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Interconnection and damping assignment control of a three-phase front end converter
Electrical Power and Energy Systems 60 (2014) 317–324
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Interconnection and damping assignment control of a three-phase front end converter Federico M. Serra ⇑, Cristian H. De Angelo, Daniel G. Forchetti Grupo de Electrónica Aplicada (GEA), UNRC, Río Cuarto, Córdoba, Argentina
a r t i c l e
i n f o
Article history: Received 19 October 2012 Received in revised form 24 November 2013 Accepted 20 March 2014 Available online 16 April 2014 Keywords: Front end converter Nonlinear control Interconnection and damping assignment Passivity based control
a b s t r a c t A new nonlinear control strategy for a three-phase front end converter used to connect renewable energy sources to the grid is proposed in this paper. The controller is designed in order to inject all the generated power into the grid, while the reactive power can be controlled to meet the power system requirements. The system is represented through its port controlled Hamiltonian model, and the controller is designed by interconnection and damping assignment. This design method allows an intuitive way to remove the undesired couplings between system dynamics while assigning the damping required to achieve the expected convergence rate. The proposed controller allows a direct control of the DC link voltage by proper selection of the controller parameters. Moreover, an integral action is added to the proposed controller in order to eliminate the steady-state error in the system variables. The proposal is validated through simulation tests performed using a realistic converter model. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction In the lasts years, electronic power converters have become important components in electrical power systems, mainly due to issues related to power quality [1–4]. Even when different converter configurations have been proposed, Voltage Source Converters (VSC) are the most used in energy conversion systems due to its bidirectional power flow capability and its low harmonic distortion [5,6]. Such power converters have allowed the increase of renewable energy sources integration into the grid [7,8]. In those applications, the VSC can be used as a Front End Converter (FEC) for wind energy or photo voltaic generation systems, where the operating mode depend on the application [9,3,10]. In the case of grid connected generation units in strong power systems, voltage and frequency are imposed by the grid, so the FEC must be synchronized and inject all the available DC power to the grid. Besides, it must be able to control the reactive power exchanged with the power system. For weak power systems or micro-grids, droop control is usually used to control the active and reactive powers that the FEC exchange with the grid, thus contributing to grid stability [11,12,10]. In order to fulfill this requirements, it is necessary that the FEC be able to control the waveform of the injected currents. Due to the actual requirements to connect power converters to the grid [13], precise control strategies are needed. Such strategies ⇑ Corresponding author. Tel.: +54 265715543445. E-mail address: [email protected] (F.M. Serra). http://dx.doi.org/10.1016/j.ijepes.2014.03.033 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
require, in several cases, a complex design including nonlinear control techniques [14]. The most used design strategies are Feedback Linearization (FL) [15,16], Sliding Mode Control (SMC) [17], and Passivity Based Control (PBC) [18,19], among others. Interconnection and Damping Assignment (IDA) is a PBC technique [20] which has been widely used to design controllers for electric power systems stabilization and control [21–24]. More specifically, for controlling VSC this technique has been utilized in rectifiers and inverters. An IDA controller for a three-phase rectifier is presented in [25] in order to obtain unity power factor, sinusoidal phase currents and constant DC output voltage. Simulation results show that the controller is stable and robust against load variations. The controller proposed in [26] is designed with similar objectives, but an integral control loop is added in order to eliminate the steady state error in the DC output. In [27] an IDA controller is designed to control the DC output even in cases of variable load. It is also shown that the controller is robust against input voltage changes. An IDA controller for a three-phase rectifier with LCL input filter is designed in [28], which is also robust to grid impedance variations. For single-phase rectifiers, an IDA controller is designed in [29], which allows controlling the DC output while obtaining unity power factor, with a reduced number of sensors. IDA is used in [30] to design a controller for a bidirectional power flow single-phase rectifier. In [31] this strategy is used to control the DC voltage in case of complex loads, considering bidirectional power flow, too.
