 8) solar cells. Fig. 6.23A illustrates a typical 72-cell PV module.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

(A)

513

Solar PV cell

≡ Solar PV module: 72 cells (12×6) (B)

(C)

String diodes Series PV modules

dc/ac inverter Three-phase ac output

Parallel strings

Figure 6.23 (A) Solar photovoltaic (PV) cell and module, (B) typical solar PV strings and array topology and (C) representation of a central PV inverter.

Series connection of solar cells and modules builds up the dc voltage rating and parallel connection builds up the current rating. A PV string consists of a number of PV modules connected in series to produce the required dc voltage, without amplification. A PV array consists of a number of parallel strings that achieve the required power rating. Some string inverters are likely to be current source inverters but in the central inverter topology, the array is connected to a common or central voltage-source inverter. Fig. 6.23B shows a representation of a large PV field and a typical central inverter topology.

6.9.3 Solar PV voltage-source inverters The basic objectives of grid-connected solar PV inverters are to: 1. Convert the dc power of the PV array into sinusoidal ac power and export it to the ac grid. 2. Regulate the dc voltage/current operating point of the PV array in order to extract maximum available dc power.

Given the large power output variations of solar PV fields, the inverters must have high-power efficiency over a wide range of input voltage and power, be reliable and cost effective. System operators nowadays require inverters to be

514

Power Systems Modelling and Fault Analysis

(A)

PV

dc/ac Three-phase

(B)

PV

dc/dc

dc/ac Three-phase

(C)

PV

dc/dc

dc/ac Three-phase

PV

dc/dc

Figure 6.24 Solar PV inverter topologies: (A) central one-stage, (B) central two-stage and (C) distributed two-stage.

grid-friendly and to contribute to grid support. This means that inverters need to have fault ride-through capability which includes fast reactive current injection during voltage dips or faults on the grid, provide voltage control and reactive power capability, frequency control, power quality, islanding detection, etc. Inverters are generally classified into single-stage and two-stage inverters as shown in Fig. 6.24. In single-stage inverters, the dc power of the PV array is directly fed to the dc/ac inverter. However, in two-stage inverters, the variable dc voltage of the array is regulated and converted to a constant dc voltage in a dc/dc converter then converted to ac by the second stage dc/ac inverter. Fig. 6.24C shows a distributed two-stage topology that may be used in smaller PV systems where it offers advantages under partial shading. However, the single-stage systems are more efficient and economical as they avoid the power losses and cost of the extra conversion stage. At the time of writing, the largest installed size of a single utility-scale solar PV generator and inverter is 2.0 MW/2.2 MVA at 1000 V dc and 4.0 MW/4.4 MVA at 1500 V dc. Most current inverters use IGBT switches although silicon carbide MOSFET switches are the latest to be introduced by some manufacturers and used at the highest power ratings.

6.9.4 Grid connection of solar PV power plant In large utility-scale solar PV installations, the most frequently used dc voltages are 600 V, 1000 V and, more recently, 1500 V. The ac output voltage of the

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

(A)

dc ≤ 1.5 kV

ac ≤ 1 kV Y Δ

515

ac MV: 10–36 kV ac

Y Δ

Y Δ

(B)

MVDC: 10–50 kV ac MV grid: 10–36 kV ac

dc: 1.6–3.5 kV

dc/dc

Y Δ

dc Distribution grid

Figure 6.25 (A) Conventional utility-scale solar PV collection approach and (B) emerging MVDC utility-scale solar PV collection approach.

voltage-source inverter, typically 350700 V, is then stepped up to medium voltage of typically 1036 kV. The current approach of utility-scale solar PV collection network is illustrated in Fig. 6.25A. Advances are currently being made in medium-voltage dc (MVDC) for utilityscale PV systems and these have very recently entered the commercial market. In these systems, dc/dc converters boost string voltages to 1050 kV dc which reduces current and hence active power losses, and increases connection distances to the main substation where a single large dc/ac inverter converts to ac for grid transmission. This approach eliminates the PV array inverters and their transformers and switchgear required by the current medium-voltage ac approach. Medium-voltage dc solutions are targeting solar PV parks in a range up to 150 MW. Fig. 6.25B illustrates a typical MVDC utility-scale solar PV collection and connection arrangement. If two inverters are connected to a single three-winding transformer, and two transformers share the same connection cable to the MV busbar, then a 100 MW solar PV plant would have 25 such parallel connections. Another step-up transformer permits connection to the high-voltage network, for example, 132 kV. Fig. 6.26 illustrates such a typical connection.

516

Power Systems Modelling and Fault Analysis

1.1 MVA 1 MW 1000Vdc

2.2 MVA 0.6/33 kV Y Δ Y

Y Δ Y 1

2 3

33 kV

25

Δ 110 MVA Y 33/132 kV

132 kV Grid

Figure 6.26 Typical arrangement of a 100 MW solar PV plant with 100 3 1.1 MVA voltagesource inverters.

6.9.5 Short-circuit current contribution of solar PV inverters The modelling and analysis of solar PV generators connected to the ac grid through dc/ac inverters and the calculation of their short-circuit current contribution are covered in Sections 6.116.14.

6.10

Technologies interfaced to the ac grid through voltage-source inverters

There are several technologies that are fully interfaced to the ac grid through a fullsize or fully rated dc/ac voltage-source inverter. These technologies include: type 4 wind turbine generators presented in Section 6.7 and solar PV generators presented in Section 6.9. Other technologies are grid-connected BESS, distributed energy resources, modern HVDC systems and variable-speed pump storage hydropower plants typically up to 100 MVA rating. In type 4 wind turbine generators, the full output of the generator is fed into a power electronics rectifier or a voltage-source converter, then through a dc link into a voltage-source inverter. In full-size converter pumped storage hydropower generators, the full output of the generator is fed into a back-to-back voltage-source

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

517

converter. In grid-connected solar PV generators and BESS, the full output is fed through the dc/ac inverter. In type 4 wind turbine generators and full-size converter pump storage hydropower generators, the generator-side converter is usually used for controlling the generator speed and the grid-side voltage-source inverter is used to export the full generator output to the ac grid as well as for voltage or reactive power control. The static converters effectively isolate the electrical generator and its specific electrical performance characteristics from the ac grid in the event of disturbances such as short-circuit faults on the ac grid network. The basic objective of the control of the solar PV voltage-source inverter is to extract the maximum available power from the solar field and export it to the ac grid but modern grid-connected inverters are also required to provide voltage or reactive power control. Like type 4 generators, the electrical performance characteristics of solar PV inverters in the event of disturbances such as short-circuit faults on the ac grid network are entirely determined by the inverter control strategy.

6.11

Modelling and analysis of grid-connected voltagesource inverters

6.11.1 General inverter model A general inverter model that represents the currently dominant technology of current-controlled voltage-source inverters is presented in this section. Since the fault current contribution of such inverters is entirely determined by the inverter control system during faults and voltage dips on the ac grid, a presentation of inverter controls becomes inevitable. Our presentation can only be general in nature since inverter control systems are proprietary and may vary among manufacturers even for the same type of technology. This treatment is, however, appropriate since it captures the main elements of the control system and hence the behaviour of the inverter under the grid disturbances of interest. In the current, widely used current-controlled voltage-source inverters, the inverter output ac current is normally controlled in order to control the active and reactive power output of the inverter. The current controllers obtain their reference values from outer control loops such as maximum power point tracking control systems used in wind turbine generators and solar PV generators. Other reference or set points are designed to deliver ac network voltage control at the inverter terminals or a remote point of common coupling. The overall cascaded control structure of a dc/ac voltage-source inverter, widely used by manufacturers today, is shown in Fig. 6.27. In this book, only basic and relevant control techniques are presented that are sufficient to establish the required understanding of the short-circuit current contribution of voltage-source inverters. Fig. 6.28 shows a grid-connected dc/ac inverter with its typical filter. The filter capacitors may be star- or delta-connected and, if the latter, then a star equivalent is

518

Power Systems Modelling and Fault Analysis

Power and voltage set points MPPT and grid level control

Reference modulation signal

Current set points Inner current controller

Outer controller

PWM switching logic

Figure 6.27 Overall cascaded control structure of a voltage-source inverter. MPPT stands for Maximum power point tracking for wind turbine generators and solar PV generators.

2

+C –

,

( ) ( )

+ C 2 –

( )

Figure 6.28 Basic circuit of a two-level grid-connected voltage-source inverter.

shown in Fig. 6.28 in order to create a neutral point and simplify the equations derived next. The following voltage equation can be written 2

3 2 er R 4 ey 5 5 4 0 0 eb

0 R 0

32 3 2 0 ir L 0 54 iy 5 1 4 0 R 0 ib

0 L 0

d iryb 1 vryb eryb 5 ½R iryb 1 ½L dt

3 2 3 2 3 i 0 vr d r 0 5 4 i y 5 1 4 vy 5 dt L ib vb

(6.40a)

(6.40b)

where eryb is the inverter three-phase output voltage; iryb is the inverter three-phase output current; vryb is the filter capacitor three-phase voltage; and R, L are the per phase filter resistance and inductance of the converter, respectively. ac current control offers advantages in terms of simplicity, stability and fast response. However, classical PI controllers have two main disadvantages. The PI controller is unable to track a sinusoidal reference current without an appreciable steady-state error and has a poor disturbance rejection capability. The reason for this is the inadequate performance of the integrator when the disturbance varies periodically. Proportional-resonant controllers can be used but this is outside the scope of this book. Therefore it is advantageous to transform Eq. (6.40) to a synchronously rotating dq reference frame where, as we discussed in Chapter 5, Modelling of rotating ac synchronous and induction machines, electrical quantities become constant dc quantities in the steady state and, for rotating machines, may vary slowly, up to a maximum of a few Hz, under transient conditions. This

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

519

q-axis b

d-axis = = =0

r-axis

y Figure 6.29 Voltage-source inverter vectors in a synchronously rotating dq reference frame.

requires back and forth transformations between the stationary three-phase ryb reference frame and the rotating dq reference frame. In the context of converter controls, the word ‘synchronously’ refers to the grid’s synchronous angular frequency ωs . For convenience, we use Fig. 6.29 and reproduce the transformations between the stationary reference ryb and the synchronously rotating dq0 reference frames using Eqs. (5.6) and (5.7). In general terms, the transformations can be written as 2 3 cosθ xd 2 4 xq 5 5 4 2 sinθ 3 1=2 x0 2

cosðθ 2 2π=3Þ 2 sinðθ 2 2π=3Þ 1=2

32 3 cosðθ 1 2π=3Þ xr 2 sinðθ 1 2π=3Þ 54 xy 5 1=2 xb

xdq0 5 ½TðθÞ xryb

(6.41a)

(6.41b)

and 2

3 2 xr cosθ 4 xy 5 5 4 cosðθ 2 2π=3Þ cosðθ 1 2π=3Þ xb

xryb 5 ½TðθÞ21 xdq0

32 3 2 sinθ 1 xd 2 sinðθ 2 2π=3Þ 1 54 xq 5 2 sinðθ 1 2π=3Þ 1 x0

(6.42a)

(6.42b)

x0 represents a zero-sequence quantity that does not exist in a balanced or isolated network as in the case of an inverter connected to the star winding side of a stardelta step-up transformer. The phase angle θ is the angle between the d axis and phase r as shown in Fig. 6.29. These transformations apply to all electrical variables.

520

Power Systems Modelling and Fault Analysis

Eq. (6.40b) can be transformed to the dq reference frame by applying the transformation given in Eq. (6.41). Premultiplying Eq. (6.40b) by ½TðθÞ, we obtain

d ½ ½ ½ iryb 1 ½TðθÞ vryb ½TðθÞ½eryb 5 TðθÞ R iryb 1 L dt

edq0

d TðθÞiryb dTðθÞ 5 ½R idq0 1 ½L 2 ½L iryb 1 vdq0 dt dt

Following some mathematical manipulations, and removing the zero-sequence component, it can be shown that the following dq equation can be obtained R 0 L 0 didq 0 1 edq 5 2 ωs L idq 1 idq 1 vdq (6.43a) 0 R 0 L dt 21 0 Expanding Eq. (6.43a), we obtain ed 5 Rid 1 L

d id 2 ωs Liq 1 vd dt

(6.43b)

eq 5 Riq 1 L

d iq 1 ωs Lid 1 vq dt

(6.43c)

where ωs 5 2πfs is grid angular frequency. The above equations show that the d and q axes voltages are interdependent due to the speed or frequency voltage crosscoupling terms ωs Liq and ωs Lid . Based on instantaneous power theory, the converter active and reactive power outputs are given by P 5 er ir 1 ey iy 1 eb ib

(6.44a)

Q 5 ery ib 1 eyb ir 1 ebr iy

(6.44b)

Eq. (6.44) can be transformed to the dq reference frame by applying the transformation given in Eq. (6.41). Following some mathematical manipulations, we obtain P5

3 ed id 1 eq iq 2

(6.45a)

Similarly, the inverter’s reactive power output is given by Q5

3 2 ed iq 1 eq id 2

(6.45b)

6.11.2 Phase-locked loop Currently, the most commonly used control strategy for a grid-connected voltagesource inverter is the decoupled d and q axis control method where the ac currents and voltages are transformed to the rotating dq reference frame and synchronised with the ac grid voltage by means of a phase-locked loop (PLL). The d axis is aligned with the measured ac grid voltage phasor and this results in a zero value of

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

(A)

521

,

Inner current controller

Outer controller

( )

(B)

= 0

+

Σ

+

1

( )

−

( ) =

Figure 6.30 (A) Control structure of a current-controlled voltage-source inverter and (B) synchronous reference frame PLL for converter grid synchronisation.

the q axis component of the measured voltage as illustrated in Fig. 6.29. This strategy enables independent control of active and reactive power, and dc and ac voltages. The voltage-source inverter is synchronised with the ac grid and injects power at the grid frequency. The reference rotating angle required for synchronisation is obtained using a PLL. A PLL is a control system that generates an output signal that has the same phase as the input signal and thus provides the rotating angle for the dq transformation in order to synchronise the converter control system with the grid frequency. There are many PLL designs that depend on specific requirements and applications but this is outside the scope of this book. For illustration, Fig. 6.30A shows the overall control structure of a current-controlled voltage-source inverter and Fig. 6.30B shows a very simple, widely used PLL design known as the synchronous reference frame PLL.

6.11.3 Inverter inner current control loop The inverter ac current is controlled in Eq. (6.43) using a reactor between the inverter voltage edq and the measured filter voltage vdq . A reference inverter voltage is generated by the inverter to control the current in the dq reference frame as follows 0 eref d 5 ed 2 ω s Liq 1 vd

(6.46a)

0 eref q 5 eq 1 ωs Lid 1 vq

(6.46b)

522

Power Systems Modelling and Fault Analysis 0

0

where the new voltage terms ed and eq represent the voltage across the inverter output filter reactor and are given by e0d 5 Rid 1 L

d id dt

(6.47a)

e0q 5 Riq 1 L

d iq dt

(6.47b)

Therefore the equation of a PI current controller acting on the current deviations from reference values is given by ð ref i e0d 5 kp iref 2 i 2 i 1 k dt d i d d d

(6.48a)

ð ref i 2 i 2 i e0q 5 kp iref 1 k dt q i q q q

(6.48b)

Substituting Eq. (6.48) in Eq. (6.46), we obtain ð ref ref i 5 k i 2 i 2 i eref 1 k dt 2 ωs Liq 1 vd p d i d d d d

(6.49a)

ð ref ref i eref 5 k i 2 i 2 i 1 k dt 1 ωs Lid 1 vq p q i q q q q

(6.49b)

Taking the Laplace transform of Eq. (6.49) where S 5 jω is the complex frequency, we obtain ki ref eref 5 k 1 id 2 id 2 ωs Liq 1 vd p d S eref q

ki ref 5 kp 1 iq 2 iq 1 ωs Lid 1 vq S

(6.50a)

(6.50b)

Fig. 6.31 shows a block diagram of the inverter inner current control loops in the synchronous dq axes reference frame. Taking the Laplace transform of Eq. (6.47), we obtain the open-loop transfer function of the inverter RL reactor as e0d 5 ðR 1 SLÞid

(6.51a)

e0q 5 ðR 1 SLÞiq

(6.51b)

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

523

+

+

Σ

+

+

−

Σ −

PWM switching logic − +

+

Σ

+

+

Σ +

Figure 6.31 Voltage-source inverter inner current control loops.

