Effects of PLL and frequency measurements on LFC problem in multi-area HVDC interconnected systems

Electrical Power and Energy Systems 81 (2016) 140–152

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Effects of PLL and frequency measurements on LFC problem in multi-area HVDC interconnected systems Elyas Rakhshani a,b,⇑, Daniel Remon a,1, Pedro Rodriguez a,b a b

Abengoa, Abengoa Research, Palmas Altas, Seville, Spain Technical University of Catalonia (UPC), Electrical Engineering Department, SEER Research Center, Barcelona, Spain

a r t i c l e

i n f o

Article history: Received 15 October 2015 Accepted 15 February 2016

Keywords: Automatic generation control Frequency measurement Phase Locked Loop HVDC interconnected systems Load frequency control

a b s t r a c t Recent advancements in power electronics have made HVDC links and renewable based generation more popular in power systems application with better grid support functionalities like frequency control and inertia emulation tasks. Conventional operation and control strategies are undergoing of different changes and all the infrastructure of future modern power system should efficiently support the delivery of ancillary services in complex scenarios of AC/DC multi-area interconnected system. The AGC system of tomorrow must be able to handle complex interactions between control areas with HVDC links and distributed generation equipment. In such scenario, the effects of wide-area interconnections, PLL (Phase Locked Loop) and frequency measurements cannot be ignored. The dynamics effects of PLL and frequency measurements are very important for HVDC operation. For obtaining an acceptable performance of AC/ DC system, the dynamic models of PLL and measurements need to be taken into account. This paper focused on the effects of PLL and frequency measurements in frequency supports of HVDC interconnected system. A novel approach for analyzing the dynamic effects of HVDC links considering PLL effects during coordination with AC system is presented and discussed. The effects of PLL are considered by introducing a second-order function. A Pade approximation method is also introduced for adding the effects of communication delays on AGC operation and the state space models are presented. The proposed model is analyzed for different multi-area test systems which contains parallel AC/DC transmission links. Ó 2016 Elsevier Ltd. All rights reserved.

Introduction In the recent years, the number of bulk power exchanges over long distances between control areas are significantly increasing. This increase is mainly due to the deregulation of power industry with the high implementation of power electronic based components in the power grid [1]. Recent developments of renewable energy integration and super-grid interconnections in modern power systems attract a lot of attentions to HVDC transmission which is known as a proven tool for dealing with new challenges of the future power system. The capability of DC systems to transmit higher power over longer distances, the possibility of interconnecting asynchronous networks, and their high efficiency have maintained the interest of both industry and academia [2,3]. Expansions of interconnected ⇑ Corresponding author at: Research Center on Renewable Electrical Energy Systems (SEER Center), GAIA building, 22, Rambla Sant Nebridi s/n, 08222 Terrassa, Barcelona, Spain. E-mail addresses: [email protected], [email protected] (E. Rakhshani). 1 Address: Technical University of Catalonia (UPC), Electrical Engineering Department, SEER Research Center, Barcelona, Spain. http://dx.doi.org/10.1016/j.ijepes.2016.02.011 0142-0615/Ó 2016 Elsevier Ltd. All rights reserved.

systems with wide area HVDC control application are leading to a complex scenario which bring more challenges in terms of communications, coordination and frequency control of interconnected multi-area systems. Interconnected power system consists of several control areas. Any mismatch between generated power and demand can cause the system frequency or tie-line power flow to deviate from their scheduled values. To eliminate these deviations, the AGC is applied to manipulate the set-points of different power generation units in each area [4]. Accordingly, the objective of AGC is to regulate the generated power from various sources in each area in a way that the frequency of power system and tie-line powers are kept within prescribed limits. The recent trends of research are through the adoption of previous concepts and conventional models considering new AC/DC complex scenarios with more application of DC interconnections and RES (Renewable Energy Systems) penetrations [5,6]. The traditional LFC models have been modified and revised to add different functionalities in the reformulation of conventional power systems. Most of those modifications are related to the AGC in a deregulated market scenario [7], different types of power plants like renewable generation [8] and recently the demand side dynamic models [9]. Therefore, the general model of multi-area