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Proposals to control VSC used in single-phase inverters using IDA can be found in [32,33]. In [32] the controller is designed to control a grid connected inverter, allowing to solve the instability problems introduced by the output filters. In [33] the system robustness against load changes is explored through experimental results. For three-phase inverters, in [34] a controller for a VSC with a pure resistive load is designed using IDA, whereas [35] presents the design of a passivity-based controller for a three-phase inverter for the connection of a photo voltaic system to the grid. The control objectives are to maintain unity power factor and control of active power to fast track the maximum power of photo voltaic array. A new control strategy for grid connected FEC based on IDA is proposed in this paper. This strategy is aimed to inject all the available DC power into the grid, while controlling the reactive power exchanged with the power system. To this aim, the system is first represented as a Port-Controlled Hamiltonian (PCH) system. In this way, the coupling between the direct and quadrature currents, and the DC voltage dynamics can be clearly seen in the so-called interconnection matrix, while the damping matrix represents the natural damping of the system. The IDA technique was chosen because it is a method based on energy, which gives a physical interpretation to the controller and it also has the advantage of showing clearly the structure of the system and the couplings between the system dynamics. Then, interconnection matrix represents the energy flow inside the system, while damping matrix represents the dissipated energy. Thus, control objective can be expressed not only in terms of system stability, but also in terms of controller performance. The design of this strategy is based on the selection of an energy function for the closed-loop system so that it allows ensuring the system stability. In this way, the control laws for the system are obtained by solving the differential equation that results from the correct choice of interconnection and damping assignments. Different from previous proposals, the particular design proposed in this paper allows a direct control of the DC link voltage dynamics using the same approach (IDA). In addition, a dynamic extension of the system through a controller with integral action is proposed, which allows eliminating the steady state error produced by parameter variation. The performance of the proposed controller is validated through simulations performed using a realistic model of the system.
Front end converter model In this section, the FEC model in Park coordinates is first presented using the power invariant transformation. Then, this model is written in the PCH form. As it can be seen in Fig. 1, the FEC is composed by an electronic power converter with IGBT (Isolated Gate Bipolar Transitors), ðS1 . . . S6 Þ, and an output RL filter. The filter inductance value is selected with the aim of achieving the adequate attenuation of the switching frequencies in the output current. Therefore, in this work we adopted L ¼ 2:5 mH. Current is comes from a slowly variant power source, which models the generation system. This current can be obtained through the quotient between the power of this source and the DC bus voltage (v dc ). The electric grid is modeled using sinusoidal voltage sources ea ; eb and ec . Model in Park coordinates The FEC model in Park coordinates (dq) can be written as follows [36],
Fig. 1. Front end converter – FEC.
Li_d ¼ RL id xdq Liq þ md v dc ed ; Li_q ¼ RL iq þ xdq Lid þ mq v dc eq ; C v_ dc ¼ is md id mq iq ;
ð1Þ ð2Þ ð3Þ
where xdq is the angular speed of the dq reference frame, which is considered equal to the grid frequency in this work; id and iq are the currents in the selected reference frame, which are obtained through the transformation of the ia ; ib and ic currents; ed and eq are the grid voltages, obtained from transformations of ea ; eb and ec ; md and mq are the modulation indexes; L and RL are the filter inductance and resistance respectively, and C is the DC link capacitor. The system (1)–(3) can be written in matrix form as follows,
2
3 2 32 3 2 3 RL xdq L md ed id Li_d 6 _ 7 6 76 7 6 7 RL mq 54 iq 5 þ 4 eq 5: 4 Liq 5 ¼ 4 xdq L md mq 0 is v dc C v_ dc
ð4Þ