Similarly, taking the Laplace transform of Eq. (6.48), we obtain ki ref 0 ed 5 kp 1 id 2 id S ki ref e0q 5 kp 1 iq 2 iq S

(6.52a)

(6.52b)

From Eqs. (6.51) and (6.52), the closed-loop transfer function of the inner current controller is given by id iref d

5

iq iref q

5

LS2

k 1 kp S i

1 R 1 kp S 1 ki

(6.53a)

and the open-loop transfer function of the inner current controller is given by id iq ki 1 (6.53b) 5 5 k 1 p ref ref S R 1 LS id 2 id iq 2 iq The above equations neglect the very small switching time delay of the inverter e2Tsw S which can be included as 1=ð1 1 Td SÞ with Td 5 Tsw =2 as presented in Section 6.2. The open-loop and closed-loop transfer functions become ki 1 1 5 kp 1 1 1 Td S R 1 LS S 2 id

id iref d

iq iref q

ki 5 kp 1 S 2 iq

1 1 1 Td S

1 R 1 LS

(6.54a)

(6.54b)

524

Power Systems Modelling and Fault Analysis

and id iref d iq iref q

5

ki 1 kp S

ð1 1 Td SÞ½LS2 1 R 1 kp S 1 ki

(6.55a)

5

ki 1 kp S

ð1 1 Td SÞ½LS2 1 R 1 kp S 1 ki

(6.55b)

The open-loop and closed-loop transfer functions are used to tune the PI controller gains kp and ki to achieve typical control objectives of fast, stable and robust response. Several tuning methods are available and can be used such as symmetric optimum, modulus optimum, closed-loop bandwidth control, root locus or gain and phase margins, etc. However, this discussion is outside the scope of this book. For our purpose, we neglect the small Td and use the pole-zero placement method where the zero is placed at the dominant pole and therefore the second-order closed-loop transfer function of Eq. (6.53a) can be written as id iref d

5

k 1 kp S ki 1 kp S 1

5 i

5 1 1 τS ð1 1 τSÞ ki 1 kp S LS 1 R 1 kp S 1 ki 2

(6.55c)

The factorised second-order characteristic equation has two poles with one matching the zero cancelling and hence

out. Expanding the factorised denominator gives ð1 1 τSÞ ki 1 kp S 5 τkp S2 1 τki 1 kp S 1

ki then equating the constants with those in the original equation LS2 1 R 1 kp S 1 ki , we obtain kp 5

L τ

(6.56a)

ki 5

R τ

(6.56b)

Therefore the PI controller gains are calculated from the inverter reactor resistance and reactance, and time constant τ of the resultant closed-loop first order lag function (id =iref d ) 5 1=ð1 1 τSÞ. The time constant τ can be related to the closedloop bandwidth as follows τ5

1 1 5 bandwidth c 3 fsw

(6.57)

where c is an empirical constant whose value is typically selected between 0.1 and 0.2, that is, the bandwidth is selected in the range of 10%20% of the inverter switching frequency. For example, for a high rating inverter with fsw 5 2000 Hz and c 5 0:1, the bandwidth is 200 Hz and τ 5 5 ms. For c 5 0:2, the bandwidth is 400 Hz and τ 5 2:5 ms. τ would be much smaller for smaller size inverters that

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

(A)

–10 0

0

10

20

Time (ms) 30 40

50

60

525

70

–0.2

_ (pu)

_ –0.4 = 2 ms –0.6

= 5 ms

–0.8

= 10 ms

–1

(B)

–10 0

0

10

20

Time (ms) 30 40

50

60

70

–0.2

_ (pu)

_ –0.4 = 2 ms –0.6 –0.8

= 5 ms = 10 ms

–1

Figure 6.32 Voltage-source inverter inner current control loop responses: (A) active current response and (B) reactive current response.

employ a much higher switching frequency. It is noted that, in practical applications, the inverter switching time delay represented as 1/(1 1 Td S) is taken into account in tuning the PI controller gains. An adequate choice of τ can provide a robust and stable current performance that also avoids a transient overshoot and hence thermal stresses on the inverter semiconductors switches. Fig. 6.32 shows the active and reactive current responses and the effect of different values of τ for given step changes in their reference currents.

6.11.4 Inverter outer control loops The d and q reference currents of the inner fast current control loops are set by slower outer control loops. With the grid voltage alignment along the d axis, the d axis reference current iref d controls the active power output and hence the dc input

526

Power Systems Modelling and Fault Analysis

voltage whereas the q axis reference current iref q controls the reactive power output or ac grid voltage. Using PI controllers, we can write the following equations: For active power control ð ref

ref

P 2 P dt iref 5 k P 2 P 1 k p1 i1 d

(6.58a)

For dc voltage control ð ref ref iref 5 k V 2 V 2 V 1 k dt V p1 dc i1 dc d dc dc

(6.58b)

For ac grid voltage control iref q

ref 5 kp2 Vrms

ð

2 Vrms 1 ki2

ref Vrms 2 Vrms dt

(6.59a)

Or for proportional voltage control only ref

iref q 5 kp3 Vrms 2 Vrms

(6.59b)

and, for reactive power control iref q

5 kp2 Q

ref

2 Q 1 ki2

ð

Qref 2 Q dt

(6.59c)

Or, after taking the Laplace transforms

iref d

ki1 ref 5 kp1 1 P 2P S

ki1 ref iref 5 k 1 2 V V p1 dc d dc S

(6.60b)

iref q

ref

ki2 ref 5 kp2 1 Vrms 2 Vrms or iref q 5 kp2 Vrms 2 Vrms S

(6.60a)

ki2 ref iref 5 k 1 Q 2Q p2 q S

(6.61a)

(6.61b)

Fig. 6.33 shows the block diagram of the converter inner and outer control loops in the synchronously rotating dq-axes reference frame. A low-pass filter represented as a simple lag is included at the outputs of the outer control loops that set the d and q current references for the inner current control loops.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

Outer control loops

−

+ −

Σ

Inner current control loops

1

+

Σ

+

Σ + −

+

1

+

Outer control loops

+

1

+

1

+

+

Σ −

−

Σ

PWM switching logic

+

527

+ +

+

Σ +

Inner current control loops

Figure 6.33 Voltage-source inverter outer and inner control loops.

6.11.5 Short-circuit current contribution of voltage-source inverters with frozen controls We have established that the current supplied by the inverter is determined by its control strategy. Before a short-circuit fault occurs on the ac grid, the inverter output current is controlled both in magnitude and phase in order to deliver the required active power and reactive power outputs. In this section, we present and explain the relationship between the current delivered by the inverter under a symmetrical short-circuit on the ac grid assuming no inverter control response to the resultant voltage and output power dips. In other words, we assume in this illustraref tion that the reference currents iref d and iq remain fixed at their prefault values during the short-circuit fault duration. Our assumption means that the converter continues to deliver the same current, in both magnitude and phase, as that supplied in the power frequency cycle just before the occurrence of the short-circuit fault. The magnitude of the converter short-circuit current contribution with respect to the short-circuit current delivered by the grid at bus i can be illustrated using Fig. 6.34. Fig. 6.34A shows a voltage-source inverter connected through a transformer to a medium-voltage distribution network typically at 1036 kV. This may then be connected to a higher voltage network through another step-up transformer. The largest inverter current prior to the short-circuit fault is that supplied by the inverter at rated active and reactive power outputs. Assume that at bus i the inverter-rated power output is Prated , rated lagging reactive power output is Qrated , rated lagging and (leading) power factor is cosð[rated Þ and the system voltage at bus s is Vs 5 Vs +0 degrees. Therefore ignoring the transformer resistance Rt , because it is much smaller than the reactance Xt , the relationship between bus i voltage and bus s voltage, is given by

528

Power Systems Modelling and Fault Analysis

(A)

Rectifier or VSC

Wind turbine generator

VSC inverter

bus i

or

(

bus s

)

Grid

)

(

+

VSC

+

Solar PV generator

bus i

(B) (

(

)

Equivalent positive sequence current source

(

)

+

bus s )

+

Three-phase short circuit fault (C)

(

) (

)

+ +

≈ (

)

Figure 6.34 Voltage-source inverter current response with frozen control: (A) inverter supplying rated current prefault, (B) inverter as a positive sequence current source during the short-circuit fault and (C) vector diagram under fault condition.

Vi 5 Vs 1 jIiðratedÞ Xt 5 Vi ejδ

(6.62a)

where δ is the load angle or angle by which Vi leads Vs . The rated inverter current IiðratedÞ delivered at bus i just before the short-circuit is 2j[rated jQrated given by IiðratedÞ 5 Prated 2 5 SratedVei e2jδ or V i

IiðratedÞ 5 IiðratedÞ e2jð[rated 2δÞ

(6.62b)

where IiðratedÞ 5 Srated Vi and Srated is the rated apparent power of the inverter. At bus s, the grid system is represented by its three-phase short-circuit current infeed and positive-sequence Xs /Rs ratio. For a solid three-phase short-circuit fault at bus i, the positive-sequence current supplied to the fault from the high-voltage grid system is given by

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

Vs Xs 1 Xt IsðscÞ 5 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp 2 jtan21 Rs 1 Rt ðRs 1Rt Þ2 1 ðXs 1Xt Þ2

529

(6.63)

Since the grid inverter continues to supply a constant, balanced three-phase current during the short-circuit fault, the inverter can be represented as a constant positive-sequence current source as shown in Fig. 6.34B. Therefore the inverter current supplied to the fault is equal to IiðratedÞ and the total short-circuit fault current is the sum of the high-voltage grid and inverter short-circuit current contributions, that is, IFaultðscÞ 5 IsðscÞ 1 IiðratedÞ

(6.64)

The grid system fault current contribution IsðscÞ is practically a reactive current since it is determined by the high X/R ratio of the faulted network and thus lags the prefault grid voltage by almost 90 degrees. The inverter current IiðratedÞ leads the grid system fault current by an angle β as shown in the vector diagram of Fig. 6.34C. Therefore IiðratedÞ does not represent a short-circuit current contribution from the inverter. Rather, its reactive component, denoted IiðscÞ , which is in phase or, in practice, sufficiently in phase with the system short-circuit current IsðscÞ , represents the inverter short-circuit current contribution. The vector diagram of Fig. 6.34C shows the above voltages and currents including the short-circuit current supplied from the grid IsðscÞ . The magnitude of the inverter positive-sequence short-circuit reactive current component IiðscÞ which is in phase with the grid system fault current IsðscÞ is given by Xs 1 Xt IiðscÞ 5 IiðratedÞ cosβ 5 IiðratedÞ cos tan21 2 ð[rated 2 δÞ Rs 1 Rt

(6.65)

For a typical modern MV/HV transformer connecting buses i and s, Xt /Rt ratio ranges from 25 to 55. These correspond to impedance phase angles tan21 ðXt /Rt Þ of 87.7 and 89 degrees, respectively. The equivalent Xs /Rs ratio of the grid can vary over a wide range depending on the type of grid connection such as underground cables, overhead lines, HV/EHV transformers and proximity of synchronous generation plant. In practice, a combination of some of these elements will better represent the grid network. Therefore typically, Xs /Rs may range from 15 to 50, which corresponds to impedance phase angles of tan21 ðXs /Rs Þ of 86.2 and 88.9 degrees, respectively. Xt For example, for Xt /Rt 5 35 and Xs /Rs 5 40, tan21 XRss 1 1 Rt D88:5 degrees, rated inverter lagging power factor of 0.95 or φ(rated) 5 cos21(0.95) 5 18.2 degrees and load angle δ 5 8 degrees, we have IiðscÞ 5 IiðratedÞ cos½88:5 2 ð18:2 2 8Þ 5 0:20IiðratedÞ . This means that for the given parameters, only 20% of the magnitude of the rated inverter current represents a reactive current and hence a short-circuit fault current contribution.

530

Power Systems Modelling and Fault Analysis

If the inverter were operating at the rated leading power factor, it can be shown that the short-circuit reactive current component of the inverter current is given by Xs 1 Xt IiðscÞ 5 IiðratedÞ cos tan21 1 ð[rated 1 δÞ Rs 1 Rt

(6.66)

Using the above parameters, IiðscÞ 5 IiðratedÞ cos½88:5 1 ð18:2 1 8Þ 5 2 0:42IiðratedÞ . However, in this case, although this reactive current has a larger magnitude, it is out of phase with the dominant current supplied from the higher voltage grid and will therefore act to reduce the total short-circuit fault current.

6.12

Grid code requirements for dynamic reactive current injection from inverters

In general, the behaviour of grid-connected voltage-source inverters under shortcircuit faults on the ac grid may be: 1. The inverter trips in less than half a power frequency cycle when its terminal voltage falls below, say, 50% of nominal voltage on any phase. 2. The inverter remains connected to the network but stops feeding any current. 3. The inverter remains connected to the network and continues to feed its prefault current. 4. The inverter remains connected to the network and its controls respond and feed a predefined reactive current response in magnitude, rise time and settling time.

The inverter behaviours under (1)(3) above are no longer acceptable by almost all network utilities in light of the substantial growth in installed capacity of wind turbine generators, solar PV generators, other nonsynchronous generators and other inverter-interfaced systems such as VSC HVDC links, BESS, etc. The majority of system operators nowadays include in their grid codes both a low-voltage fault ride-through requirement, that is, the ability to remain connected for a specified duration and a reactive current injection requirement. A high-voltage ride-through requirement is also generally required. At the time of writing, most grid codes worldwide require a fast, positivesequence reactive current injection but are silent on any negative-sequence reactive current injection. A few grid codes have very recently begun to require a negativephase sequence current injection under unbalanced voltage dips at the converter terminals in addition to the positive-phase sequence current requirement. In effect, this is a requirement for unbalanced reactive current injections at the three-phase output terminals of the inverter. The aim is to ensure that inverters still contribute reactive currents during phase-to-phase faults to assist with fault detection and, in general, avoid potential over-voltages on healthy phases during unbalanced earth faults. Fig. 6.35 shows positive- and negative-sequence reactive current injection requirements of a typical high-voltage grid code.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

Δ (

) or Δ (

531

)

Inductive current 1

=

= 6

0.8 0.6

=

0.4

Voltage drop

0.2 0

–1

–0.8

–0.6

–0.4

–0.2 –0.2 0

0.2

–0.4

Δ Δ

= 2 ( (

) or )

0.4

Voltage rise

–0.6 –0.8 –1

Capacitive current

Figure 6.35 Typical grid code requirement for dynamic positive- and negative-sequence reactive current injections.

Within a voltage dead band of typically 6 5% around the initial voltage, the inverter reactive current output depends on the grid code requirement for voltage control and reactive power under normal system operating conditions. The reactive current requirement could be (1) zero for a unity power factor or zero reactive power requirement; (2) a constant for a constant reactive power requirement; or (3) a continuously controlled value for a continuous voltage control requirement. In Fig. 6.35, we show the latter case. Worldwide, grid code requirements for the rise time in the response of the reactive current typically range from 1.5 to 3 cycles, or 30 to 60 ms in a 50 Hz system. The settling time requirement typically ranges from 3 to 10 cycles (60 to 200 ms in a 50 Hz system). Referring to Fig. 6.35 and using 1 and 2 signs to denote positive-sequence and negative-sequence quantities, respectively, we can write Δi1 5 k1 Δu1

(6.67a)

1 1 i1 5 i1 i 1 k Δu

(6.67b)

Δi2 5 k2 Δu2

(6.68a)

i2 5 k2 u2

i2 i 50

k 5 k1 5 k2 5 1

Δu 5 Δu2 5

u2 i 50

(6.68b)

Δi1 Δi2 5 1 Δu Δu2

(6.69a)

1 U 2 U 1 i

Un U2 Un

(6.69b)

(6.69c)

532

Power Systems Modelling and Fault Analysis

where Δi1 is the change in positive-sequence current; i1 is the positive-sequence 2 current; i1 i is the initial positive-sequence current; Δi is the change in negativesequence current; i2 is the negative-sequence current; i2 i is the initial negativesequence current; Δu1 is the change in positive-sequence voltage; Δu2 is the change in negative-sequence voltage; U 1 is the positive-sequence voltage; Ui1 is the initial positive-sequence voltage measured over a sufficiently long time period such as 1 minute; U 2 is the negative-sequence voltage; andUn is the nominal system voltage. The factor or gain k is taken to be equal in the positive-sequence and negativesequence current injection and may typically range from 2 to 6 as illustrated in Fig. 6.35 where the maximum injected reactive current is assumed to be 1 pu of rated current. Where priority is given to the required reactive current injection over the active current, the active current has to be recalculated and adjusted in order to avoid converter transient current overload. With both positive- and negative-sequence current injections, the following simplified and slightly conservative approximation can be used as a limiting condition 1 i j 1 ji2 j 5 imax

(6.70a)

where imax is the maximum transient current limit of the inverter. However, a more accurate expression may be used by evaluating the transient currents of the inverter in the stationary ryb reference frame as follows ir;max 5

q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2 i1 1 ði2 Þ2 1 2i1 i2 cosðθi1 2 θi2 Þ

(6.70b)

iy;max 5

q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ

2 i1 1 ði2 Þ2 1 2i1 i2 cosðθi1 2 θi2 2 2π=3Þ

(6.70c)

ib;max 5

q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ

2 i1 1 ði2 Þ2 1 2i1 i2 cosðθi1 2 θi2 1 2π=3Þ

(6.70d)

where θi1 is the phase angle of the positive-sequence current phasor i1 and θi2 is the phase angle of the negative-sequence current phasor i2 . Where only positive-sequence reactive current, denoted Δiq , is injected, the available maximum active current is given as i1 d;max

5

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 1 i2max 2 Δi1 q 1iq;i

2

(6.70e)

1 where i1 d;max is the limited positive-sequence active current; Δiq is the injected 1 positive-sequence reactive current; and iq;i is the initial positive-sequence reactive current.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

533

Example 6.6 Consider a grid-connected solar PV inverter having a maximum transient current rating imax 5 1:1 pu of rated current, k1 5 2 and k2 5 0. The initial active and reactive currents are 0:9 pu and zero, respectively. A three-phase fault on the ac grid results in a symmetrical voltage dip at the inverter terminals of 2 0:4 pu. Calculate the inverter active and reactive steady state current responses. The injected reactive current is iq1 5 2 3 ð 2 0:4Þ 5 2 0:8 pu and the positive-sequence qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 5 1:12 2 ð 20:8Þ2 5 0:755 pu. active current has to be limited to id;max For a voltage dip of 2 0:5 pu, the injected reactive current is iq1 5 2 3 ð 2 0:5Þ 5 2 1 pu qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 5 1:12 2 ð 21Þ2 5 0:4583pu. and the active current is limited to id;max If k1 5 3, then for a voltage dip of 2 0:4 pu, the injected reactive current is 1 iq 5 3 3 ð 2 0:4Þ 5 2 1:2 pu. However, this has to be limited to the inverter’s maximum transient current rating of 1:1 pu. The active current is limited to zero.