E. Rakhshani et al. / Electrical Power and Energy Systems 81 (2016) 140–152

AGC system is adapting to meet the necessary changes in modern scenarios of the power systems. It should be noted that AGC facilities could potentially be used not only to realize emergency control for frequency stability, also to coordinate with modern HVDC transmission and FACTS devices for long-term dynamics control [10,11]. As it was explained, the power grid is experiencing the increased needs for important issues, e.g. enhancing the bulk power transmission for a very long distance, reliable integration of large-scale renewables with multi-area interconnected systems and more flexibility and controllability in power flow at the transmission system. Renewable energy sources, which are located far away from consumption centres, are now driving forces for the development of new transmission concepts. Challenging projects usually include integration of large-scale offshore wind farm which are located far from shore, or the integration of solar energy from the Middle East and North Africa [12]. In those projects, it is claimed that DC transmission systems are more technically and economically convenient than AC [12]. Moreover, another driving force for the development of new transmission concepts is power market integration, which means trading of power over long distances. One major challenge for the power system under deregulation is to implement communication and controllers into suitable levels of system operation and control [13]. The widespread application of communication systems in the power system control causes unavoidable time delays. Considering the characteristic of the communication channel, LFC scheme is a typical time delay system. From stability analysis and controller design point of view, it is very important to identify the maximum range of delay which allows a power system equipped with an LFC scheme to remain stable. Recently, time delay has been considered in the design of load frequency controller in different case studies [13,14]. But there is not a lot of work considering a scenario with complex AC/DC interconnection and communication delays of LFC loop at the same time. As it was mentioned, in order to enhance the controllability, one important application of HVDC transmission line is its operation in parallel with an existing AC transmission line. Thus, they can act as AC/DC parallel links interconnecting any two control areas. In literature, AGC of a two-area power system interconnected via AC/DC parallel link, is carried out with different control approaches for better dynamic responses [14–21]. But in none of them a complete model of DC link for AGC application is considered. In all the presented research work, the part related to PLL dynamics and frequency measurements of HVDC stations is also missed. The phase-locked loop (PLL) is typically used for angle reference generation for the traditional line-commutated converter (LCC)-based HVDC and the newer voltage-sourced converter (VSC)-based HVDC transmission applications [22]. This angle reference is used for generating the firing pulses for the insulated-gate bipolar transistor (IGBT) switches of the VSC. Results from recent research works show that the gains of the PLL parameters can greatly affect the operation of the VSC-HVDC converter. The efficiency and maximum power limits of VSC system can be affected by PLL gains and damping [23,24]. In this paper, a novel approach for analyzing the impacts of PLL and frequency measurements of HVDC links on the stability of multi-area load frequency control (LFC) systems is proposed. The used PLL model is based on a second-order function describing the PLL dynamics and its control gains. This function is added in a DC link which is implemented by a Supplementary Power Modulation Controller (SPMC). This controller is presented in a coordinated manner for controlling the HVDC set-points during AGC operation. Since the importance and application of HVDC links are increasing, it was necessary to introduce a more detailed model of HVDC

141

station dynamics for LFC studies. In fact, the main objective of this paper is to introduce and analyze the dynamic effects of PLL as an important part of VSC stations. It should be noted that the effects of the time delay in the LFC loop are also considered and sensitivity analysis is performed. A complete state space model of multiarea AGC system with a parallel AC/DC line considering PLL and delay effects is presented. Stability regions of the system for different values of time delay in LFC loop are obtained and presented. The proposed model will be very essential and useful for further studies in frequency and active power regulation with HVDC links. In the following sections, the dynamic model of multi-area AGC system with AC/DC connection will be explained in Section ‘Multiarea AC/DC interconnected system’. Then the complete model of AGC with PLL model and communication delays of LF loop is presented in Section ‘AC/DC interconnected control system with communication delay and PLL’. System analyses with the simulations in different test systems (two and four areas) are given in Sectio n ‘Simulation results’ and finally the paper is concluded by Section ‘Conclusion’. Multi-area AC/DC interconnected system The load frequency control and AGC issue is well discussed in power system control literatures [1–4]. To understand the concept of LFC problem, the Area Control Error (ACE) is introduced. In a two-area power system model, the ACE for ith area is defined as follows:

ACEi ¼ bi Df i þ DPij

ð1Þ

where the DPij is the net tie-line power flow variation between two areas (DP ij ¼ DP ji ), f is the system’s frequency; bi is referred as the frequency bias and is generally referred to the tie-line bias control. Therefore, in a normal two-area AC link we have:

DPij ¼ DPtie;AC

ð2Þ

The state space presentation of ith area could be as follow:

 K pi
DPmi  DPLi  DPtie;AC 1 þ sT pi n X ¼ DPm;ik

Dxi ¼ DPmi

ð3Þ ð4Þ

k¼1

where DPLi ði ¼ 1; 2Þ is local load deviation, K pi is the power system gain, T pi is the power system time constant, DP m;ik ðk ¼ 1; 2Þ is the output of generation units and:

K pi ¼

1 ; Dsysi

T pi ¼

M sysi 2Hsysi =x0 ¼ Dsysi Dsysi

ð5Þ

Considering that Hsysi and Dsysi are inertia and damping. The rest of the variables could be defined as follows:

DPm;ik ¼

  1 Dx i  K Ii DP refi 1 þ sT tg;ik Ri  2p

ð6Þ

where Rk ðk ¼ 1 : 4Þ is considered as droop for each generation company (GENCO), T tg is the overall time constants of turbine and governor in each GENCO and is equal to ðT tik þ T gik Þ [5]. The reference of generation units in ith area will be based on ACE and could be considered like this:

  ACEi 1 bi Dxi þ DPtie;AC ¼ s 2p s T ij ¼ ½Dxi  Dxj  s

DPrefi ¼

ð7Þ

DPtie;AC

ð8Þ

and T ij is also the synchronization power coefficient [1].

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parallel VSC-HVDC stations. In this study, the dynamics of fast transient HVDC power electronic parts is neglected because the time constant of electronic parts is much smaller than mechanical part involved in the dynamic analysis of power system. The SPMC is designed as a higher level damping controller to improve the performance of power system during load changes. The frequency deviations are used as a control signal for the VSC-HVDC units to control the power flow by changing the duty cycles of converters. The coordinated control strategy for this kind of HVDC link can be written as follows:

DxDC ¼ K fi Dxi þ K AC DPtie;AC þ K fj Dxj

ð9Þ

Considering that DxDC is the control signal for the DC link (the desired DC power reference), K fi , K fj and K AC are control gains. The HVDC link is also represented in the form of a transfer function:

Fig. 1. Diagram of SPMC to the power system with parallel AC/DC links.

DPDC ¼ Gp xDC 1 Gp ðsÞ ¼ 1 þ sT dc

In order to model the HVDC link for dynamic analysis and implementing of the damping controller on the interconnected systems, the concept of Supplementary Power Modulation Controller (SPMC) is used [11]. The block diagram of the SPMC for modeling the VSC-HVDC in AGC power system is shown in Fig. 1. The power flow through the VSC-HVDC link is modulated based on the frequency variations at the two sides of the DC link and the variations of the AC power flow between the areas with the

ð10Þ ð11Þ

where T DC is the time constant of the HVDC unit and DP DC is real DC power flow through the system. The DxDC is the input signal of HVDC which will be generated by frequency deviations of each interconnected areas and AC tie-line power deviations (in a case

Fig. 2. Supplementary control of AGC considering delay function.

13 12.5 14

Total damping of modes

12 13 12

15 10 5

11.5

10

11

11

8

10 9 8 10

10

10 6

8

8

4

6 8

6

6

td2

2

2 2

0

0

10 9.5

4

4 4

10.5

td1

9

2

td2 0

0

Fig. 3. Relationships between the time delay (sec) in each area and the damping of the system.

td1

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2

t d1=0.91 s

1.5 1

Imag

0.5 0 -0.5 -1 -1.5 -2 -9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

Real Fig. 4. Eigenvalue trajectory of dominant poles over time delay changes (arrow direction indicates increase in time delay).

Fig. 5. A typical control structure of grid-connected converters.

of any parallel AC line with HVDC transmission link). As mentioned in [1,5] for this type of higher level control design, the proper time response could be between 100 ms to 500 ms. In this study, the time constant is assumed 100 ms for T DC . The total tie-line power variation in AC–DC system becomes:

These aspects are briefly explained and presented in the final proposed model for the AGC process.

DPij ¼ DPDC þ DP tie;AC

In LFC practice, measuring and data communication for tie-line power and ACE signals (with the response characteristics of generator units) will take a time about 2 s or more in decision cycles of the LFC systems. In a new environment, the communication delays in the LFC analysis are becoming a more important challenge due to the restructuring, expanding of the system with long distance HVDC links. As shown in Fig. 2, a time delay can be expressed as ess in Laplace domain. In Laplace domain, the first-order Pade approximation of delay block can be used as:

ð12Þ

where DPDC is a power modulation by HVDC link [5]. AC/DC interconnected control system with communication delay and PLL The frequency control model must be detailed and flexible enough to include delay and PLL measurement effects in AC/DC multi-area interconnected power system control procedure. Especially considering the high penetration of power electronic based generation and HVDC links, the effects of PLL will be extremely important and it is necessary to be included in the control models.

Communication delays

estd ¼

1  0:5 t d s 1 þ 0:5 t d s

ð13Þ

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Fig. 6. Small signal model of a basic PLL.

Fig. 7. Proposed two-area model for HVDC interconnected system with SPMC structure and PLL dynamics.