In order to design the controller for this system using the IDA technique, the system must be represented through its PCH model [20]. PCH model The PCH model of a dynamic system can be written as,
x_ ¼ ½JðxÞ RðxÞ
@HðxÞ þ gðxÞu þ f; @x
ð5Þ
where x is the state vector, u is the input vector, JðxÞ is the interconnection matrix, RðxÞ is the damping matrix, HðxÞ is the energy function of the system, gðxÞ is the input matrix and f is an external disturbance. In this work, the state vector is defined as follows:
x ¼ ½ x1
T x3 ¼ Lid
x2
Liq
C v dc
T
;
ð6Þ
and the input vector is,
mq T :
u ¼ ½ md
ð7Þ
The interconnection and damping matrices are defined from (4) as follows:
2
xdq L 0
0
6 J ¼ 4 xdq L 0 2
RL
0
3
0
7 0 5;
0
0
0
ð8Þ
3
6 R¼40
RL
7 0 5;
0
0
0
ð9Þ
where J is anti-symmetric and R is symmetric positive semi-definite, that is,
J ¼ JT
and R ¼ RT P 0:
ð10Þ
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Besides, for this system, matrices J and R do not depend on the states or the inputs. The input matrix is,
2
v dc
v dc 75:
ð11Þ
iq
The system energy function HðxÞ can be obtained as the sum of the energy stored in the filter inductances and in the DC link capacitor,
HðxÞ ¼
2 2 Liq C v 2dc 1 x21 x22 x23 Li ¼ dþ þ þ þ ; 2 L L C 2 2 2
ð12Þ
@HðxÞ ¼ id @x
iq
v dc
T
j13 7 j23 5; 0
0 j23
R1
0
6 Ra ¼ RTa ¼ 4 0
R2
0
0
0
ð20Þ
3
7 0 5: R3
ð21Þ
In order to ensure that the system states (x) tend to its reference value (x ) asymptotically, we may choose Hd ðxÞ as,
Hd ðxÞ ¼
1 T 1 ð P Þ; 2
ð22Þ
2
:
ð13Þ
3 L 0 0 6 7 P ¼ 4 0 L 0 5; 0 0 C
ð14Þ
with ¼ x x . The time derivative of Hd ðxÞ is,
Finally, the external disturbance f is given by
eq
is
T
;
where ed and eq are the measured grid voltages and is is the DC current which comes from the generation system. Controller design The FEC operating mode considered in this paper shows the main objective of injecting all the power available in the DC side (the generated power) to the grid, while controlling the reactive power exchanged with the power system. One way to perform this power control is by controlling the currents id and iq ; this is achieved by modifying the modulation indexes md and mq . The objective of injecting all the available DC power into the grid can be achieved by maintaining the DC link voltage constant. Furthermore, this voltage control can be performed with the id current control through the design method. Moreover, the reactive power control can be made by directly controlling the iq current. Therefore, the controller design has to ensure that the state variables of the system (5) reach the reference vector,
x ¼ Lid
j12
j13 2
3
where,
then,
f ¼ ed
0 6 Ja ¼ JTa ¼ 4 j12
3
0
6 gðxÞ ¼ 4 0 id
2
C v dc
T
H_d ðxÞ ¼ T P1 Rd P1 < 0;
2
0
½J R
0
7 0 5:
0
0
@Ha ðxÞ @Hd ðxÞ ¼ ½Ja Ra þ gðxÞu þ f; @x @x
Ha ðxÞ ¼ Hd ðxÞ HðxÞ:
ð16Þ
ð27Þ
RL id þ xdq Liq R1 ðid id Þ þ ed
v dc
;
ð28Þ
:
ð29Þ
RL iq xdq Lid R2 ðiq iq Þ þ eq
v dc
In addition, id can be obtained from (26) and (3) considering id ¼ id ; iq ¼ iq and eq ¼ 0,
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 1 4 ed ed D 2 ¼ þ 4iq 5; 2 RL RL RL
id
so that x be a minimum of Hd ðxÞ,
@ 2 Hd ðxÞ @x2
ð26Þ
The solution of (26) results in the control laws for md and mq ,
@Hd ðxÞ ; @x
ð25Þ
where,
mq ¼
@Hd ðxÞ ¼ 0; @x x¼x
3
Using (5) and (16), the following differential equation can be obtained,
with the desired dynamic response, while ensuring closed-loop stability. We propose to design a control law u, such that the dynamics of closed-loop system can be described by a PCH system with the form [20],
x_ ¼ ½Jd ðxÞ Rd ðxÞ
xdq L 0
0
6 Ja ¼ 4 xdq L
md ¼
:
ð24Þ
and Rd must be positive definite matrix. This can be achieved by properly choosing the elements of Ra . The elements of the matrix Ja can be chosen to cancel the existing coupling between the state variables,
ð15Þ
Liq
ð23Þ
ð30Þ
with,
> 0;
ð17Þ
x¼x
where Hd ðxÞ is the desired energy function for the system, and Jd ðxÞ and Rd ðxÞ are the desired interconnection and damping matrices, which can be written as,
Jd ðxÞ ¼ JðxÞ þ Ja ðxÞ;
ð18Þ
Rd ðxÞ ¼ RðxÞ þ Ra ðxÞ:
ð19Þ
Matrices Ja ðxÞ and Ra ðxÞ are used to synthesize the proposed control strategy. For this particular case these matrices can be written in generic form as follows:
D ¼ 4v dc is þ R3 v dc v dc :
ð31Þ
Then, the equations for the error dynamics are,
RL þ R1 id ; L RL þ R2 _ iq ¼ iq ; L R _ v dc ¼ 3 v dc : C
_ i
d
¼
ð32Þ ð33Þ ð34Þ
From which R1 ; R2 and R3 can be calculated so that to obtain the desired convergence speed.