6.13

Short-circuit current contribution of voltage-source inverters during balanced three-phase voltage dips

6.13.1 Dynamic reactive current control In Section 6.11, we presented the modelling and controls of the grid-connected voltage-source inverters and in Section 6.12, we presented typical modern grid code requirement for dynamic, positive- and negative-sequence reactive current injections under balanced and unbalanced voltage dips. In this section, we present inverter short-circuit current contribution under balanced or symmetrical three-phase short-circuit faults on the ac grid. These faults result in balanced voltage dips at the inverter terminals and hence only a positive-phase sequence voltage. The set point of the positive-sequence reactive current injection in the inner current control loop iref is modified as shown in q the block diagram of Fig. 6.36. This is calculated to meet the grid code +

Σ

Min ,

−

1 1+

+ −

1

Σ

Δ

Δ

,

Σ

+

+

,

1+

Figure 6.36 Typical voltage-source inverter block diagram for dynamic reactive current control.

534

Power Systems Modelling and Fault Analysis

positive-sequence reactive current injection requirement discussed in Section 6.12. The limited active current set point iref d is calculated as shown in Fig. 6.36. In this block diagram, the measured actual positive-sequence voltage is compared to the positive-sequence voltage that existed over a short period of typically 1 minute before the fault or voltage dip. In practical controller implementation, the k factor gain is adjustable and the output of this block Δiref q is added to the initial reactive current value to obtain the reactive current set point. This is shown in Fig. 6.36.

6.13.2 Examples of inverter positive-sequence short-circuit current contribution We now illustrate the short-circuit current contribution of a typical voltage-source inverter used in a utility-scale solar PV power plant and illustrated in Fig. 6.37. The inverter data are as follows: Apparent power, SN Rated voltage, VN Rated current, IN Rated transient overcurrent, Imax Rated power factor Rated MW, PN and Mvar, QN Inverter reactor impedance Inverter capacitive filter Inverter transformer

1.1 MVA 600 V 1058 A 1164 A 0.92 (leading and lagging) 1.012 MW, 0.430 Mvar Resistance 0.0004 Ω, reactance 0.017 Ω Resistance 0.065 Ω, capacitor 321 μF (star equivalent) Impedance 7.2% on 1.1 MVA, X/R 5 8.6

Following the onset of the ac grid fault, it is assumed that the inverter control starts to increase the current after a 20 ms time delay. This delay is due to the requirement to detect the fault by measuring and processing the three-phase rms voltages at the inverter terminals. The inner controller response is tuned such that the current reaches 90% of its reference value after 14.3 ms.

PV

Inverter output

Inverter reactor

Inverter LV/MV transformer

Grid MV

Figure 6.37 Solar PV voltage-source inverter equivalent.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

535

Example 6.7 A solid three-phase short-circuit fault is applied on the inverter transformer medium-voltage terminals at t 5 0. The initial active and reactive power outputs of the inverter are 0.1 pu (of rated MVA) and zero, respectively. The gain or k factor of the injected positive-sequence reactive current is 2. The results are shown in Fig. 6.38. Inverter LN voltages (V)

(A)

500 250 0 –250 –500

00 –80 –60 –40 –20

0

00 –80 –60 –40 –20

0

–1

20

40

60

80 100 120 140 160 180 200

Inverter currents (A)

ms

–1

20

40

60

80 100 120 140 160 180 200

ms

Positive sequence voltage (pu)

(B)

0.1 pu 00 –80 –60 –40 –20

0

–1

00

Positive sequence reactive current (pu)

–1

0

–8

0

–6

0

–4

20

40

60

80 100 120 140 160 180 200

ms

0

–2

0

20

40

60

80

0

10

0

12

0

14

0

16

0

18

0

20

ms

–1.1 pu

Figure 6.38 Voltage-source inverter response to a three-phase fault on transformer MV side with initial active power output of 0:1 pu: (A) inverter instantaneous voltages and currents and (B) positive-sequence voltage and reactive current.

536

Power Systems Modelling and Fault Analysis

Example 6.8 The applied fault is similar to that in Example 6.7. but the inverter initial active power output is 0.92 pu and reactive power output compensates the inverter transformer reactive power losses. The results are shown in Fig. 6.39.

500 250 0 –250 –500 00 –80 –60 –40 –20 –1

0

00 –80 –60 –40 –20

0

20

60

40

80 100 120 140 160 180 200

ms

Inverter currents (A)

Inverter LN voltages (V)

(A)

–1

20

60

40

80 100 120 140 160 180 200

ms

Positive sequence voltage (pu)

(B)

0.1 pu 00 –80 –60 –40 –20

0

–1

00

–1

0

–8

0

–6

0

–4

20

40

60

80 100 120 140 160 180 200

ms

0

–2

0

20

40

60

80

0

10

0

12

0

14

0

0

18

16

0

20

Positive sequence currents (pu)

ms

–1.1 pu

Figure 6.39 Voltage-source inverter response to a three-phase fault on transformer MV side with initial active power output of 0:92 pu: (A) inverter instantaneous voltages and currents and (B) positive-sequence voltage, active and reactive currents.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

6.14

537

Short-circuit current contribution of voltage-source inverters during unbalanced voltage dips

6.14.1 Inverter model in positive- and negative-sequence synchronous reference frames An unbalanced short-circuit fault on the ac grid such as a single-phase to earth fault or a two-phase fault results in unbalanced voltage dips at the inverter terminals. As presented in Chapter 2, Symmetrical components of faulted three-phase networks containing voltage and current sources, unbalanced phase voltages give rise to sequence voltages at the inverter terminals that consist of positive- and negativesequence voltage components but no zero-sequence voltage component because of the delta-star winding connections of the inverter transformer. In grid codes where negative-sequence reactive current injection requirement is not explicitly made, inverter manufacturers usually suppress the negative-sequence component and inject a positive-sequence current component only, that is, balanced three-phase currents. Where a negative-sequence current injection in proportion to negative-sequence voltage change is required as illustrated in Fig. 6.35, then this requires that the inverter negative-sequence voltage be measured as well as its positive-sequence voltage. Measuring, monitoring and controlling positive- and negative-sequence currents require a dual-sequence control scheme with a positive-sequence dq1 rotating reference frame as presented in Section 6.11 and a negative-sequence dq2 reference frame rotating in the opposite direction. Fig. 6.40 shows a dual-sequence reference with a positive-sequence voltage vector rotating at synchronous grid frequency ωs , and a negative-sequence voltage vector rotating in the opposite direction to the positivesequence vector, that is, at 2 ωs . Therefore following the derivation presented in

q+

q− + d+ − −

r-axis d−

Figure 6.40 Voltage-source inverter vectors in double synchronously rotating reference frames dq1 and dq2 .

538

Power Systems Modelling and Fault Analysis

Section 6.11 and using Fig. 6.31 for the inverter positive-sequence current control scheme, it can be shown that Fig. 6.41 represents the positive- and negative-sequence inner current control system in the double synchronously rotating dq reference frames. The negative-sequence current reference iref q;2 is calculated in accordance with the grid code requirement such as that given in Eq. (6.68). When grid codes do not explicitly require negative-sequence current injection, it is common practice that negativesequence current suppression is employed in the negative-sequence PI controller which implies the following: ;2 iref 50 d

(6.71a)

;2 iref 50 q

(6.71b)

In fact, typically, this negative-sequence current control or rather suppression is always active in the presence or otherwise of voltage dips on the grid.

,

+

+ +

Σ

−

−

Positive sequence

,

Σ

,

− ,

+

Σ

+

+ +

Σ

,

+ + ,

+ −

−

Σ

+ + +

Σ

,

Σ

−

−1

,

−

− ,

+

Σ

−

+ Σ +

,

Negative sequence

Figure 6.41 Voltage-source inverter positive- and negative-sequence inner current control in the double synchronously rotating dq reference frames.

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539

As illustrated in Fig. 6.41, the stationary-phase voltages and currents are transformed into positive-sequence dq1 axes and negative-sequence dq2 axes voltages and currents using a positive PLL angle θ and a negative PLL angle 2 θ, respectively. It can be seen that where the stationary-phase voltages vryb and currents iryb are unbalanced, the transformation results in a double-frequency component 2ωs in all d and q axis voltages and currents and hence inverter active and reactive power outputs. This result would be obtained from the basic synchronous reference frame PLL (SRF-PLL) structure shown in Fig. 6.30 although this simple structure is fast and precise under balanced grid voltages having a positivesequence voltage component only. However, it is not sufficiently accurate under unbalanced grid voltages and is sensitive to the presence of harmonic voltage distortions. Many PLL structures have been developed, such as the double decoupled synchronous reference frame PLL (DDSRF-PLL), instantaneous power theory PQ-PLL, sinusoidal signal integrator SSI-PLL, double second-order generalised integrator DSOGI-PLL, notch-filter-based PLL, etc. The well-established notchfilter-based PLL uses a second-order band-stop filter which, as the name implies, is tuned to reject the required double-frequency component leaving unaltered other frequencies but, this improvement is accompanied by a slower convergence of output phase angle. The types, design and tuning of PLLs under various grid voltage characteristics are outside the scope of this book. For our purposes, it suffices to state that double-frequency components should be filtered or removed from the positive- and negative-sequence d and q axes quantities before these are fed to the PI controllers in the inner current control loops.

6.14.2 Examples of inverter positive- and negative-sequence short-circuit current contributions We now illustrate two examples of inverter short-circuit current contribution for an unbalanced two-phase short-circuit fault at the MV side of the inverter transformer. In both examples, the gain or k factor of the injected positive-sequence and negative-sequence reactive currents is 2. The inverter data are as used in Section 6.13.2 and the inverter’s initial active and reactive power outputs are 0.92 pu and a value that compensates the inverter transformer reactive power losses. Example 6.9 The inverter feeds positive-sequence current only and negative-sequence current is suppressed. The inverter’s response to the solid two-phase short-circuit with positive-sequence current contribution is shown in Fig. 6.42.

540

Power Systems Modelling and Fault Analysis

(A)

Inverter LN voltages (V)

500 250 0 –250

14 0 16 0 18 0 20 0

0

0 12

80

10

60

40

0

20

ms

12 0 14 0 16 0 18 0 20 0

0 10

80

60

40

0

20

–6 0 –4 0 –2 0

–8

–1

ms

1.2 1 0.8 0.6 0.4 0.2

12 0 14 0 16 0 18 0 20 0

80 10 0

60

40

0

20

–8 0 –6 0 –4 0 –2 0

0

0

–1 0

Positive & negative sequence voltages (pu)

(B)

0

1800 1400 1000 600 200 –200 –600 –1000 –1400 –1800 00

Inverter currents (A)

–1

00 –8 0 –6 0 –4 0 –2 0

–500

ms

12 0 14 0 16 0 18 0 20 0

0 80 10

60

40

20

–1 00 –8 0 –6 0 –4 0 –2 0 0

ms Positive sequence currents (pu)

0

–0.2 –0.4 –0.6 –0.8 –1 –1.2

Figure 6.42 Voltage-source inverter response to a two-phase fault on transformer MV side with only positive-sequence current injection: (A) inverter instantaneous voltages and currents and (B) positive- and negative-sequence voltages, and positive-sequence active and reactive currents.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

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Example 6.10 The inverter feeds both positive- and negative-sequence currents. The inverter’s response to the solid two-phase short-circuit with both positive- and negative-sequence short-circuit current contributions is shown in Fig. 6.43. Inverter LN voltages (V)

(A)

500 250 0 –250 –500 00 –80 –60 –40 –20

0

00 –80 –60 –40 –20

0

00 –80 –60 –40 –20

0

00 –80 –60 –40 –20

0

20

40

60

80 100 120 140 160 180 200

ms

Inverter currents (A)

–1

–1

20

40

60

80 100 120 140 160 180 200

ms

Positive & negative sequence voltages (pu)

(B)

Positive & negative sequence currents (pu)

–1

–1

20

40

60

80 100 120 140 160 180 200

ms

20

40

60

80 100 120 140 160 180 200

ms

Figure 6.43 Voltage-source inverter response to a two-phase fault on transformer MV side with both positive- and negative-sequence current injections: (A) inverter instantaneous voltages and currents and (B) positive- and negative-sequence voltages, and positive-sequence active and reactive currents.

542

6.15

Power Systems Modelling and Fault Analysis

Sequence network representation of voltage-source inverters during balanced and unbalanced shortcircuit faults

We have shown that the short-circuit current contribution of voltage-source inverters is determined by its current control strategy. The inverter can quickly deliver a prespecified, constant ac short-circuit current to meet the grid code requirements of system operators while remaining within its transient current limit of typically 1.11.2 pu. Clearly, as seen from the external grid network, after around 3540 ms from the onset of the grid fault, the inverter behaves as a constant ac current source in parallel with an infinite impedance since the per unit reactance of its shunt filter capacitance is very large and can be conservatively neglected. Where grid codes do not specify negative-sequence current injection during grid voltage dips, the inverter acts as a constant positive-sequence current source in the positive-sequence network that feeds balanced three-phase reactive currents during both balanced and unbalanced voltage dips on the network. The inverter can be represented as an open-circuit in the negative- and zero-sequence networks. Fig. 6.44B shows the positive-, negative- and zero-sequence representations of the inverter that injects a positive-sequence reactive current only. However, where grid codes do specify both positive- and negative-sequence current injections, the inverter acts as a constant positive-sequence current source in the positive-sequence network and a negative-sequence current source in the negative-sequence network under unbalanced grid voltage dips. The inverter is (A)

Voltage-source inverter

Solar PV Or

Type 4 wind turbine generator

Or HVDC etc.

(B)

= 0

= 0

(C)

= 0

Figure 6.44 Voltage-source inverter positive, negative and zero sequence representation during voltage dips: (A) Some types of inverter input sources, (B) inverter injects positivesequence current only and (C) inverter injects both positive- and negative-sequence currents.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

543

represented as an open-circuit in the zero-sequence network. Fig. 6.44C shows the positive-, negative- and zero-sequence representations of such an inverter.

6.16

Short-circuit currents in ac grids containing mixed synchronous generators and inverters

6.16.1 General Based on the material presented in Chapter 2, Symmetrical components of faulted three-phase networks containing voltage and current sources, Chapter 5, Modelling of rotating ac synchronous and induction machines, and this chapter, we will now present techniques for the calculation of short-circuit currents in ac grid networks containing mixed conventional synchronous and induction generators, inverterinterfaced generators and other systems such as batteries and HVDC. Large-scale network analysis of such mixed systems is presented in Chapter 7, Short-circuit analysis techniques in large-scale ac power systems. Fig. 6.45A shows a typical utility-scale solar PV plant that exports its power to the ac grid at 132 kV nominal system voltage. The grid is represented by its shortcircuit current infeeds or its positive-, negative- and zero-sequence impedances and their associated X/R ratios. The 50 voltage-source inverters, their step-up (A)

2.2 MVA 2 MW PV VSC 1000Vdc

2.2 MVA 0.6/33 kV

33 kV 110 MVA 630 mm2 132 kV 33/132 kV 120 MVA

Grid #10 Short-circuit fault 10×2.2 MVA Inverters

(B) 100 MW PV

110 MVA VSC

ac grid

+ +

+ 33 kV

132 kV

=

+

=

+

=

+

Figure 6.45 100 MW solar PV park of 50 3 2.2 MVA Voltage-source inverters connected to an ac grid containing conventional synchronous generators and used in the analysis of balanced and unbalanced short-circuit currents: (A) Solar PV park network and (B) equivalent representation.

544

Power Systems Modelling and Fault Analysis

transformers and connecting 33 kV cables to the 33 kV collection busbar are aggregated and represented as a single equivalent as shown in Fig. 6.45B. The susceptances of the 33 and 132 kV cables are neglected in the short-circuit current calculations but not in the initial load flow calculations. The solid short-circuit fault under consideration is applied at the 132 kV busbar.