Fig. 8. The configuration of the power system with HVDC link.

Then the transfer function (13) can be transformed into state space equation in the form:

(

y ¼ xd;i xd;i þ 2 ud;i  u_ d;i x_ d;i ¼ 2 t t d

ð14Þ

d

where t d is the time delay, xd;i is the delayed state variable of reference power and as shown in Fig. 2, the input signal, ud;i , is

the references signals for generation units which are coming from ACE signal. This signal is consists of tie-line power flow and grid frequency deviation signals which is related to another state of the global multi-area system. Therefore, the final states considering the delay of ACE signal for ith area could be extracted as follows:

Dx_ d;i ¼

2 2 b xd;i þ DPref ;i  i Dxi þ DP DC þ DPtie;AC td td 2p

ð15Þ

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E. Rakhshani et al. / Electrical Power and Energy Systems 81 (2016) 140–152 Table 1 Control parameter for studies model.

Table 2 Eigenvalue comparisons for different systems.

Parameters

Value

Kf 1 Kf 2 K AC fi (i=1,2) xni (i=1,2)

0.35 0.1 2 1.5 3.5

More analysis is performed to identify the maximal range of time delay for the two-area studied system in this paper. Fig. 3, shows a sensitivity analysis for total damping of oscillatory modes over a wide range of variation for time delays in LFC loops. The 3-D plot in Fig. 3 shows where the damping of the system could be maximum for different values of time delays. Based on these results, the time delays should be less than 2 s to have the highest damping. It should be noted that, the results of eigenvalue analysis in Fig. 4, are also leading to the same result by Fig. 3. It is obvious that with increasing the time delay, the dominant modes are moving to the right side of s-plane. In this case study, for values higher than 0.91 s, the eigenvalues will move to the right side as a sign of instability of the system.

Modes

AC system

AC/HVDC system

AC/HVDC system with PLL and delay

k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11 k12 k13 k14 k15 k16

0.0275 + 0.8256i 0.0275  0.8256i 0.3423 + 0.3995i 0.3423  0.3995i 0.7023 1.8271 2.1073 2.6872 2.6316 – – – – – – –

21.0096 1.3381 + 0.9485i 1.3381  0.9485i 0.3266 + 0.4318i 0.3266  0.4318i 0.6653 3.1995 2.1709 2.6888 2.6316 – – – – – –

18.3859 6.4064 + 2.6769i 6.4064  2.6769i 4.9162 2.3178 2.6939 2.6316 0.6069 + 0.9533i 0.6069  0.9533i 1.2500 0.2436 + 0.4667i 0.2436  0.4667i 0.7198 + 0.0936i 0.7198  0.0936i 0.1824 0.2288

are coming for higher level control actions like droop frequency control. Related current references could be obtained based on the concept of Instantaneous Active-Reactive Control in the reference generation block [22]. The inner current loop controller is used for providing the reference voltage for converter. The angle of the grid voltage is provided by a PLL. This PLL can be used for estimating the frequency. This estimation is very important for proper action of inner current controller. The power references are coming from higher level control. As explained before, the dynamics of converter and HVDC could be modeled as a first-order transfer function with a proper time constant imitating the control time of its components. In this study, in order to evaluate the dynamic effects of PLL and delayed

Model of PLL dynamics Usually the Phase Locked Loop (PLL) is one of the methods for synchronizing the converter to the grid and measuring the frequency. Its dynamics can be approximated by a first-order or second-order model [22]. A general control structure of gridconnected converter with DC link and PLL is shown in Fig. 5. As shown in Fig. 5, reference values for active and reactive powers

Frequency deviation in area 1

0.1

AC System AC/DC System AC/DC with PLL and delay

0.05

0

-0.05

(A) Area1 -0.1

5

10

15

20

25

30

Frequency deviation in area 2

time (s) 0.1

AC System AC/DC System AC/DC with PLL and delay

0.05

0

-0.05

(B) Area2 -0.1

5

10

15

20

time (s) Fig. 9. Dynamic response of frequency in both areas.

25

30

E. Rakhshani et al. / Electrical Power and Energy Systems 81 (2016) 140–152

DC power modulation respons (p.u.MW)

146

(A)

x 10

-3

5

AC/HVDC System AC/HVDC with PLL and Delay

0 -5 -10 -15 -20 5

10

15

20

25

30

time (s)

AC power deviation (p.u.MW)

(B)

AC System AC/HVDC System AC/HVDC with PLL and Delay

0.01

0.005

0

-0.005

-0.01

5

10

15

20

25

30

(C)

Total tie line power deviation (p.u.MW)

time (s)

AC System AC/HVDC System AC/HVDC with PLL and Delay

0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 5

10

15

20

25

30

time (s) Fig. 10. Dynamic response of transmitted power for different systems, (A) DC line power, (B) AC line power, (C) Total tie-line power (DPtie,12).

measurements, more detailed model is considered by adding additional transfer functions that model the dynamics of PLL and delayed measurements. For defining a proper model imitating the dynamics of PLL in higher level control analysis, a brief review is presented over the normal components of PLL. The basic structure of PLL is presented in Fig. 6. It contains of Phase Detector (FD), Loop filter (LF) and Voltage-Controlled Oscillator (VCO). Considering unitary values for PD and VCO gains of this closedloop system (kpd ¼ kv co ¼ 1), will give the following characteristic transfer functions. Open-loop transfer function: k

F OL ðsÞ ¼ PDðsÞ  LFðsÞ  VCOðsÞ ¼

kp s þ Tpi s2

ð16Þ

Therefore, the close-loop transfer function is: k

HðsÞ ¼

kp s þ Tp F OL ðsÞ i ¼ 1 þ F OL ðsÞ s2 þ kp s þ kp

ð17Þ

Ti

This second-order transfer function can be written in a normalized way as follows:

HPLL ðsÞ ¼

s2

2fxn s þ x2n þ 2fxn s þ x2n

ð18Þ

where

sffiffiffiffiffi k xn ¼ p and Ti

f¼

pffiffiffiffiffiffiffiffiffi kp T i 2

ð19Þ

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E. Rakhshani et al. / Electrical Power and Energy Systems 81 (2016) 140–152

Frequency deviation in area 1

0.1

AC System AC/DC System AC/DC with PLL and delay

0.05

0

-0.05

-0.1

(A) Area1 5

10

15

20

25

30

Frequency deviation in area 2

time (s) AC System AC/DC System AC/DC with PLL and delay

0.05

0

-0.05

-0.1

(B) Area2 -0.15 5

10

15

20

25

30

time (s) Fig. 11. Response of frequency deviations in two-area system.

The approximated time constant of this second-order function could be s ¼ fx1 n for 1% of steady state response [22].

Therefore, considering the input signals (Dxi ) for ith area (i ¼ 1 : 2) of two-area interconnected system, complete state equations of these new states could be as follows:

Modeling considering the PLL measurement dynamics

Dx_ 1;Pi ¼ Dx1;Pi

For estimating or measuring the frequency, different components like PLL can be used. These components will introduce some delay with specific dynamics to the system [22]. Therefore, it would be important to take into account these effects during frequency regulation of the transmission systems. The modified model of two-area interconnected power system with SPMC control method adding the PLL is depicted in Fig. 7. As explained before, in the second-order transfer function, there is one zero in the numerator. This zero will exhibit some overshoots in the system responses. The relationship between input and output signals in this second-order system could be identified as follows:

Dx_ 2;Pi ¼ b1;i Dxi þ b2;i DP mj þ b3;i DPtie;AC þ b4;i DP DC þ b5;i Dx1;Pi

€ þ 2fxn y_ þ x2n y ¼ 2fxn u_ þ x2n u y

Based on presented information about adding a new secondorder system for PLL and communication delays, the linearized mathematical presentation of the studied two-area system could be extracted as follows:

ð20Þ

The input signal uðtÞ is the grid frequency Dxi which is related to other state of the global multi-area system and the output signal will consist of two states variables.



Dx1;P DY P ¼ Dx2;P



ð21Þ

Based on classic control concepts, this second-order system could be represented by a set of two linear state equations.



Dx_ 1;P Dx_ 2;P





¼

0

1

x2n

2fxn



  0 Dx1;P þ 2fxn Dx2;P

0

x2n



Du_ Du



ð22Þ

ð23Þ

þ b6;i Dx2;Pi þ b7;i DPLi

ð24Þ

where

2fi xni 2f xni K pi þ x2ni ; b2;i ¼ i T pi T pi 2fi xni K pi 2fi xni K pi ¼ ; b4;i ¼ T pi T pi 2fi xni K pi ¼ x2ni ; b6;i ¼ 2fi xni ; b7;i ¼ T pi

b1;i ¼ b3;i b5;i

Dx_ ¼ ADx þ BDu

ð25Þ

where A is the new state matrix considering the dynamics of PLL and measurements in the system with parallel AC/HVDC link control:



A¼

A11

A12

A21

A22





; ð1616Þ

B¼

B11 B21



ð162Þ

and all the sub-matrices are as follows:

ð26Þ

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E. Rakhshani et al. / Electrical Power and Energy Systems 81 (2016) 140–152