6.16.2 Steady-state and transient characteristics of inverter short-circuit reactive currents As discussed in Section 6.12, the inverter is modelled as a voltage-dependent current source in the positive-sequence and negative-sequence networks. This represents the linear region on Fig. 6.35. Outside this region, the inverter is represented as a constant current source equal to its maximum transient current rating. Mathematically, using Fig. 6.35, we can write

Ic 5

8 > > kΔVc > < > > > : Imax

Imax k Imax ΔVc $ k ΔVc ,

(6.72)

where Ic is the steady-state short-circuit reactive current supplied by the inverter; k is the k factor given in Section 6.12; ΔVc is the voltage change at the inverter terminals; and Imax is the maximum transient current rating of the inverter. The above equation applies to both positive-sequence and negative-sequence reactive currents and, as presented in Section 6.11, represents the final steady-state current. This current is dependent on the inverter terminal voltage during the fault but this injected current also modifies the inverter terminal voltage. This means that the steady-state value of the injected reactive current can be determined by an iterative calculation process until the change in current magnitude is within a prespecified tolerance. Based on the grid code requirement shown in Fig. 6.35 with k 5 2, the injected current is linearly dependent on the inverter terminal voltage up to a voltage dip of 0:5 pu. For larger voltage dips, the injected current becomes constant and equal to the maximum transient current the inverter is designed to deliver. In Fig. 6.45A, the inverter transformer impedance is typically 6%10% on rating and the 33/132 kV transformer impedance is typically 12%24% on rating, giving a total impedance to the fault point of 18%34%. With a typical value of maximum transient current of 1.11.2 pu on rating, the inverter terminal voltage could range from 0.2 to 0.41 pu. Therefore in such cases, the injected inverter steady state current would be constant and equal to the maximum inverter transient current rating. In Section 6.11, we showed that the actual value of inverter current from the onset of fault or voltage dip is dependent on the measurement and detection time of the voltage dip and the response of the current control system. In practical fault current analysis, calculations of initial make-and-break short-circuit currents are required. This is presented in detail in Chapter 8, International standards for

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

545

short-circuit analysis in ac power systems. For now, we simply state that the former current is required to be calculated at half cycle of power frequency after the onset of the fault, that is, 10 and 8.33 ms in 50 and 60 Hz power systems, respectively. The latter correspond to the minimum short-circuit current break time, which is typically in the range of 1.56 cycles of power frequency. The inverter inner current control loop is very fast and the injected reactive current variation with time following detection of the voltage dip, assuming a welldesigned fast-acting control system with no overshoot, can be approximated as follows: 8 0 t # Tm < ic ðtÞ 5 Ic 1 2 e2ðt2Tm Þ=τ c Tm , t # Tm 1 4τ c : Ic t . Tm 1 4τ c

(6.73)

where ic ðtÞ is inverter current variation with time from onset of voltage dip; Tm is the measurement and detection time delay of the rms voltage dip during which the inverter injected reactive current is taken as zero. Tm is typically 1020 ms; τ c is the effective time constant of the inner current control loop, typically 38 ms; and Ic is the maximum steady-state injected reactive current given in Eq. (6.72). To illustrate use of the above equation, consider the following: Tm 5 20ms, τ c 5 5ms and a minimum circuit-breaker opening time t 5 30ms. Therefore the inverter current contribution to the initial make fault current is obviously zero, and the contribution to the break fault current is 0:865Ic . For a minimum circuit-breaker opening time t 5 50ms, the contribution to the break fault current is Ic . For τ c 5 8ms, the contributions to the break fault current become 0:713Ic and 0:976Ic for t 5 30 and 50ms, respectively. In all phasor short-circuit current analysis to follow, the ac grid includes conventional rotating machines that are represented as voltage sources behind their subtransient or transient impedances. However, the technique is extended in Chapter 7, Short-circuit analysis techniques in large-scale ac power systems, and applied to the general analysis of large-scale networks containing mixed conventional machines and inverters. The inverter reactive current calculated in the analysis represents its injected current immediately upon fault occurrence, that is, at maximum voltage dip at the inverter terminals. However, the inverter injected current increases the inverter terminal voltage and hence reduces the voltage change. This causes a reduction in injected current according to Eq. (6.72), which, in turn, reduces the inverter voltage. This calculation can be carried out iteratively until the change in injected current is within a small tolerance.

6.16.3 Analysis of three-phase short-circuit fault currents The positive-sequence equivalent circuit of Fig. 6.45B is shown in Fig. 6.46A. The grid is represented as a voltage source behind an equivalent impedance and the inverter is represented as a voltage-dependent positive-sequence current source.

546

Power Systems Modelling and Fault Analysis

(A)

+

=

+

=

F

=

+

Three-phase solid short circuit fault

+

(B)

F ( )

(C)

(D) +

+

F

( )

+

F ( )

Figure 6.46 Analysis of three-phase faults: (A) system representation, (B) voltage-source inverters do not inject currents, (C) voltage-source inverters inject positive-sequence reactive current and (D) grid fault current.

Inverters do not contribute a fault current The inverter is represented as an open circuit in the positive-sequence equivalent circuit as shown in Fig. 6.46B. The positive-sequence fault current contribution of the ac grid is given by 1 IFðgÞ 5

Vg1 Zg1

(6.74a)

and the asymmetric fault current is given by " !# pﬃﬃﬃ 1 2 2πft iF ðtÞ 5 2IFðgÞ 1 1 exp X=R g

(6.74b)

where X=R g is the positive-sequence X/R ratio calculated from the positivesequence equivalent impedance Zg1 of the ac grid, that is,

X=R

5 g

Xg1 R1 g

(6.74c)

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

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Inverters contribute a positive-sequence current As presented in Chapter 2, Symmetrical components of faulted three-phase networks containing voltage and current sources, the total short-circuit fault current at the 132 kV busbar is the superposition of the short-circuit current contributions of the voltage and current sources, that is, the inverter and grid contributions as shown in Fig. 6.46C and D.

ac grid voltage source is short-circuited The equivalent circuit is shown in Fig. 6.46C. The positive-sequence fault current supplied by the inverters with the ac grid voltage source short-circuited, that is, excluding the ac grid current contribution is given by 1 5 Ic1 IFðcÞ

(6.75a)

Inverter current source is open-circuited The equivalent circuit is shown in Fig. 6.46D. The positive-sequence fault current supplied by the ac grid with the inverter source current open-circuited, that is, excluding the inverter contributions, is given in Eq. (6.74a). The total positive-sequence fault current is given by 1 1 IF1ðcÞ IF1 5 IFðgÞ

(6.75b)

Assuming the short-circuit fault occurs at zero voltage on the voltage waveform, then the ac grid fault current will also include a dc current component that can be calculated from the X/R ratio ‘seen’ at the point of fault. The current injected by the inverter is a symmetrical ac current with no dc current component. Therefore the total asymmetric fault current is given by iF ðtÞ 5

pﬃﬃﬃ 1 2 2πft pﬃﬃﬃ 2IF ðgÞ 1 1 exp 1 2IF1ðcÞ fX=Rgg |ﬄﬄﬄ{zﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ac grid contribution

(6.75c)

Inverter contribution

where X=R g is as given in Eq. (6.74c). Combining the ac fault current components, we have " # i pﬃﬃﬃ pﬃﬃﬃh 1 2 2πft 1 1

iF ðtÞ 5 2 IF ðgÞ 1 IFðcÞ 1 2IFðgÞ exp X=R g

(6.75d)

1 1 5 0, IF1 5 IFðgÞ and For calculation of initial make fault current, we have IFðcÞ

" ( )# pﬃﬃﬃ 1 2 2πft

iF ðtÞ 5 2IFðgÞ 1 1 exp X=R g

(6.76a)

548

Power Systems Modelling and Fault Analysis

For the calculation of break fault current, we have " # i pﬃﬃﬃ pﬃﬃﬃh 1 2 2πft 1 1

iF ðtÞ 5 2 IF ðgÞ 1 IFðcÞ 1 2IFðgÞ exp X=R g

(6.76b)

For the next iteration in the calculation of the fault current, the required inverter terminal voltage is calculated using Fig. 6.46C as follows:

Vc1 5 Ic1 Zt1 1 ZT1

(6.76c)

6.16.4 Analysis of single-phase short-circuit fault currents In deriving the single-phase short-circuit fault current at the 132 kV busbar, we consider three practical cases and derive the total symmetrical and asymmetrical fault current for each case.

Inverters do not contribute a fault current The inverter is represented as an open-circuit current source in the positive-, negative- and zero-sequence circuits. The three sequence equivalent circuits and their interconnection are shown in Fig. 6.47.

= 0

+

F

( )

F

( )

F

( )

= 0

= 0

+

( )

Figure 6.47 Analysis of one-phase faults: voltage-source inverters do not inject currents.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

549

The ac component of the short-circuit fault current is calculated as follows: IF 5 3IF1 5 3IF2 5 3IF0 5

3Vg1 Zg1 1 Zg2 1

Zgo ZTo Zgo 1 ZTo

(6.77a)

and the asymmetric fault current is given by " !# pﬃﬃﬃ 2 2πft iF ðtÞ 5 2IF 1 1 exp X=R F

(6.77b)

The Thevenin’s fault point X=R F ratio is calculated from the series combination of the positive-, negative- and zero-sequence fault point equivalent impedances. From Fig. 6.47, we obtain

ZoZo Imaginary Zg1 1 Zg2 1 Z o g1 TZ o g T X=R F 5 Zgo ZTo 1 2 Real Zg 1 Zg 1 Z o 1 Z o

g

(6.77c)

T

Inverters contribute positive-sequence current only The inverter is represented as a positive-sequence current source in the positivesequence circuit but as an open-circuit in the negative- and zero-sequence circuits. The three sequence equivalent circuits and their interconnections are shown in Fig. 6.48A. The positive-sequence fault current can be calculated using the superposition theorem.

Inverter current source is open-circuited The resultant equivalent circuit is shown in Fig. 6.48B. The positive-sequence fault current supplied by the ac grid with inverter current source open-circuited, that is, excluding the solar PV converter contribution is given by 1 IFðgÞ 5

Vg1 Zg1 1 Zg2 1

Zgo ZTo Zgo 1 ZTo

(6.78a)

ac grid voltage source is short-circuited The resultant equivalent circuit is shown in Fig. 6.48C. The positive-sequence fault current supplied by the solar PV inverter with ac grid voltage source shortcircuited, that is, excluding the ac grid contribution is given by

550

Power Systems Modelling and Fault Analysis

(A)

+ ( )

F

( )

F

( )

F

( )

= 0

= 0

+

( )

(B)

(C) +

( )

( )

( )

F

F + ( )

+ ( )

( )

+

Figure 6.48 Analysis of one-phase faults: (A) voltage-source inverters inject positivesequence reactive current and (B and C) illustration of the application of the superposition theorem.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

Zg1 Ic1

1 IFðcÞ 5

Zg1 1 Zg2 1

551

(6.78b)

Zgo ZTo Zgo 1 ZTo

By superposition, the total positive-sequence fault current is given by 1 1 1 IFðcÞ IF1 5 IFðgÞ

IF1 5

(6.78c)

Vg1 1 Zg1 Ic1 Zg1 1 Zg2 1

(6.78d)

Zgo ZTo Zgo 1 ZTo

The single-phase fault current including ac grid and inverter contributions is given by

1 1 1 3IFðcÞ 5 IF 5 3IF1 5 3IFðgÞ

3 Vg1 1 Zg1 Ic1 Zg1 1 Zg2 1

Zgo ZTo Zgo 1 ZTo

(6.79a)

The Thevenin’s fault point X/R ratio is as given in Eq. (6.77c). The asymmetric single-phase fault current is given by iF ðtÞ 5

pﬃﬃﬃh 1 i 2 2πft pﬃﬃﬃh 1 i 2 3IFðgÞ 1 1 exp 1 2 3IFðcÞ fX=RgF |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} ac Grid Contribution

(6.79b)

Inverter Contribution

Or, combining the ac fault current components, " # i pﬃﬃﬃh i pﬃﬃﬃh 1 2 2πft 1 1

iF ðtÞ 5 2 3IFðgÞ 1 3IFðcÞ 1 2 3IFðgÞ exp X=R F

(6.79c)

For the next iteration in the calculation of the fault current, it can be shown that the inverter positive- and negative-sequence voltages, calculated using Fig. 6.48B and C, are given by 1

1

Vc1 5 Ic Zt1 1 ZT 1

Vc2 5

Zgo ZTo Zgo 1 ZTo ZoZo Zg1 1 Zg2 1 Z o g1 TZ o g T

2 Zg2 Zg1 1 Zg2 1

Zgo ZTo Zgo 1 ZTo

Zg2 1

Vg1 1 Zg1 Ic1

Vg1 1 Zg1 Ic1

(6.79d)

(6.79e)

552

Power Systems Modelling and Fault Analysis

Inverters contribute positive- and negative-sequence currents The inverter is represented as a positive-sequence current source in the positivesequence circuit, a negative-sequence current source in the negative-sequence circuit and an open-circuit in the zero-sequence circuit. The three sequence equivalent circuits and their interconnections are shown in Fig. 6.49A. Again, the positivesequence fault current can be calculated using the superposition theorem.

(A) + ( )

F

( )

( )

F

( )

+

= 0

(C) ( )

F

+

( )

F ( )

+

(B) +

( )

( )

F

+

( )

( )

( )

F

( )

= 0

( ) ( )

F

( )

Figure 6.49 Analysis of one-phase faults: (A) voltage-source inverters inject positive- and negative-sequence reactive currents and (B and C) illustration of the application of the superposition theorem.

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Inverter positive- and negative-sequence current sources are open-circuited The resultant equivalent circuit is as previously shown in Fig. 6.47. The positivesequence fault current supplied by the ac grid with both inverter current sources open-circuited, that is, excluding the inverter contributions, is given in Eq. (6.78a).

ac grid voltage source is short-circuited The resultant equivalent circuit contains two current sources as shown in Fig. 6.49B. Various methods can be used to solve for the required currents and voltages in large-scale power networks and these methods are presented in Chapter 9, Network equivalents and practical short-circuit current assessments in large-scale ac power systems. Here, we choose to use the straightforward superposition method. With the negative-sequence current source Ic2 open-circuited, the positivesequence fault current is given in Eq. (6.78b). With the positive-sequence current source Ic1 open-circuited, as shown in Fig. 6.49C, the positive-sequence fault current is given by 1 IFðc2Þ 5

Zg2 Ic2 Zg1 1 Zg2 1

(6.80a)

Zgo ZTo Zgo 1 ZTo

The total inverter current contribution is given by 1 1 1 IFðcÞ 5 IFðc1Þ 1 IFðc2Þ 5

Zg1 Ic1 1 Zg2 Ic2 Zg1 1 Zg2 1

Zgo ZTo Zgo 1 ZTo

(6.80b)

Finally, by superposition, the total positive-sequence fault current supplied by the ac grid and inverters is given by 1 1 1 IFðcÞ IF1 5 IFðgÞ

(6.81a)

Using Eqs. (6.79a) and (6.80b), we obtain IF1 5

Vg1 1 Zg1 Ic1 1 Zg2 Ic2 Zg1 1 Zg2 1

(6.81b)

Zgo ZTo Zgo 1 ZTo

The single-phase fault current supplied by the inverter and ac grid is given by

IF 5 3IF1 5

3 Vg1 1 Zg1 Ic1 1 Zg2 Ic2 Zg1 1 Zg2 1

Zgo ZTo Zgo 1 ZTo

The Thevenin’s fault point X/R ratio is as given in Eq. (6.78c).

(6.81c)

554

Power Systems Modelling and Fault Analysis

Where Zg1 5 Zg2 , Eq. (6.81c) becomes IF 5 3IF1 5

h

i 3 Vg1 1 Zg1 Ic1 1 Ic2 2Zg1 1

(6.81d)

Zgo ZTo Zgo 1 ZTo

The asymmetric single-phase fault current is given by pﬃﬃﬃh 1 i 2 2πft pﬃﬃﬃh 1 i iF ðtÞ 5 2 3IFðgÞ 1 1 exp 1 2 3IFðcÞ fX=RgF |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ ﬄ} |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} ac grid contribution

(6.82a)

Inverter contribution

or, combining the ac fault current components, " # i pﬃﬃﬃh i pﬃﬃﬃh 1 2 2πft 1 1

iF ðtÞ 5 2 3IFðgÞ 1 3IFðcÞ 1 2 3IFðgÞ exp X=R F

(6.82b)

For the next iteration in the calculation of the fault current, it can be shown that the inverter positive- and negative-sequence voltages, calculated using Figs. 6.48B, C and 6.49C are given by Vc1 5 Ic1

Zt1 1 ZT1

1

Zgo ZTo Zgo 1 ZTo ZoZo Zg1 1 Zg2 1 Z o g1 TZ o g T

Zg2 1

2 4V 1 1 Z 1 I 1 2 g g c

3

Zg1 Zg2 Zg1 1 Zg2 1

Zgo ZTo Zgo 1 ZTo

Ic2 5

(6.82c) 2

1

Vc2 5 Ic Zt1 1 ZT 2

"

Zg2 Zg1 1 Zg2 1

Zgo ZTo Zgo 1 ZTo

! # o o Z Z g T Vg1 1 Zg1 Ic1 2 Zg1 1 o I2 Zg 1 ZTo c (6.82d)

6.16.5 Analysis of two-phase short-circuit fault currents In deriving the two-phase short-circuit fault current at the 132 kV busbar, we consider two practical cases and derive the total symmetrical and asymmetrical fault current for each case.

Inverters contribute positive-sequence current only The inverter representation is as described in Section 6.16.3. The two sequence equivalent circuits and their interconnection are shown in Fig. 6.50. The positivesequence fault current can be calculated using the superposition theorem.

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555

(A)

F

F

+

+

F

= 0

(B)

(C) ( )

+

+

+

F

+

( )

F

Figure 6.50 Analysis of two-phase faults: (A) voltage-source inverters inject positivesequence reactive current and (B and C) illustration of the application of the superposition theorem.

Inverter current source is open-circuited The resultant equivalent circuit is shown in Fig. 6.50B. The positive-sequence fault current supplied by the ac grid with inverter current source open-circuited, that is, excluding the solar PV inverter contribution, is given by 1 IFðgÞ 5

Vg1 Zg1 1 Zg2

(6.83a)

ac grid voltage source is short-circuited The resultant equivalent circuit is shown in Fig. 6.50C. The positive-sequence fault current supplied by the solar PV inverter with ac grid voltage source shortcircuited, that is, excluding the ac grid contribution, is given by 1 IFðcÞ 5

Zg1 Ic1 Zg1 1 Zg2

(6.83b)

By superposition, the total positive-sequence fault current is given by 1 1 1 IFðcÞ IF1 5 IFðgÞ

IF1 5

Vg1 1 Zg1 Ic1 Zg1 1 Zg2

(6.83c)

(6.83d)

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Power Systems Modelling and Fault Analysis

The fault current flowing on phase B for a fault on phases Y and B including ac grid and inverter contributions is given by pﬃﬃﬃ 1 1 i j 3 V1 pﬃﬃﬃ 1 pﬃﬃﬃh 1 g 1 Zg Ic I F 5 j 3I F 5 j 3 I FðgÞ 1 I 1 FðcÞ 5 2 Z1 g 1 Zg

(6.83e)

or, where Zg1 5 Zg2 , ! pﬃﬃﬃ 3 Vg1 1 1 Ic IF 5 j 2 Zg1

(6.83f)

The Thevenin’s fault point X/R ratio is calculated with current sources opencircuited and voltage sources short-circuited. From Fig. 6.50A, we obtain

Imaginary Zg1 1 Zg2 X=R F 5 Real Zg1 1 Zg2

(6.83g)

The asymmetric fault current is given by iF ðtÞ 5

pﬃﬃﬃhpﬃﬃﬃ 1 i 2 2πft pﬃﬃﬃhpﬃﬃﬃ 1 i 2 3IFðgÞ 1 1 exp 1 2 3IFðcÞ fX=RgF |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ac grid contribution

(6.84a)

Inverter contribution

Or, combining the ac fault current components, " # i pﬃﬃﬃhpﬃﬃﬃ i pﬃﬃﬃpﬃﬃﬃh 1 2 2πft 1 1

iF ðtÞ 5 2 3 IFðgÞ 1 IFðcÞ 1 2 3IFðgÞ exp X=R F

(6.84b)

For the next iteration in the calculation of the fault current, it can be shown that the inverter positive- and negative-sequence voltages, calculated using Fig. 6.50B and C, are given by

Vc1 5 Ic1 Zt1 1 ZT1 1 Vc2 5

i Zg2 h 1 Vg 1 Zg1 Ic1 1 2 Zg 1 Zg

i Zg2 h 1 1 1 V 1 Z I g c Zg1 1 Zg2 g

(6.84c)

(6.84d)

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Inverters contribute positive- and negative-sequence currents The inverter representation is as described in Section 6.16.3. The two sequence equivalent circuits and their interconnection are shown in Fig. 6.51A. Again, the positive-sequence fault current can be calculated using the superposition theorem.

Inverter positive- and negative-sequence current sources are open-circuited The resultant equivalent circuit is shown in Fig. 6.51B. The positive-sequence fault current supplied by the ac grid with both inverter current sources open-circuited, that is, excluding the inverter contributions, is as given in Eq. (6.82a).

ac grid voltage source is short-circuited The equivalent circuit contains two current sources. We first open-circuit the negative-sequence current source Ic2 and calculate the positive-sequence fault current using Fig. 6.51C as follows: 1 IFðc1Þ 5

Zg1 Ic1 Zg1 1 Zg2

(6.85a)

Next, we open-circuit the positive-sequence current source Ic1 and calculate the positive-sequence fault current using Fig. 6.51D as follows 1 2 IFðc2Þ 5 2 IFðc3Þ 5

2 Zg2 Ic2 Zg1 1 Zg2

(6.85b)

By superposition, the positive-sequence fault current supplied by the inverters is given by 1 1 1 IFðcÞ 5 IFðc1Þ 1 IFðc2Þ 5

Zg1 Ic1 2 Zg2 Ic2 Zg1 1 Zg2

(6.85c)

Finally, by superposition, the total positive-sequence fault current is given by IF1 5

Vg1 1 Zg1 Ic1 2 Zg2 Ic2 Zg1 1 Zg2

(6.85d)

The two-phase fault current flowing on phase B for a fault on phases Y and B including ac grid and inverter contributions is given by i pﬃﬃﬃ Vg1 1 Zg1 Ic1 2 Zg2 Ic2 pﬃﬃﬃ 1 pﬃﬃﬃh 1 1 IF 5 j 3IF 5 j 3 IFðgÞ 1 IFðcÞ 5j 3 Zg1 1 Zg2

(6.85e)

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Power Systems Modelling and Fault Analysis

(A)

F

F

F

(B)

F

F

=0

=0

(C)

F

F

=0

(D)

F

F

=0

Figure 6.51 Analysis of two-phase faults: (A) voltage-source inverters inject positive- and negative-sequence reactive currents, (BD) illustration of the application of the superposition theorem.

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or, where Zg1 5 Zg2 , # pﬃﬃﬃ " pﬃﬃﬃ 1 j 3 Vg1 1

2 1 Ic 2 Ic I F 5 j 3I F 5 2 Zg1

(6.85f)

The Thevenin’s fault point X/R ratio is given in Eq. (6.83c). The asymmetric fault current is given by iF ðtÞ 5

pﬃﬃﬃhpﬃﬃﬃ 1 i 2 2πft pﬃﬃﬃhpﬃﬃﬃ 1 i 2 3IFðgÞ 1 1 exp 1 2 3IFðcÞ fX=RgF |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ ﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ac grid contribution

(6.86a)

Inverter contribution

or, combining the ac fault current components, " # i pﬃﬃﬃpﬃﬃﬃ pﬃﬃﬃpﬃﬃﬃh 1 2 2πft 1 1

iF ðtÞ 5 2 3 IFðgÞ 1 IFðcÞ 1 2 3IFðgÞ exp X=R F

(6.86b)

For the next iteration in the calculation of the fault current, it can be shown that the inverter positive- and negative-sequence voltages, calculated using Fig. 6.50B and C, are given by

Vc1 5 Ic1 Zt1 1 ZT1 1

i Zg2 h 1 1 1 2 V 1 Z I 1 I g c c Zg1 1 Zg2 g

(6.86c)

Vc2 5 Ic2 Zt1 1 ZT1 1

i Zg2 h 1 1 1 2 V 1 Z I 1 I g g c c Zg1 1 Zg2

(6.86d)

Similar steps can be followed to derive the expressions for the two-phase to earth short-fault current. However, we leave this to the interested researcher/reader.

6.16.6 Examples Example 6.11 Fig. 6.52 shows a typical grid code reactive current injection requirement curve for a currentcontrolled voltage-source inverter. The curve shows the implemented dynamic reactive current support response in the case of voltage dips at the inverter terminals. Discuss if the inverter can be represented as a voltage source behind a reactance, like a synchronous generator, as opposed to a current source representation. Referring to our discussion in Section 6.12, the grid code curve can be mathematically described as

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Power Systems Modelling and Fault Analysis

2

Synchronous generator curve with X” = 1/ k

I (pu)

1.6

1.1 pu Grid code curve

1.2 0.8

Vc

0.4 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

V (pu) Figure 6.52 To compare the representation of current-controlled voltage-source inverter and a synchronous generator, used in Example 6.11.

I5

Imax k ðV o 2 V Þ

V , Vc V $ Vc

(6.87a)

where V o is the inverter initial voltage in pu; V is the inverter voltage during the voltage dip in pu; k is the gain that defines the slope of the currentvoltage (IV) relationship; and Vc is the inverter terminal voltage level below which the inverter reaches and maintains a constant current equal to its transient limit Imax . Given the following inverter parameters: V o 5 1 pu, Imax 5 1:1 pu, k 5 2:2 and Vc 5 0:5 pu, Eq. (6.87a) becomes

I5

1:1 2:2ð1 2 V Þ

V , 0:5 V $ 0:5

(6.87b)

Eq. (6.87b) is shown as the grid code curve on Fig. 6.52. The k factor has a physical unit of a susceptance and its inverse is therefore a reactance. If the inverter is represented as a synchronous generator, that is, a voltage source behind a sub1 transient reactance equal to X 00 5 1k 5 2:2 5 0:4545 pu, then the following equation for a synchronous generator applies I5

1 ð1 2 V Þ X00

V ,1

(6.87c)

The synchronous generator curve is shown on Fig. 6.52. We conclude that the inverter can be represented as a voltage source behind a reactance equal to 1=k for V $ 0:5 pu, that is, for more remote faults. However, for V , 0:5 pu, that is, faults that are electrically closer to the inverter, a synchronous generator representation of a voltage source behind a reactance equal to 1=k results in an overestimate in the short-circuit reactive current infeed of the inverter. The closer the electrical distance from the inverter to the fault, the larger the overestimate.

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Example 6.12 Consider the ac grid network shown in Fig. 6.53A. The grid consists of current-controlled voltage-source inverters that are designed to inject positive-sequence reactive currents. There are no conventional synchronous generators in this grid. Discuss the fault current characteristics for the balanced three-phase fault shown at F. For simplicity, the load current is neglected. Fig. 6.53B shows the three-phase equivalent circuit with balanced three-phase current sources producing positive-sequence currents only. Using the superposition theorem and Fig. 6.53C, the fault current is given by 1 1 I1 F 5 IFð1Þ 1 IFð2Þ 5

2 jXc1 I1 2 jXc2 I1 1 1 1 2 jXc1 1 Z1 2 jXc2 1 Z2

In practical networks, the capacitive reactances of lines and cables dominate compared to the series inductive impedances of lines, cables and transformers. Hence 1 1 I1 F DI1 1 I2

Therefore the currents injected by the inverters minus the negligibly small network capacitive charging currents are fed to the fault. (A)

VSC1

Fault F

VSC2

(B)

(C)

Figure 6.53 Three-phase fault in a power system containing voltage-source inverters only, used in Example 6.12: (A) System, (B) illustration of fault in a three-phase circuit and (C) positive sequence equivalent circuit.

562

Power Systems Modelling and Fault Analysis

Example 6.13 Using the example ac grid network shown in Fig. 6.53A, discuss the fault current characteristics for an unbalanced two-phase fault at F. Consider two cases where the inverters are designed to inject only positive-sequence currents, or both positive- and negative-sequence currents. Case 1: Inverters inject balanced phase currents, that is, positive-sequence currents The three-phase system representation with a two-phase fault is shown in Fig. 6.54A and the inverters are represented by balanced three-phase current sources. The sequence network representation is shown in Fig. 6.54B. The expressions of the positive- (and negative-) sequence 2 fault current I1 F (and IF ) can be easily calculated but we leave this simple exercise for the interested reader. Again, with the dominant capacitive reactances, we can easily write 2 I1 F 5 2 IF D

1 1 1

I 1 I2 2 1

and the phase fault current is given by pﬃﬃﬃ IF 5 j 3I1 F Dj

pﬃﬃﬃ 3 1 1

I 1 I2 2 1

However, it is important to note that these positive- and negative-sequence fault currents and hence phase fault current can only flow in the shunt capacitances of the network as

(A)

(B)

Positive-sequence network

Negative-sequence network

Figure 6.54 Two-phase fault in a power system containing voltage-source inverters only, which inject only positive-sequence currents, used in Example 6.13: (A) Illustration of the fault in a three-phase circuit and (B) connection of positive and negative sequence networks.

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illustrated on Fig. 6.54B. If, theoretically, these capacitances are considered nonexistent, no shunt path would exist and hence no fault current could flow. Case 2: Inverters inject positive- and negative-sequence currents. The three-phase system representation of Fig. 6.55A shows three-phase inverter currents that are not balanced. The sequence network is shown in Fig. 6.55B. The expressions of the pos2 itive- (and negative-) sequence fault current I1 F (and IF ) can be easily calculated but, again, we leave this simple exercise to the interested reader. Again, with the dominant capacitive reactances, we can easily obtain 2 I1 F 5 2 IF D

1 1 1 1 2 2

I 1 I2 2 I1 1 I2 2 1 2

and the phase fault current is given by pﬃﬃﬃ pﬃﬃﬃ 1 3 1 1 2 2 I1 1 I2 2 I1 1 I2 IF 5 j 3IF Dj 2 However, as in Case 1, these positive- and negative-sequence fault currents and hence phase fault current can only flow in the shunt capacitances of the network as illustrated in Fig. 6.55B. If, theoretically, these capacitances are considered nonexistent, no shunt path would exist and hence no fault current could flow. In other words, the feeding of both positive- and

(A)

(B)

Positive-sequence network

Negative-sequence network

Figure 6.55 Two-phase fault in a power system containing voltage-source inverters only, which inject positive- and negative-sequence currents, used in Example 6.13: (A) Illustration of the fault in a three-phase circuit and (B) connection of positive and negative sequence networks.

564

Power Systems Modelling and Fault Analysis

(A)

(B)

Positive-sequence network

Negative-sequence network

Figure 6.56 Two-phase fault in a power system containing voltage-source inverters only which inject only positive-sequence currents and including induction motor loads, used in Example 6.13: (A) Illustration of the fault in a three-phase circuit and (B) connection of positive and negative sequence networks. negative-sequence currents by inverters in a system that consists of only inverters provides no additional benefits under two-phase fault conditions. In practical power system networks supplying mixed loads, the load mix includes directly connected induction motors mostly connected to the LV network and some connected to the MV network. As we presented in Chapter 5, Modelling of rotating ac synchronous and induction machines, these motors supply a positive-sequence current and present a low negative-phase sequence impedance under unbalanced faults. The motor’s initial positive-sequence and negative-sequence impedances are equal to the motor’s transient impedance. The positivesequence reactance increases with time whereas the negative-sequence reactance is constant. Therefore an equivalent impedance that represents the aggregate motors’ impedance in series with the intervening network impedances should be included in the positive- and negativesequence network representation. To illustrate, consider the network of Fig. 6.56A supplying a load at point F through stepdown transformers. The equivalent sequence network of Fig. 6.53B changes to that shown in Fig. 6.56B. This shows that the induction motor load will initially feed a small positive-sequence fault current and will present a shunt path in the negative-sequence circuit that allows the flow of some negative-sequence current. Overall, this effect is relatively small unless the induction motor composition within the load mix is very high over the whole system.

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Example 6.14 In Fig. 6.53A, replace the second inverter VSC2 with a conventional synchronous generator as shown in Fig. 6.57A. The generator is represented as a balanced three-phase voltage source. Discuss the fault current characteristics for the unbalanced two-phase fault shown at F. For simplicity, the load current is neglected but its effect is as discussed in Example 6.13. The three-phase circuit representation is shown in Fig. 6.57B and the sequence equivalent circuit is shown in Fig. 6.57C. The positive-sequence, negative-sequence and phase fault currents can easily be derived using the superposition theorem but we leave this to the interested reader. Fig. 6.57C shows that the negative-sequence circuit has a low impedance shunt path that corresponds to the generator negative-sequence impedance. In this circuit topology, the presence of the conventional voltage-source synchronous generator ensures that a large negative-sequence current and phase fault current flow even though the converter injects positive-sequence current only. In other words, in this example, we show that the feeding of negative-sequence current by the inverter provides no additional benefits.

(A)

VSC1

Two-phase fault F

Synchronous generator

(B)

(C)

Positive-sequence network

Negative-sequence network

Figure 6.57 Two-phase fault in a mixed power system of voltage-source inverters and synchronous generation, voltage-source inverters inject only positive-sequence currents, used in Example 6.14: (A) One-line diagram, (B) illustration of the fault in a three-phase circuit and (C) connection of positive and negative sequence networks.

566

Power Systems Modelling and Fault Analysis

Example 6.15 Consider the 100 MW solar PV power park shown in Fig. 6.45A. The data of this system is: Inverter: 2.2 MVA, 0.6 kV, reactor impedance 15% on 2.2 MVA and X/R 5 50, no. of inverters 5 50, plant rating 110 MVA, 100 MW, inverter transient current limit 1.1 pu. Inverter transformer: 0.6/33 kV, 2.2 MVA, impedance 10% on 2.2 MVA and X/R ratio 5 10. Plant transformer: 33/132 kV, 100 MVA, positive-sequence impedance 17.75% and X/R ratio 5 43, zerosequence impedance 16.5% and X/R ratio 5 30.7. 132 kV cable: Length 1.3 km, positive-sequence impedance (0.033 1 j0.1245) Ω/km, zero-sequence impedance (0.099 1 j0.06586) Ω/km. 132 kV ac grid fault infeeds: Source voltage 1 pu, three-phase fault current infeed: 20 kA, X/R 5 57; one-phase fault current infeed: 24 kA, X/R 5 48. The k factor of the inverters dynamic current injection is k 5 2. Calculate the short-circuit fault currents for a solid fault on the 132 kV busbar. The time of interest is 50 ms after fault inception and the inverters are assumed to have reached their steady-state reactive current magnitude in accordance with Eq. (6.75). The fault currents are to be calculated for threephase, two-phase and one-phase to earth faults. For the unbalanced faults, consider inverters that supply positive-sequence current only, and both positive- and negative-sequence currents with equal priority. The equivalent circuit of the 100 MW solar PV power plant of Fig. 6.45A is shown in Fig. 6.58. Solution We will use the equations developed in Sections 6.16.36.16.5 for the cases of three-phase fault, one-phase fault and two-phase fault, respectively. Base MVA and voltage are 100 MVA and 132 kV. 132 kV cable: positive-sequence and zero-sequence impedances are (0.0246 1 j0.0929)% and (0.0738 1 j0.0491)%. ac grid source equivalent: positive-sequence and negative-sequence impedance are (0.03836 1 j2.186)%, zero-sequence impedance is (0.03715 1 j1.093)%.