2

A11

6 6 6 0 6 6 1 6 6 2pR1 ðT t1 þT g1 Þ 6 6 6 2pR2 ðT1t2 þT g2 Þ 6 6 6 0 ¼6 6 6 0 6 6 b1 6 6 2p 6 6 0 6 6 T 12 6 2p 4 0 2

A12

6 6 6 6 6 6 6 6 6 6 ¼6 6 6 6 6 6 6 6 6 4 2

A21

K p1 T p1

K p1 T p1

0

0

0 0

K p1 T p1

K p1 T p1

0

1 T p2

0

0

K p2 T p2

K p2 T p2

0 0

K p2 T p2

K p2 T p2

0

0

1 ðT t1 þtg1 Þ

0

0

0

0 0

0

0

K I1 ðT t1 þT g1 Þ

0

0

1 ðT t2 þtg2 Þ

0

0

0 0

0

0

K I1 ðT t2 þT g2 Þ

1 2pR3 ðT t3 þT g3 Þ

0

0

1 ðT t3 þtg3 Þ

0

0 0

0

0

0

0

1 ðT t4 þt g4 Þ

0 0

0

0

0

1 2pR4 ðT t4 þT g4 Þ

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0

0

0

0

Kf 1 T DC

Kf 1 T DC

Kf 2 T DC

Kf 2 T DC

b1 2p

6 6 0 6 6 6 ¼6 0 6b 6 1;1 6 4 0 0

2

A22

0

1 T p1

0 6 0 6 6 6 0 ¼6 6b 6 5;1 6 4 0 2

0

0

0

0

0

0 0

1

1

0

0

0

0

0

0 0

1

1

0

0

0

0

0

0 0

0

0

0

0

0

0

0

0

0 0

K AG T DC

1 T DC

0

3

The proposed AC/DC model with PLL and time delay is simulated and analyzed in this section. In order to evaluate the performance of the system different test system are used. All the analyses are performed in Matlab platform and the system parameters of two and four areas are presented in appendix.

ð104Þ

3

0

0

0

2 Td

0

1

1

2 T d2

0

b2 2p

0

0

0

0

0

2 Td

1

1

0

2 T d2

0

0

0

0

0

0

0

0

0

0

0 0 b1;2

b2;1 0 0

b2;1 0 0

0 0 b2;2

0 0 b2;2

0 0 0

0 0 0

b3;1 0 b3;2

b4;1 0 b4;2

0 0 0

0

0

0 0

1

0

0

b6;1

0

0

0

1

0

b6;2

0

0

3

b5;2

0 0

7 7 7 5

b7;2

3 7 7 7 7 7 7 7 7 5 ð64Þ

2 K

;

B11

6 6 6 6 ¼6 6 6 6 4

p1

T p1

0

0

K p2 T p2

0 .. . 0

0 .. . 0

7 7 7 7 07 7 07 7 7 05 0 ð612Þ

3 7 7 7 7 7 7 7 7 5

;

ð122Þ

ð42Þ

It should be noted that, the control parameters of Eq. (9) which are implemented in the proposed model with PLL and delay function could be obtained using optimization theory. Usually, it would be possible to define a cost function for obtaining the optimum values for these control gains. These gains could be defined based on optimization theory by minimizing the following cost function [5]:

Z J¼

½ACE21 þ ACE22 dt

ð1012Þ

Simulation results

0

0

7 7 7 7 7 0 7 7 7 0 7 7 7 K I2 7 7 ðT t3 þT g3 Þ 7 K I2 7 7 ðT t4 þT g4 Þ 7 7 0 7 7 7 0 7 7 7 0 7 5 0 0

;

0

0

3

tion solver that could be used as a classical optimization which implements the Sequential Quadratic Programming (SQP) and the interior-point method for optimization of LFC problem [5].

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

0

0

0

b2 2p  T212p

0

6b 6 7;1 B21 ¼ 6 4 0

0

0

ð27Þ

This cost function is the regular function which is based on the ISE (Integral of Squared Error) method [5]. It should be noted that the ACE is the area error which consists of frequency and tie-line deviations in each area. The control parameter for this case study could be obtained using the Eq. (27) and classical FMINCON function in Matlab software. FMINCON (Find minimum of constrained nonlinear multivariable function) is a generic non-linear optimiza-