Fault 33 kV 132 kV Grid

110 MVA VSC 100 MW PV

Figure 6.58 Equivalent network used in Example 6.15.

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Three-phase fault ac grid current infeed 5 45:73e2j89 pu. Immediately upon fault occurrence, inverters ‘see’ a voltage change of 2100% and their steady-state current is 1:1e2j90 pu. The total ac fault cur rent is 46:83e2j89 pu or 20:48e2j89 kA. The inverter current of 1:1e2j90 pu raises the inverter positive-sequence terminal voltage to 0:3e2j2:8 pu and represent the final solution since the change in voltage remains greater than 0:55 pu. Two-phase fault The calculations of the short-circuit fault currents in the case of the inverters feeding positivesequence current only are left for the interested reader. The solution for the case where the inverters feed both positive- and negative-sequence currents of equal priority is given below. The positive-sequence fault current from the ac grid is 22:86e2j89 pu. The change in positive- and negative-sequence voltages at the fault point and at the inverter terminal immediately upon fault occurrence are 2 0:5 and 0:5 pu; respectively. With equal priority and k 5 2, the injected positive- and negative-sequence inverter currents are 2 j0:55 and j0:55 pu, respectively. The positive- (negative-) sequence fault current due to injected positive- (negative-) sequence inverter current is 2 j0:275 pu ðj0:275 puÞ and the total positive-sequence fault current from the inverters is 2 j0:55 pu. The total positive-sequence fault current from the ac grid and inver ters is 23:4e2j89 pu and the total phase B fault current is 40:55ej0:98 pu or 17:74ej0:98 kA. The injected positive-sequence inverter current of 2 j0:55 pu raises the inverter positivesequence terminal voltage from 0:5 to 0:65 pu. Conversely, the injected negative-sequence inverter current of j0:55 pu reduces the inverter negative-sequence terminal voltage from 0:5 to 0:35 pu. The inverter dynamic reactive current controller responds to these voltage changes and adjusts the injected positive- and negative-sequence currents. In this case, the full steady-state current magnitudes will still be supplied. The phase voltages at the inverter terminal are calculated as vr 5 1ej0 pu, vy 5 0:57e2j153:5 pu and vb 5 0:55ej152:4 pu. One-phase to earth fault We will provide the solution of the short-circuit fault currents for the case where inverters feed both positive- and negative-sequence currents. The positive-sequence fault current from the ac grid is 18:56e2j88:8 pu. The positive- and negative-sequence voltages at the inverter terminal immediately upon fault occurrence are calculated as 0:59e2j0:1 pu and 0:41e2j179:8 pu giving equal magnitude 2j179:8 changes of 0:41e . With k 5 2 and equal priority, the injected positive- and negativesequence currents are each equal to 2 j0:55 pu. The positive- and negative-sequence fault currents due to injected positive- and negative sequence inverter currents are each equal to 0:223e2j89:8 pu and the total positive-sequence 2j89:8 fault current from the inverters is 0:446e pu. The total positive-sequence current flowing into the fault from the ac grid and inverters is 19:0e2j88:8 pu and the total phase R fault current is 56:9e2j88:8 pu or 24:9e2j88:8 kA. The injected positive- and negative-sequence inverter currents of 2 j0:55 pu raise the inverter positive-sequence terminal voltage from 0:59 to 0:75 pu and reduce the inverter negative-sequence terminal voltage magnitude from 0:41 to 0:27 pu. The inverter dynamic reactive current controller responds to these voltage changes and adjusts the injected positive- and negative-sequence currents to 2 j0:56 and 2 j0:54 pu, respectively. The phase voltages at the inverter terminal immediately upon fault occurrence are vr 5 0:19e2j0:76 pu, vy 5 0:87e2j96:27 pu, vb 5 0:872ej96:25 pu and after inverter current injec tions vr 5 0:477e2j2:0 pu, vy 5 0:90e2j105:3 pu, vb 5 0:92ej105 pu.

568

6.17

Power Systems Modelling and Fault Analysis

Grid-forming voltage-source inverters

6.17.1 Background to grid-following and grid-forming inverters In Sections 6.10 and 6.11, we presented modelling and analysis of the currently dominant technology of grid-connected voltage-source inverters which are designed to operate as controlled, balanced or unbalanced three-phase current sources. These inverters are also known as grid-following since they follow the already-established voltage and frequency of the ac grid. In subsequent sections, we present the emerging technology of voltage-source inverters that are controlled in such a way that they operate as controlled, balanced, three-phase voltage sources behind an impedance during steady-state and transient conditions that involve relatively small over-currents. These inverters are also known as grid-forming inverters. Broadly speaking, grid-forming inverters can be controlled to mimic or replicate certain features of real synchronous generators and there are known as virtual synchronous machines (VSM or VISMA) inverters. Other grid-forming inverters employ droop control strategies. The latter are used in low-voltage microgrid applications and small island power systems. However, they can also be applied in large interconnected ac grids as an alternative to VSM inverters. Grid-forming inverters may have a flexible control system that enables seamless transition from grid-connected to islanded modes of operation. When operating as a balanced three-phase voltage-source, grid-forming inverters present their filter reactor impedance in the positive-sequence system and this same impedance as an inherent negative-sequence impedance in the negative-sequence system. This means that, just like real synchronous machines, the inverter supplies without delay inherent positive-sequence and negative-sequence currents under unbalanced network conditions. The latter reduces negative-sequence voltages and hence grid phase voltage unbalance.

6.17.2 Emerging challenges in power systems dominated by grid-following voltage-source inverters New challenges in the operation of power systems with high penetration of gridfollowing inverter-interfaced generators have recently emerged in a number of power systems and these issues are expected to appear in other power systems as the rate of penetration of current-controlled inverters increases. Some of the main challenges are presented next. Grid-following current-controlled voltage-source inverters do not contribute to the overall power system inertia. Only rotating machines make this contribution through the stored kinetic energy in their rotating masses. Reduced system inertia causes an increase in the initial rate of change of system frequency following sudden tripping of generation power infeed or load. This introduces a risk of further and cascade generation trippings that could lead to widespread customer demand disconnections. Grid-following inverters cannot supply an instantaneous or half cycle short-circuit current like rotating machines that have a stored electromagnetic energy. These

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569

inverters supply almost zero current at half cycle after the occurrence of a voltage dip or short-circuit fault on the ac grid due to the need to measure, process and detect the voltage dip, switch to reactive current injection mode then deliver the response of the inner current control system as discussed in Sections 6.11, 6.13 and 6.16.2. The delay in the supply of reactive current in the first 1520 ms in inverter dominated power systems might slightly prolong the operation time of some types of protection relays. However, the practical significance of a short delay of one to two cycles will vary among power systems. The effect will be negligible in modern power systems that already employ very fast circuit breakers with opening times of around one cycle compared to three- or four-cycle circuit breakers. Another modern practice is the removal of trip relays and hence the reduction of overall protection time of 1020 ms. At the time of writing, further research is required in this area. Both grid-following and grid-forming inverters make a small and limited contribution to short-circuit current levels on the power system. Their maximum steadystate short-circuit current contribution of typically 1.11.2 pu of rated current can be contrasted with 2.23.5 pu supplied by synchronous machines for faults on the higher voltage side of their generator-transformers. Grid-following inverters do not allow operation in very weak parts of networks because the extent of dq cross-coupling discussed in Section 6.10 becomes amplified besides the issues caused by network voltage unbalance and harmonic voltage distortion. Also, operation of such inverters alone, that is, without synchronous generation, in islanded networks is not possible. There are other technical challenges in the operation of power systems and particularly small- to medium-sized systems that result from high penetrations of gridfollowing inverter-interfaced systems. These include voltage stability, synchronising torques, supersynchronous control interactions, increased phase voltage unbalance and harmonic amplifications. However, these topics are outside the scope of this book. We will therefore restrict our attention to the short-circuit current contribution of inverter-interfaced generators.

6.17.3 Control structure of grid-forming inverters At the time of writing, the application of grid-forming VSM inverters and inverters based on droop control strategies in large utility-grid is still emerging although the latter are now being offered by some manufacturers. However, inverters based on droop control strategies have already been implemented in microgrids and small island power systems that comprise battery energy storage, wind and/or solar PV generators and/or conventional diesel generators. Importantly, and as discussed in Section 6.17.1, currently, there is no consensus on the preferred grid-forming inverter approach, VSM or droop-control based, in large-scale power systems. Moreover, for VSM inverters, there is no consensus on a unified set of features of a real synchronous machine a VSM inverter should emulate. The literature shows varying complexity of synchronous machine models used. Accordingly, there are significant differences in the implementation of control system structures and designs of VSM inverters.

570

Power Systems Modelling and Fault Analysis

Solar PV

Battery ES Wind

dc/dc

dc

dc/dc VSC ac/dc

Grid Gate signals

HVDC

Grid-forming algorithm and control

Fuel cell, etc.

Figure 6.59 Conceptual grid-forming voltage-source inverter structure.

Some research proposes a control system that essentially provides a delayed ‘synthetic’ inertia rather than the true instantaneous inertial response of a real synchronous machine. Some implementations use an inner current-control loop requiring current references while others do not. Some use a PLL in various forms for deriving the grid voltage angle and/or frequency and some have dispensed with the PLL apart from initial synchronisation to the grid, like a real synchronous machine. Some research dispenses completely with the PLL and proposes that initial synchronisation to the grid may be done by the inverter control system. At the time of writing, manufacturers and suppliers of grid-connected voltagesource inverters are beginning to embrace the utilities requirement for gridconnected, grid-forming inverters in large-scale power systems. As with other similar historical developments, it is inevitable that the preferred functional performance, structure and design of grid-forming inverters will emerge over the next few years. Grid-forming inverters can interface a range of technologies to the ac grid such as type 4 wind turbine generators, solar PV generators, battery energy storage, fuel cells, certain distributed energy resources, HVDC links, etc. Fig. 6.59 illustrates the conceptual structure of a grid-forming inverter and various potential applications.

6.18

Grid-forming virtual synchronous machine (VSM) inverters

6.18.1 Possible VSM inverter features of real synchronous machines The dynamic behaviour of a real synchronous machine is exhibited through the dependence of grid frequency on machine rotor speed, the machine having a

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571

rotating mass and stored kinetic energy, delivering real inertial response without delay, controlling active power/frequency and reactive power/voltage, providing synchronising and power oscillation damping torques, and enabling seamless parallel operation of many machines in a grid of any size. Under grid voltage dips, a real synchronous machine feeds a short-circuit current to the fault that is dependent on the external impedance to the fault point immediately and without any delay. Like a real synchronous machine, a grid-forming VSM inverter should have a virtual rotor with mass and damping and exhibits an inertia constant whose value can be set by the user. Such a VSM inverter should also feed without delay positive- and negative-sequence short-circuit currents under network voltage dips. The VSM should have an inherent negative-sequence impedance that allows the flow of negative-sequence current from the network. Moreover, and like a real synchronous machine, the grid-forming VSM inverter should contribute to the damping of harmonic voltage distortion on the network by presenting an inherent frequencydependent harmonic impedance. Therefore as seen from the grid, and apart from high-frequency harmonics in the IGBT bridge output voltage, the electrical behaviour of a grid-forming VSM inverter should replicate some of the essential characteristics of a real electromechanical synchronous machine. To achieve this objective, the control strategy of a grid-forming VSM inverter should control the amplitude, frequency and phase angle of the inverter output voltage. It may also be desirable to be able to synchronise and operate without the need for a PLL. The PLL can produce erroneous signals under large three-phase grid voltage dips at the inverter terminal and it can go unstable in weak power grids with a high grid impedance. Consequently, the removal of a PLL from a gridforming inverter may be desirable.

6.18.2 A model of grid-forming VSM inverters The common feature found in almost every grid-forming VSM inverter implementation is the representation of a real synchronous machine rotor swing dynamics and hence inertial response and sometimes electrical damping. The delivery of a predefined inertial response of typically H 5 2 MWs/MVA to H 5 8 MWs/MVA requires a suitable and sufficient amount of energy storage on the inverter’s dc busbar which may take the form of ‘power’ rather than ‘energy’ batteries or ultracapacitors that can store the required amount of energy over the delivery timescale required which may be in the order of 1025 s in practical power systems. In this edition of our book, we have chosen a grid-forming VSM inverter control structure that is based on a third-order dynamic model of a synchronous machine and hence the following subset of functionalities: 1. a virtual governor-turbine model; 2. a virtual rotor model with inertia and damping; and 3. a virtual field or internal voltage model.

This grid-forming VSM inverter control structure is shown in Fig. 6.60.

572

Power Systems Modelling and Fault Analysis

PWM switching

Park transformation Virtual field voltage model

Inverse park

PQ measurements Virtual rotor model

Virtual governorturbine model

Figure 6.60 Possible virtual synchronous machine (VSM) inverter control structure.

+

+

−

− − −

+

Virtual rotor

Virtual rotor Virtual governorturbine

+

Virtual field voltage

Virtual governorturbine

Virtual field voltage

Figure 6.61 Representation of a three-phase grid-forming virtual synchronous machine (VSM) inverter.

Using Fig. 6.28 of a three-phase voltage-source inverter, a three-phase gridforming VSM inverter may be represented as shown in Fig. 6.61A and B, where the three-phase IGBT bridge is represented by a balanced three-phase voltage source behind a fixed impedance equal to the inverter actual reactor impedance.

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VSM inverter’s virtual governor-turbine model The virtual governor-turbine model of the VSM inverter can be arbitrarily chosen to represent the dynamics of a heavy-duty gas turbine but without any complexities such as ambient temperature dependence, etc. The virtual governor-turbine model uses a steady-state speed or frequency droop slope Df and low-pass filters representing a governor with a time constant Tg and a gas turbine with a time constant Tt . The dynamic equations can be written as ref

ref

Pm 5 Pe 1

1 ref f 2f Df

(6.88a)

Tg

dP v ref 5 Pm 2 Pv dt

(6.88b)

Tt

dP m 5 Pv 2 Pm dt

(6.88c)

or using the Laplace operator S 5 d=dt, we have Pv 5

1 ref P 1 1 STg m

(6.89a)

Pm 5

1 Pv 1 1 STt

(6.89b) ref

where f is virtual frequency of the VSM inverter in pu; f is frequency set point in pu; Df is frequency droop slope in pu frequency deviation per pu power deviaref tion or in physical units of Hz/MW; P is electric power set point in pu; andP m is virtual mechanical power of VSM inverter in pu. Fig. 6.62A shows the transfer function block diagram of the virtual governorturbine of the VSM inverter.

VSM inverter’s virtual rotor model with inertia and damping The virtual rotor model of the VSM inverter is based on the electromechanical swing equation of a real synchronous generator and, with electrical damping, is given by

dΔω 1 5 P m 2 P e 2 kdm Δω dt 2H

(6.90a)

dδ 5 ωs Δω dt

(6.90b)

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Power Systems Modelling and Fault Analysis

(A) +

1

Σ

−

+

1

1

Σ +

̅ (B)

+

−

1

Σ

+

1

Σ

+

−

+

1

+

Σ

(C) − +

Σ

+ −

Σ

̅

1

̅

1

Figure 6.62 Possible virtual synchronous machine inverter controllers: (A) virtual governorturbine model, (B) virtual rotor model with electrical damping and (C) virtual AVR field voltage model.

or, using the Laplace operator S 5 dtd , Eq. (6.90) can be written as Δω 5 δ5

1 P m 2 P e 2 kdm Δω 2HS

ωs Δω S

(6.91a) (6.91b)

and δr 5 δ 1 kdd ωs Δω

(6.91c)

where t is time in seconds; ω is VSM virtual rotor angular speed in pu; H is VSM virtual inertia constant in MWs/MVA; P m is VSM virtual mechanical power in pu; P e is VSM measured electric power in pu; kdm is mechanical damping factor term in pu power per pu speed deviation; kdd is electrical damping factor term in pu angle per pu speed deviation; δ is VSM virtual rotor angle in electrical radian; and ωs 5 2πfs is rated angular speed in electrical rad/s. The beneficial damping of the physical damper windings of a real synchronous machine is replicated by the addition of a feed-forward term that superimposes a virtual rotor angle change proportional to rotor speed deviation through a virtual damping factor kdd in accordance with Eq. (6.91c). This factor can be set to a high value to achieve a targeted level of high damping that may even dispense with the need for additional power system stabilisers that are used as part of the excitation control systems of real synchronous machines.