Analysis with two-area system The power system is assumed that contains of two areas and each area includes two generation companies (GENCO) and one Distributed companies (Discos) as a load. It also consists of parallel AC and HVDC lines. The block diagram for two-area load frequency control (LFC) system is shown in Fig. 8. In this simulation, it is assumed that a 0.03 p.u. load step change at t = 3 s happens. It is also assumed that the time constant of delay in LFC loop is around 0.5 s and for having a fair comparison for higher level application, a range between 100 ms and 500 ms delay could be assumed for the time response of second-order function of PLL. It is assumed that this time constant is associated PLL dynamics and its measurements together [22]. The control parameters of studied power system are given Table 1. The dynamic behaviors for several systems are presented and compared. The comparisons are mainly related to the normal case with AC line, AC/DC lines and AC/DC with PLL and delay effects. The frequency deviations in both areas are presented in Fig. 9. The blue2 line is related to the normal AC system and the green line shows the response of AC/DC system. It is clear that by adding the DC link the damping of the system is increasing. But considering the response of the system with PLL and delay effects (black trace), the overshoot and settling times are increased and the performance of the system is decreased. From these responses, it could be concluded that adding a PLL is affecting the dynamic performance of the system. This issue will be clearer considering the presented results in Fig. 10 and Table 2. In Fig. 10, the power deviations in AC/DC transmission lines are presented. The results are compared for the AC/DC system with and without PLL. It is obvious by adding PLL, more efforts with 2 For interpretation of color in Figs. 9 and 15, the reader is referred to the web version of this article.

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(A)

AC System AC/DC System AC/DC with PLL and delay

AC power deviation (p.u.MW)

0.01

0.005

0

-0.005

-0.01

5

10

15

20

25

30

time(s)

DC power modulation respons (p.u.MW)

(B)

AC/DC System AC/DC with PLL and delay

0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02

5

10

15

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25

time(s) Fig. 12. Dynamic performance of AC/DC link power, considering PLL effects (black trace), (A) AC link power, B) DC link power.

Fig. 13. Single-line diagram of studied four-area interconnected system.

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x 10

12

AC/DC System AC/DC with PLL and Delay

10

Pdc (p.u.)

8 6 4 2 0 -2 -4 -6

0

10

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time (s) Fig. 14. Topological connections of four areas system. Fig. 16. Dynamic performance of DC link power.

more oscillation will appear. Another comparison regarding eigenvalue of the systems with and without PLL effects are presented by Table II. It is clear that by adding PLL dynamics, the number of oscillatory modes will increase from 4 to 8 modes which deteriorate the dynamic performance of the system by increasing the oscillations and settling time of the system. Some oscillatory modes are also close to zero.

50 40

AC System

30

AC/DC System

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AC/DC with PLL

10

PLL effects under parameter variations

0

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AC System AC/DC System AC/DC with PLL and Delay

0.06 0.04 0.02 0 -0.02 -0.04 -0.06

(A) Area1

-0.08 -0.1 0

10

20

30

40

50

∆ω1

∆ω2

∆ω3

parameter, xn , is changing to imitate different time constant of PLL and measurement dynamics. The time response of PLL is inversely proportional to xn [22]. As explained in Section ‘Model of PLL dynamics’, for lower values of xn , the PLL function will have a

AC System AC/DC System AC/DC with PLL and Delay

0.4 0.2 0 -0.2 -0.4

(C) Area3

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60

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time (s)

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0.1 0.05 0 -0.05 -0.1

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frequency deviation in area 4 (rad)

frequency deviation in area 2 (rad)

20

time (s)

0.15

-0.2

∆ω4

Fig. 17. Comparisons of settling times for different studied systems.

frequency deviation in area 3 (rad)

frequency deviation in area 1 (rad)

In order to analyze the dynamic effects of PLL, a more numerical analysis is presented. These analyses are performed to check the response of the system for various changes in the parameters of PLL second-order function. In this comparisons, it is assumed that both PLLs (each one located in one area) are using the same values for their parameters (f1 ¼ f2 ¼ f ¼ 1 and xn1 ¼ xn2 ¼ xn1 ). The frequency responses in area 1 and 2 are presented in Fig. 11. It is assumed that the value for f is equal to 1 and the other

AC System AC/DC System AC/DC with PLL and Delay

0.04 0.02 0 -0.02 -0.04

(D) Area4

-0.06 -0.08

0

10

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30

time (s)

time (s) Fig. 15. Response of frequency deviations in four-area system.