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575

Eq. (6.91) can be represented as a transfer function block diagram as shown in Fig. 6.62B.

VSM inverter’s virtual field or internal voltage control model The virtual field or internal voltage regulator model of the VSM inverter is chosen to represent a high static gain or low-voltage droop, a fast-acting static exciter and the field winding of a real synchronous machine. In this model, the measured reactive power output is multiplied by the voltage droop slope Dv then subtracted from the voltage error which is the voltage reference minus the measured feedback voltage. The error signal is then taken directly to the exciter with a low-pass filter with a time constant Te then to the virtual field winding with a time constant Tf . The dynamic equations can be written as e ref rotor 5 V

ref

2 V 2 Dv Q

(6.92a)

Te

de x 5 e ref rotor 2 e x dt

(6.92b)

Tf

de rotor 5 e x 2 e rotor dt

(6.92c)

or using the Laplace operator S 5 d=dt, we have ex 5

1 e ref 1 1 STe rotor

(6.93a)

1 ex 1 1 STf

(6.93b)

e rotor 5

ref

where V is terminal voltage of VSM inverter in pu; V is voltage set point in pu; Dv is voltage droop slope in pu voltage deviation per pu reactive power deviation; Te is virtual VSM exciter time constant in seconds; and Tf is virtual VSM field time constant in seconds. Fig. 6.62C shows the transfer function block diagram of the virtual field voltage of the VSM inverter.

6.19

Grid-forming voltage-source inverters using droop control

As discussed in Section 6.17, grid-forming inverters employing a control strategy based on frequency and voltage droops or slopes schemes are an alternative to gridforming VSM inverters. The potential use of these inverters with sufficient dc

576

Power Systems Modelling and Fault Analysis

Grid-forming inverter

A

Figure 6.63 Power flow from a grid-forming inverter through an inductive impedance.

energy storage in large utility-scale wind and solar PV power generation systems presents an attractive application on technical grounds and perhaps economic grounds in the not too distant future. Since the inverter lacks physical inertia, the basis of droop control can be derived from the general relationship among active power, frequency, voltage and reactive power with powers flowing through an inductive impedance as shown in Fig. 6.63. Using phasor notation, the grid-forming inverter is represented as a voltage source E 5 Eejδ and supplies an apparent power S 5 P 1 jQ 5 VI to the ac bus through a complex inductive impedance Z 5 R 1 jX 5 Zejϕ . Using I 5 ðE 2 VÞ=Z and V 5 V+0 degrees, we can write after a little algebra

V R P 5 2 Q Z X

X 2R

Z 2 5 R2 1 X 2

Ecosδ 2 V Esinδ

(6.94a) (6.94b)

Eq. (6.94a) shows that P and Q are coupled through R and X if neither is negligibly small. This means that both active and reactive powers are dependent on both voltage difference and power angle. This is the case in low-voltage microgrids where the network resistance can be larger than the reactance but both are significant. There are several droop control approaches proposed in the literature to resolve the power coupling challenges in microgrids that discuss conventional droop, reverse/opposite droop, indirect operation, virtual impedance and adaptive droop. However, such a discussion is outside the scope of this book. In our case, we focus on the potential application of droop control on inverters for use in higher voltage grids that are characterised by high X=R ratios so that the resistance R can be neglected. Substituting R 5 0 in Eq. (6.94), we obtain the following decoupled equations P5

EV sinδ X

(6.95a)

Q5

V ðEcosδ 2 V Þ X

(6.95b)

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and for a small power angle, sinδDδ and cosδD1, we can write δD

X P EV

E 2 VD

X Q V

(6.96a)

(6.96b)

Eq. (6.96b) shows that by regulating reactive power flow Q, the voltage difference and hence ac bus voltage V can be controlled. Also, Eq. (6.96a) shows that by regulating active power flow P, the power angle δ can be controlled. Droop control mimics speed governor control on conventional synchronous power plant which provides speed or frequency regulation where frequency is ‘drooped’ against active power. At the same time, the voltage V is drooped against the measured reactive power Q. Using Eq. (6.96), the conventional f =P and V=Q droop equations for grid-forming inverters are given by

f 5 f ref 1 Df P ref 2 P 0 P0 5 δ5

1 P 1 1 STt

fs f S

V 5 V ref 1 Dv Q ref 2 Q 0 Q0 5

1 Q 1 1 STex

(6.97a) (6.97b)

(6.97c) (6.98a) (6.98b)

From Eqs. (6.97) and (6.98), we obtain e ref i ðtÞ 5 V sinð2πft 1 δ 1 αi Þ

(6.99)

where f ref , fs and f are set point, nominal and measured frequencies, respectively; P ref and P are set point and measured powers, respectively; Q ref and Q are set point and measured reactive powers, respectively;Df and Dv are frequency and voltage droop slopes as defined previously; and e ref is reference internal voltage of the i inverter generated by a voltage-controlled oscillator using V , f and δ, and αi 5 0; 232π and 2π 3 for phase r, y and b respectively.

Δf =fs ΔV=Vn Df 5 2ΔP=P where Pr is rated power, Dv 5 2ΔQ=Q where Vn is nominal voltage r r and Qr is rated reactive power. In large grids, typical droop slopes are 3%4% for Df and 2%4% for Dv . For example, in the context of droop controlled inverters,

578

Power Systems Modelling and Fault Analysis

Df 5 3%, means that a 100% change in power P causes a 3% change in frequency f , and Dv 5 4% means that a 100% change in reactive power Q causes a 4% change in voltage V. Eqs. (6.97) and (6.98a) are illustrated graphically in Fig. 6.64A and B, respectively. A negative Pr is used to illustrate a battery source for the grid-forming inverter. Similar to a VSM inverter, Tt and Tex time constants in Eqs. (6.97b) and (6.98b) can be selected to represent the mechanical time constants of a steam or gas turbine, and a modern fast-acting static exciter for a synchronous generator, respectively. Fig. 6.65 shows a simple control structure of a grid-forming inverter employing droop controls and the necessary over-current limitation strategy required similar to a VSM grid-forming inverter.

(A)

(B)

0

0

Figure 6.64 Illustration of droops of a droop-controlled grid-forming inverter: (A) f =P droop and (B) V=Q droop.

PWM switching

Control and fault current limitation

+

+

Σ + Σ + +

Σ

+

1 −

Σ −

1

Voltage/reactive power droop control

Figure 6.65 Possible grid-forming inverter droop controllers.

Instantaneous PQ measurements

Frequency/power droop control

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With droop controls, a voltage-source inverter becomes grid-forming, acts as a voltage source and provides inherent inertia, frequency and voltage control. Moreover, and like conventional synchronous generators, the use of similar droops enables parallel inverters to share their active and reactive power outputs in proportion to their ratings. This droop control is simple and reliable and requires no communication among inverters, and, arguably, simpler than the control structure of a VSM inverter. The supervisory controller of each inverter provides settings for the reference frequency, reference voltage and slopes.

6.20

Natural short-circuit current contribution of gridforming inverters

Since the inverter is controlled to behave as a true three-phase balanced voltage source, the fault current output under voltage dips on the ac grid is instantaneous and its phase angle with respect to the inverter initial voltage is determined by the network characteristics, namely the external inverter impedance to the fault point. This characteristic is inherently highly inductive in grid-connected inverters although it may be more resistive in low-voltage microgrids as well as marine and aeronautical power systems. The inverter can inherently deliver the desirable instantaneous short-circuit current during the fault detection period without the delay necessitated by measurements and controls required by grid-following inverters.

6.20.1 Natural three-phase short-circuit current of grid-forming inverters Fig. 6.66A shows a grid-forming inverter represented as a balanced three-phase voltage source behind a fixed impedance equal to its output filter impedance R 1 jωs L. The natural short-circuit current contribution of the inverter is the potential current that would be supplied without any current limitation. Earth faults are not considered due to the absence of a zero-phase sequence path. (A)

(B) −

+

−

+

−

+

−

+

−

+

−

+

Figure 6.66 Three-phase faults at grid-forming inverter (A) terminals and (B) through an external grid impedance.

580

Power Systems Modelling and Fault Analysis

The fundamental frequency component of the balanced three-phase voltage source of the inverter is given by pﬃﬃﬃ (6.100) ei ðtÞ 5 2Esinðωs t 1 δi Þ i 5 r; y; b where E is internal rms voltage of inverter; ωs 5 2πfs in rad/s, fs is power frequency in Hz; and δi is internal voltage phase angle of inverter in radian. δy 5 δr 2 2π 3 and δb 5 δ r 1 2π=3 For a solid three-phase short-circuit fault at the inverter terminals, the natural short-circuit current is given as 2 0 13 9 8 > > > > ω L > > > sin4ωs t 1 δi 2 tan21 @ s A5 > > > > > pﬃﬃﬃ > > R = <

2E (6.101) ii t 5 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 0 13 L > > 2t= R > R 2 1 ð ω s LÞ 2 > > > ω s L A5 > > > > > > 2 sin4δi 2 tan21 @ e > > ; : R The natural fault current consists of a steady-state ac current component and a transient dc current component that decays exponentially with a dc time constant equal to L=R. In the case of a fault through an external grid impedance Re 1 jωs Le , shown in Fig. 6.66B, the fault current can be calculated using Eq. (6.101) by replacing R with ðR 1 Re Þ and L with ðL 1 Le Þ.

6.20.2 Natural two-phase short-circuit current of grid-forming inverters This case is shown in Fig. 6.67A. For a solid two-phase clear-of-earth short-circuit fault that occurs on phases y and b, we can write ey ðtÞ 2 eb ðtÞ 5 2Riy ðtÞ 1 2L (A)

diy ðtÞ dt

(6.102a) (B)

−

+

−

+

−

+

−

+

−

+

−

+

Figure 6.67 Two-phase faults at grid-forming inverter (A) terminals and (B) through an external grid impedance.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

581

and, using Eq. (6.100), we have ey ðt Þ 2 eb ðt Þ 5 2

pﬃﬃﬃpﬃﬃﬃ 3 2Ecosðωs t 1 δr Þ

(6.102b)

Substituting Eq. (6.102b) in Eq. (6.102a), the current solution is given by 2 0 13 9 8 > > > > ω L > 4ωs t 1 δr 2 tan21 @ s A5 > > > 2 cos > > > > pﬃﬃﬃ pﬃﬃﬃ > > R = < 3 2E qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ iy ðt Þ 5 2 0 13 L > > 2 2t= R > R 2 1 ð ω s LÞ 2 > > > > > 21 @ω s LA5 > > 4 > > 1 cos δr 2 tan e > > ; : R ib ðtÞ 5 2 iy ðtÞ

(6.103a)

(6.103b)

As in the case of a three-phase short-circuit fault, the fault current consists of a steady-state ac current component and a transient dc current component that decays exponentially with a dc time constant equal to L=R. In the case of a fault through an external grid impedance Re 1 jωs Le , shown in Fig. 6.67B, the fault current can be calculated using Eq. (6.103) by replacing R with ðR 1 Re Þ and L with ðL 1 Le Þ.

6.21

Over-current limitation strategies of grid-forming inverters

6.21.1 General For typical utility-scale grid-forming inverters, the inverter total output filter impedance is typically 0.050.15 pu, giving an unlimited ac rms fault current of 6.720 pu for a solid three-phase fault at the inverter output terminals. The combined impedance of the inverter output filter and inverter medium-voltage transformer is typically 0.130.25 pu, giving an unlimited ac rms fault current on the medium-voltage busbar of 47.7 pu. For an effective dc time constant (L 1 Le )/ðR 1 Re Þ of around 45 ms and maximum dc current offset, the prospective first peak instantaneous current can reach 7.213.9 pu. However, as discussed in Section 6.17.2, if allowed to flow, these current magnitudes would destroy the inverter unless the current is very quickly limited to the inverter transient current limit of typically 1.11.2 pu. The instantaneous fault current could exceed 1.5 pu even for remote grid faults that result in a voltage dip at the inverter terminals of more than 10%14%. Therefore a fault, or more generally, overcurrent limiting strategy is an essential prerequisite for the application grid-forming inverters. The inverter output current limiting strategy needs to operate effectively and reliably under all types of balanced, unbalanced, close-up and remote short-circuit

582

Power Systems Modelling and Fault Analysis

faults, as well as large phase angle jumps perhaps up to 60 degrees. The phase angle of the limited current should not be changed. The current limitation strategy should act to limit the instantaneous asymmetric current which may include a different magnitude of dc current component depending on the fault instant on the voltage waveform. The dc current magnitude could be zero for a fault close to voltage peak or maximum for a fault close to zero voltage. Different dc fault current magnitudes can occur in the case of multiphase faults. These objectives mean that the action to limit the converter over-current magnitude may have to be completed within typically a couple of milliseconds after the onset of the dip/fault. Integrated gate commutated thyristor (IGCT) switches and injection enhanced gate thyristor (IEGT) switches are used in medium-voltage high-power converters. IEGT switches can handle short-circuit fault current and enable the converter to deliver up to two or three times rated current for a short period of time. IGCT switches, however, cannot handle a fault current more than around 1.5 pu but this would still offer practical benefits. One application of IGCT- and IEGT-based highpower converters is in doubly fed induction machines and full-size converterinterfaced machines used in variable-speed pumped storage hydropower stations.

6.21.2 Current limitation in inverters using single voltage controller structure Several strategies for preventing inverter over-current are described below.

Strategy 1: Clipping PWM voltage reference This strategy can be used with either VSM or droop-controlled grid-forming inverters where the magnitude of the internal voltage is controlled or clipped in the time domain if the inverter current in any phase exceeds the maximum instantaneous current limit of the inverter. The strategy is illustrated in the block diagram in Fig. 6.68. This is a simple and effective method but the clipped current waveform is distorted and contains odd harmonic orders of decreasing amplitude, for example, third, fifth and seventh harmonics, as well as some high-frequency small-amplitude oscillations around the clipped current mean. The transient harmonic currents exist

Figure 6.68 Grid-forming inverter over-current limitation strategy 1: clipping PWM voltage reference.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

=

Inverse park

+

583

+

Figure 6.69 Grid-forming inverter over-current limitation strategy 2: limiting internal voltage in dq reference frame then clipping.

during the clipping period only, which should generally occur for a few cycles only until the current drops below the threshold. The effects of the current distortion on the operation of protective current transformers and protection relays need careful assessment.

Strategy 2: Limiting internal voltage in dq reference frame This strategy is based on the VSM inverter control structure shown in Fig. 6.60. The strategy is illustrated in the block diagram in Fig. 6.69. The magnitude of the inverter internal voltage is limited or reduced immediately upon current build-up to a maximum value given by edq

Limited

5 vdq Unlimited 1 ωs LImax

(6.104)

where edq Limited is the limited internal positive-sequence voltage of inverter; vdq is the unlimited positive-sequence output voltage of inverter; Imax is the Unlimited maximum allowed positive-sequence current of inverter; and X 5 ωs L is the reactance of the inverter filter reactor. In effect, this approach limits the difference between the inverter internal voltage and the inverter terminal voltage, that is, the voltage drop across the filter reactor to a maximum value given by the maximum inverter current limit and the reactance of the filter reactor which is a constant given by ωs LImax . However, limiting edq in accordance with Eq. (6.104) is not sufficient if the current contains dc current component. Additional limitation by clipping the PWM voltage reference of the inverter eryb in the ryb reference frame is required which, as in strategy 1, causes some harmonic distortion in the current waveform.

Strategy 3: Static virtual resistor Again, this strategy is based on the VSM structure shown in Fig. 6.60 and is illustrated in the block diagram in Fig. 6.70. It is based on the principle of adding a static virtual resistor in series with the inverter filter resistor in order to reduce or limit the internal voltage of the VSM inverter to a value given by

584

Power Systems Modelling and Fault Analysis

Inverse park

Figure 6.70 Grid-forming inverter over-current limitation strategy 3 with static virtual resistance.

R edq Limited 5 edq Unlimited 2 vr 0

0 i Rvr dq

(6.105)

where edq Limited is the limited internal positive-sequence voltage of inverter; edq Unlimited is the unlimited internal positive-sequence voltage of inverter; idq is the positive-sequence output current of inverter; and Rvr is virtual resistor in series with inverter filter resistor. The addition of a virtual series resistor to the output filter of the inverter maintains and replicates the inherent fault current waveform of a real synchronous machine. However, a sufficiently large value of the virtual resistor Rvr , for example, 2040 times the resistance of the output filter may be required. This should also reduce the dc time constant and hence the dc fault current component as well as the ac component of the fault current but the latter may depend on the proximity of the fault location to the inverter terminals and the X=R ratio of the inverter external circuit. Being virtual, no physical active power losses occur in Rvr , so the greater the value of the virtual resistor Rvr , the greater the reduction in fault current magnitude. However, a disadvantage is that the static resistor Rvr is always present in the operation and control of the inverter. High values in weak ac grids have to be avoided to avoid inverter angle instability that may be caused by increased coupling between active and reactive power and hence reduced power swing damping. Therefore if used on its own, the magnitude of fault current reduction achievable may not be sufficient and additional reduction may be required.

Strategy 4: Transient virtual resistor Like strategy 1, this strategy can be applied to VSM and droop-controlled inverter control structures and is illustrated in the block diagram in Fig. 6.71A. However, the resistor added in series with the inverter filter resistor is transient and hence nonlinear. By definition, the virtual resistance should be zero for inverter currents up to the rated current then a resistance value that increases with inverter current magnitude up to inverter current limit may be effective. This strategy is particularly useful if the inverter is slightly oversized so that its current limit is greater than its

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

(A)

0

π −

585

<

+

(B)

(C)

Figure 6.71 Grid-forming inverter over-current limitation strategy 4 with transient virtual resistance: (A) possible implementation, (B) virtual resistor/current relationship and (C) equivalent circuit for maximum virtual resistor.

rated current. An example would be an inverter having a rated current of 1 pu and a maximum rms current limit of 1.2 pu or better. An example of a nonlinear behaviour of the transient resistor is illustrated in Fig. 6.71B. Mathematically, we can write

Rvr ðjI jÞ 5

0 mjI j

jI j # Irated Irated , jI j # Ilimit

(6.106a)

In the limiting range, the voltage across the virtual resistor is given by Vvr 5 jI jRvr ðjI jÞ or Vvr 5 mjI j2

(6.106b)

It is possible to calculate the maximum value of the virtual resistor at the maximum inverter current limit Ilimit . Assuming a three-phase fault at the inverter terminals, as illustrated in Fig. 6.71C, the maximum resistance of the virtual resistor is given by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðV 1 Þ2 RvrðmaxÞ 5 2 ð ω s LÞ 2 2 R jIlimit j2

(6.107)

For example, for a typical large inverter with filter impedance ωs L 5 0:1 pu and ðωs LÞ=R 5 40, an inverter positive-sequence voltage of 1 pu and a current limit jIlimit j 5 1:2 pu, RvrðmaxÞ 5 0:825 pu.

586

Power Systems Modelling and Fault Analysis

The transient virtual resistor method can also be applied in the cascaded VSM inverter control structure discussed in Section 6.21.3.

Strategy 5: Combination of strategy 1 and strategy 4 In this strategy, inverter current clipping is implemented for the first few cycles of over-current, only then is a transient virtual resistor introduced for the remaining duration of the over-current for as long as the over-current condition remains.

6.21.3 Current limitation strategies in inverters using cascaded control structure These strategies are based on the use of a VSM or droop-controlled grid-forming inverter that uses a cascaded voltagecurrent control structure as illustrated in Fig. 6.72 for a VSM inverter. In this structure, the voltage control loop produces current references to a second current control loop giving an inherent benefit where current limitation can be applied directly to the input current references of the current controller. However, generally, tuning of the cascaded controllers in such a structure to ensure stability under all operating conditions is not straightforward. Two current limiting strategies are considered in the cascaded voltagecurrent control structure.

Strategy 1: Saturation of individual current references In this current limiting method, the input current references to the current controller ref iref d and iq are limited using a simple saturation block function since these are dc

PWM switching Park transformation

Inverse park VSM algorithm Current controller

Governor-turbine Virtual rotor Current limitation

Voltage controller

measurements

Figure 6.72 Grid-forming inverter overcurrent limitation in cascaded control structure.

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587

(A)

Voltage controller

Voltage controller

(B)

Figure 6.73 Grid-forming inverter overcurrent limitation in cascaded control structure by saturation of individual current references: (A) Block diagram and (B) vector representation.

quantities in the synchronously rotating dq reference frame. Fig. 6.73A illustrates this method. However, since the inverter current limit given by ref Ilimit 5 Imax

5

r ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ iref d;max

2

1 iref q;max

2

(6.108)

ref the choices of iref d;max and iq;max are not obvious. Either the d axis current or q axis current could saturate without the inverter current reaching its limit depending on the nature of ac grid disturbance. Also, a saturation of only one of the reference currents changes the dq current phase angle from the angle that existed before saturation. This is undesirable as it risks imposing a different current phase angle in the stationary reference frame to the desired natural short-circuit current of the inverter. The current limitation is illustrated in Fig. 6.73B where iref q is saturated.

Strategy 2: Saturation of magnitude of current reference In this current limiting method, the magnitude of the iref dq reference current is limited and its phase angle is kept unchanged. Before saturation, the reference current magnitude and phase angle are given by

588

Power Systems Modelling and Fault Analysis

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2ﬃ ref 1 iref iref idq 5 q d θref dq

21

5 tan

iref q

(6.109a)

! (6.109b)

iref d

The current magnitude is then directly limited to the maximum current limit of the inverter and the same angle θref dq is used to calculate the reference values of the limited d and q axes currents as follows: ref ref iref d;limited 5 cosðθdq Þidq;limited

(6.110a)

ref ref iref q;limited 5 sinðθdq Þidq;limited

(6.110b)

The method is illustrated in Fig. 6.74A and can be implemented as shown in Fig. 6.74B. (A)

(B) =

Voltage controller

+

= tan

Current controller Figure 6.74 Grid-forming inverter overcurrent limitation in cascaded control structure by saturation of current reference magnitude: (A) Vector representation and (B) Block diagram.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

6.22

589

Symmetrical components sequence equivalent circuits of ‘grid-forming’ inverters

Under steady-state and dynamic power frequency grid conditions that do not cause the inverter current to exceed its maximum current limit, a grid-forming inverter is controlled to behave as a balanced, three-phase voltage source behind its output filter reactor impedance as shown in Fig. 6.75B. If the ac grid is balanced, then the inverter can be represented in the sequence domain as a positive-phase sequence voltage source behind its impedance as shown in Fig. 6.75C. Under relatively small phase voltage unbalance on the ac grid or remote unbalanced grid fault disturbances that do not cause the inverter current to exceed its maximum limit, the inverter continues to behave as a positive-sequence voltage source in the positive-sequence system and presents its output impedance in the negative-sequence system. The corresponding sequence representations of the gridforming inverter are shown in Fig. 6.76. Under large phase voltage unbalance on the ac grid, balanced and unbalanced disturbances such as grid faults that cause activation of inverter current limitation, the inverter behaviour and representation changes. In the current limitation strategies described in Section 6.21, when the inverter current is limited, the inverter controller will effectively control and maintain a constant output current equal to the maximum inverter current limit. Control action takes place under all current limitation strategies described in Section 6.21 whether by clipping the current in the time domain, reducing the inverter internal voltage directly with current clipping, insertion of transient virtual resistance, direct saturation of reference currents or a combination of current clipping and virtual transient resistance. (A)

(B)

+ Gate signals

−

Controls

+

−

=ℎ −

+

(C)

+ −

Figure 6.75 Grid-forming inverter representation in normal steady-state and balanced system conditions: (A) balanced three-phase voltage source, (B) positive-sequence voltage source behind filter reactor impedance and (C) positive-phase sequence equivalent circuit.

590

Power Systems Modelling and Fault Analysis

With time domain current clipping and current reference saturation, the gridforming inverter loses control of its voltage and the fundamental frequency component of the nonlinear current is limited or effectively controlled to a constant value. This means that the grid-forming inverter changes from a controlled voltage source to a controlled current source. With a transient virtual resistance, the inverter voltage controller remains active during the disturbance and controls the internal voltage with a variable nonlinear output impedance. In time domain simulations, the inverter can still be modelled as a controlled voltage source behind a modified impedance. However, in steady-state sequence representation, and since the control of the internal voltage effectively limits the output current, the inverter can be represented as a constant current source. Therefore with over-current limitation activated under balanced symmetrical voltage dips, the grid-forming inverter behaves as a balanced, positive-sequence, constant current source behind its filter shunt capacitance. The sequence equivalent circuit is shown in Fig. 6.77A, where the inverter reactor impedance and shunt filter capacitance are neglected. Under unbalanced two-phase faults, the grid-forming inverter will deliver limited, that is, constant phase fault currents or limited positive- and negative-sequence currents and is represented as positive- and negativesequence constant current sources as shown in Fig. 6.77B. (A)

(B)

(C)

+ −

Figure 6.76 Grid-forming inverter equivalent circuits under small unbalance and disturbances and no over-current limitation activation: (A) positive-sequence, (B) negativesequence and (C) zero-sequence equivalent circuits. (A)

(B)

Figure 6.77 Grid-forming inverter representation with over-current limitation activation: (A) positive-sequence current source under symmetrical three-phase voltage dips and (B) positive- and negative-sequence current sources under two-phase faults.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

6.23

591

Examples

Example 6.16 Consider a three-phase grid-forming inverter having the following data: Rated apparent power: 2.2 MVA, ac rated voltage: 0.6 kV, rated power factor 0.909. Rated rms current: 2117A, maximum transient rms current: 2328:7 A. Filter reactor: resistance 0.00049 Ω, inductance 78.1 μH. Inverter transformer: 0.6 kV/33 kV, rated power: 2.2 MVA, impedance 10%, X/R 5 10. Equivalent impedance from the inverter transformer medium-voltage terminals to a point A on the ac grid is 0:01638 Ω with X/R 5 20. Nominal system frequency is 50 Hz. Initial inverter output is 0.8 MW. Calculate the natural short-circuit current, that is, without limitation, supplied by the gridforming inverter for a solid three-phase short-circuit fault at point A. The fault current on one phase should include maximum dc current offset. Solution Using base quantities of 2.2 MVA and 0.6 kV, the base impedance is calculated as 0.1636 Ω. Filter reactor impedance is calculated as ð0:003 1 j0:15Þ pu, inverter transformer impedance is ð0:01 1 j0:0995Þ pu and combined impedance to the fault point is ð0:0129 1 j0:24944Þ pu. The natural unlimited three-phase fault currents of the grid-forming inverter are shown in Fig. 6.78.

8 6 4 2 0 –2 –4 –6 –8 –80

–40

0

40

R

80

Y

B

120

160

Currents (pu)

200

ms

Figure 6.78 Grid-forming inverter: natural unlimited fault currents.

Example 6.17 The inverter used in Example 6.16 employs the fault current limitation ‘strategy 1’ presented in Section 6.21.2. The currents are clipped when each reaches an instantaneous value of pﬃﬃﬃ 2 3 2328:7 5 3293:3A which corresponds to 1.555 pu of rated rms current. The limited fault currents are shown in Fig. 6.79 excluding the small-amplitude high-frequency oscillations that would be superimposed on the clipped current waveform.

592

Power Systems Modelling and Fault Analysis

(A) 8 6 4 2 0 –2 –4 –6 –8 –40 –20

0

20

40

60

80

Limited ir

100 120 140 160 180 200

Unlimited ir (pu)

ms

(B) 8 6 4 2 0 –2 –4 –6 –8 –40 –20

0

20

40

60

80

Limited iy

100 120 140 160 180 200

Unlimited iy (pu)

ms

(C) 8 6 4 2 0 –2 –4 –6 –8 –40 –20

0

20

40

Limited ib

60

80

100 120 140 160 180 200

Unlimited ib (pu)

ms

Figure 6.79 Grid-forming inverter with over-current clipping strategy showing unlimited and limited currents: (A) Phase r currents, (B) phase y currents and (C) phase b currents.

Example 6.18 The inverter used in Example 6.16 employs the fault current limitation ‘strategy 2’ presented in Section 6.21.2. With maximum transient fault current supplied by the inverter, the voltage drop across its filter impedance is 0:165 pu, its terminal voltage is 0:221 pu, its internal voltage is

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators (A) 1.5 1 0.5 0 –0.5 –1 –1.5 –40 –20

0

20

40

R (B)

60

Y

80

B

100 120 140 160 180 200

ms

Internal voltages (pu)

8 6 4 2 0 –2 –4 –6 –8 –40 –20

0

20

40

R (C)

60

Y

80

B

100 120 140 160 180 200

Limited currents (pu)

ms

8 6 4 2 0 –2 –4 –6 –8 –40 –20

ms 0

20

40

60

Unlimited R

80

100 120 140 160 180 200

Limited R

Currents (pu)

8 6 4 2 0 –2 –4 –6 –8 –40 –20

ms 0

20

40

Unlimited Y

60

80

100 120 140 160 180 200

Limited Y

Currents (pu)

8 6 4 2 0 –2 –4 –6 –8 –40 –20

ms 0

20

40

Unlimited B

60

80

100 120 140 160 180 200

Limited B

Currents (pu)

Figure 6.80 Grid-forming inverter with limited inverter internal voltage edq : (A) Threephase inverter voltage waveforms, (B) three-phase limited fault currents and (C) comparison of natural or unlimited and limited fault currents for all phases R, Y and B.

593

594

Power Systems Modelling and Fault Analysis

reduced from 1 to 0:385 pu and the dc time constant of the dc component of fault current is 62 ms. The limited fault currents are shown in Fig. 6.80 where it is shown that the limited currents according to Eq. (6.104) will still substantially exceed the maximum current limit.

Example 6.19 The inverter used in Example 6.16 employs the fault current limitation ‘strategy 5’ presented in Section 6.21.2. The inverter rms current limit is assumed to be 1.2 pu of rated current. The current is clipped during the first 100 ms at an instantaneous value of 1.696 pu then the transient 8 6 4 2 0 –2 –4 –6 –8 –50

ms 0

50

100

150

Limited ir : clipped then Rvr

200

250

300

Unlimited ir (pu)

8 6 4 2 0 –2 –4 –6 –8 –50

ms 0

50

100

150

200

250

Limited iy : clipped then Rvr

Unlimited iy

0

200

300

(pu)

8 6 4 2 0 –2 –4 –6 –8 –50

ms 50

100

Limited ib : clipped then Rvr

150

250

300

Unlimited ib (pu)

Figure 6.81 Grid-forming inverter with over-current clipping in the first 100 ms then insertion of transient virtual resistor: comparison of limited and unlimited currents in all three phases.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

595

virtual resistor takes over and limits the current to 1 pu rated or 1.414 pu instantaneous. The per unit value of the virtual resistor that limits the rms fault current to 1 pu is around six times the filter reactance. This reduces the dc time constant of the fault current to 1.2 ms and effectively eliminates this component. The limited fault currents are shown in Fig. 6.81.

Further reading Papers Kalamen, L., et al., Asynchronous generators with dynamic slip control, Transactions on Electrical Engineering, Vol. 1, No. 2, 2012. Kalsi, S.S., et al., Calculation of system fault currents due to induction motors, Proceedings of IEE, Vol. 118, No. 1, 1971. Muller, S., et al., Doubly fed induction generator systems for wind turbines, IEEE Industry Applications Magazine. Sava, G., et al., Comparison of active crowbar protection schemes for DFIGs wind turbines, 16th International Conference on Harmonics and Power Quality INSPEC No. 14399161, Romania, May 2014. Vicatos, M.S., et al., Transient state analysis of a doubly-fed induction generator under threephase short-circuit, IEEE Transactions on Energy Conversion, Vol. 6, No. 1, 1991. Niiranen, J., Voltage dip ride through of a doubly fed generator equipped with an active crowbar, Nordic Wind Power Conference, Chalmers University of Technology, Sweden, March 2004. Seman, S., et al., Analysis of 1.7 MVA doubly fed induction generator during power systems disturbances, presented at NORPIE June 2004, Norwegian University of Science and Technology, Trondheim, Norway. Kong, X., et al., Fault current study of wind turbine driven doubly-fed induction generator with crowbar protection, Elektrotehniski Vestnik, Vol. 81, No. Nos 12, 2014, 5763. Pannell, G., et al., Analytical study of grid-fault response of wind turbine doubly fed induction generator, IEEE Transactions on Energy Conversion, Vol. 25, No. 4, 2010. Sulla, F., et al., Symmetrical and unsymmetrical short-circuit current of squirrel-cage and doubly-fed induction generators, Electric Power Systems Research, Vol. 81, 2011, 16101618. Ling, Y., et al., Rotor current transient analysis of DFIG-based wind turbines during symmetrical voltage dips, Energy Conversion and Management, Vol. 76, 2013, 910917. Zubiri, G.M., et al., Impact on the power system protection of high penetration of wind farms technology, B5_204, CIGRE 2010. Zhong, Q., et al., Synchronverters: inverters that mimic synchronous generators, IEEE Transactions on Industry Electronics, Vol. 58, No. 4, 2011. Zhong, Q., Virtual synchronous machines a unified interface for smart grid integration, IEEE Power Electronics Magazine. Hesse, R. et al., Microgrid stabilisation using the virtual synchronous machine (VISMA), International Conference on Renewable Energies and Power Quality, April 2009, Spain. Hesse, R., et al., Comparison of methods for implementing virtual synchronous machine on inverters, International Conference on Renewable Energies and Power Quality, March 2012, Spain.

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Power Systems Modelling and Fault Analysis

D’Arco, S., et al., Virtual synchronous machines classification of implementations and analysis of equivalence to droop controllers for microgrids, PowerTech, IEEE Grenoble 2013. Engler, A., Applicability of droops in low voltage grids, International Journal of Distributed Energy Resources, DER Journal, No. 1, January 2005. Unruh, P., et al., Distributed grid-forming inverters in power grids, 7th International Workshop on Integration of Solar into Power Systems, October 2017, Germany. Ofir, R., et al., Droop vs. virtual inertia: comparison from the perspective of converter operation mode, Proceedings of the IEEE International Energy Conference, June 2018, Cyprus.

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

Modelling of voltage-source inverters, wind turbine and solar photovoltaic (PV) generators

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