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higher time constant and obviously for higher value of xn , the lower time response could be expected. In fact, with lower values of xn the system performance will be deteriorated and more oscillatory modes will appear. Higher values of xn is a ticket for a fast reacting system and in such situation, the real positive effects of the HVDC supplementary control could be visible, while very high xn obviously lets in more noises. The dynamic responses for transmitted power in AC and DC links are also presented in Fig. 12. Based on the presented results, it is obvious that the effects of PLL could be explained based on its time response. When the time constant is too high, (for example more than 7 s which means xn has smaller values like 0.1), it could be observed that the system will be unstable or more close to the behavior of normal AC system and the real effects of DC link will be disappeared which is the results of PLL failure. Additionally, it is obvious that for the higher values of xn , the system starts to show the positive effects of DC link with more damping and acceptable dynamic performance. The DC link control is more effective with proper PLL but with more oscillation.

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of the time delay of the LFC loops with sensitivity analysis for defining the marginal values was also presented. The proposed approach could be easily implemented for any type of interconnected power systems with different size and characteristics. All the models are evaluated with two and four area test systems. Based on the presented information, it became clear that the positive effects of DC link for damping the oscillation are significantly depends on the suitable performance of PLL. Any type of malfunction in PLL could bring instability problems to the HVDC systems. Therefore, it is necessary to take into account the dynamic model of PLL in AC/DC interconnected models for LFC studies. Acknowledgment This work was partially supported by Spanish Science Ministry of Economy and Competitiveness under the projects ENE201348428-C2-2-R and ENE2014-60228-R. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the host institutions or funders.

Analysis with four-area system As shown in the Fig. 13, generic four-area power system based on a 12-bus multi machine benchmark which consists of four Genco (Generation Company) and four Disco (Distribution Company) is implemented and the total load in the system is assumed to be 1500 MW. The basic parameters of this system are shown in Appendix. In this model, there are 6 tie-line connections which two of them are the parallel AC/DC lines. The studied HVDC link is located in parallel with the AC line between area 1 and 3. In this scenario, it is assumed that there is a load change around 0.1 p.u. in area 3 and the effects of this change on dynamic response considering normal AC system, AC/DC system and VSP based AC/DC system are discussed and compared. In order to show the interconnection of different areas, the topological connections of four-area test system is also presented in Fig. 14. The frequency deviations in all areas of the test system are presented in Fig. 15. The blue line is related to the normal AC system, the red line shows the response of AC/DC system, and the black trace is the response of the AC/DC system with more detailed model of DC link adding PLL and also the LFC loop delays. The same as previous scenario, it is also predictable that a more damped system is achievable using DC link in parallel with AC line, while adding the effects of delays and PLL will bring more oscillatory response. It should be noted that, because of the characteristics of this four-area model, the main improvement with DC link is more related to the dynamic response of area 1 and area 3. The areas which are connected by DC links shown in Fig. 14. The dynamic response of DC link is also presented in Fig. 16. For more detailed observation, settling time of the frequency deviations are also presented in Fig. 17. Based on the presented comparisons, it is obvious that by implementing the DC link with SPMC approach considerable improvement in settling times is obtained. While after implementing the delays and PLL effects, a little increase on settling times is observable. Conclusion In this paper, a new approach for modeling and analyzing the dynamic effects of PLL in multi-area AGC interconnected system is proposed. Due to the importance of HVDC links in the future modern power systems, it was important to review and proposed a more detailed dynamic model for LFC analysis considering the second-order function of PLL for each station of DC link. The effects

Appendix A Parameters of two-area interconnected test systems with AC/DC tie lines [4,5]: Tt1 = Tt2 = 0.32 s, Tg1 = Tg2 = 0.08 s, R1 = R2 = 2.3 (Hz/pu), KP1 = KP2 = 102 (Hz/pu), TP1 = TP2 = 25 s, B1 = B2 = 0.425 (pu/Hz), T12 = 0.214 s, Tdc = 0.1 s Parameters of the four-area interconnected test systems with AC/DC tie lines [5]: Kp1 = 76 (Hz/pu), Kp2 = 142 (Hz/pu), Kp3 = 140 (Hz/pu), Kp4 = 115 (Hz/pu), Tp1 = 14.5 s, Tp2 = 19.1 s, Tp3 = 9.40 s, Tp4 = 9.12 s, R1 = R2 = R3 = R4 = 3 (Hz/pu), B1 = 0.416 (pu/Hz), B2 = 0.377 (pu/Hz), B3 = 0.378 (pu/Hz), B4 = 0.388 (pu/Hz), Tt1 = Tt2 = Tt3 = Tt4 = 0.32 s, Tg1 = Tg2 = Tg3 = Tg4 = 0.08 s, TDC = 0.1 s, T12 = 0.029 s, T13 = 0.143 s, T14 = 0.0099 s, T21 = 0.029 s, T23 = 0.0205 s, T31 = 0.143 s, T32 = 0.0205 s, T34 = 0.0089 s, T41 = 0.0099, T43 = 0.0089 s.

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