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Integral action The control laws (28) and (29) are dependent on the parameters of the system. Therefore a steady-state error in the state variables may appear due to parameter variation. To cope with this problem an integral action is added to the proposed controller, by extending the system as proposed in [37,38]. The PCH model of the system with the output equation can be written as,
x_ ¼ ½JðxÞ RðxÞ
@HðxÞ þ gðxÞu þ f; @x
ð35Þ
@HðxÞ : @x
y ¼ gT ðxÞ
ð36Þ
The PCH model of the additional controller is,
u_ ¼ uca ;
ð37Þ
@Hca ðuÞ @Hca ðuÞ yca ¼ g ðuÞ ¼ ; @u @u
ð38Þ
T
where u is the state variable of the additional controller, uca is the input vector, yca is the output vector and Hca ðuÞ is the energy function of the integral controller. The energy function of the closed loop system with the additional controller is,
HT ðx; uÞ ¼ Hd ðxÞ þ Hca ðuÞ;
ð39Þ
and,
Hca ðuÞ ¼
1 T 1 u P u: 2
ð40Þ
The closed-loop system with the additional controller is a PCH system of the form,
2 3
@HT ðx;uÞ ½Jd ðxÞ Rd ðxÞ gðxÞ x_ @ðxÞ 4 5: ¼ @HT ðx;uÞ gT ðxÞ 0 u_ @ðuÞ
ð41Þ
Solving (41), the new control laws are obtained,
md ¼
RL id þ xdq Liq R1 ðid id Þ þ ed
v dc
mq ¼
id
ud ;
ð42Þ
uq :
ð43Þ
RL iq xdq Lid R2 ðiq iq Þ þ eq
v dc
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 1 4 ed ed D 2 ¼ þ 4iq 5; 2 RL RL RL
ð44Þ
objective of guaranteeing that all the available power on the DC link can be injected into the grid, id is calculated using (44) and iq is defined in terms of the reactive power to be injected into the grid. To achieve the control objectives mentioned above, the reference frame must be oriented on the d axis grid voltage, such that eq ¼ 0. In this work we used a phase locked loop (PLL) based on a generalized second-order integrator to achieve such synchronization with the grid voltage [39]. Fig. 2 shows a block diagram for the proposed control strategy. As it can be appreciated, the proposed control scheme requires measurement of the complete state vector and the disturbance vector. Even when those variables are can be easly measured, state observers [40,41] could be combined with the proposed controller in order to reduce the number of sensors, if desired.
Simulation results With the aim of evaluating the FEC behavior under different operating conditions, numerical simulations where performed using a complete converter model of the SimPowerSystems package, in Matlab™. This model considers the actual behaviour of the converter, including conduction and switching losses, modeled through the equivalent resistances and capacitances of the switches (IGBTs). In the simulated system, the input DC current, is , is obtained from a generation systems which extracts the energy from a – usually – slowly variant source. Then, the generation system is considered a power source PðtÞ such that,
PðtÞ ¼ is v dc ;
ð48Þ
from which is possible to obtain is from the DC link voltage (v dc ) and the generated power PðtÞ. The controller parameters used in the simulations are shown in Table 1 and the specifications of the FEC and the grid, are listed in Table 2. In Section ‘Input power change’, the system behavior under changes in the input power is shown. Section ‘Reactive power change’ shows the proposed control performance when a change in the reactive power reference is applied, by changing the quadra ture current component (iq ). Section ‘Change of the power flow direction’ shows the system performance in the face of a change in the power flow direction. Finally, Section ‘Change of system parameters’ shows the control performance under changes in the system parameters.
with,
D ¼ 4v dc is þ R3 v dc v dc ;
ð45Þ
where,
ud ¼ K 1 uq ¼ K 2
Z Z
v dc ðid id Þdt þ K 1 v dc ðiq iq Þdt þ K 2
Z Z
id ðv dc v dc Þdt;
ð46Þ
iq ðv dc v dc Þdt;
ð47Þ
and K 1 ; K 2 are the integral gains. In this way, the added integral action allows eliminating the steady state error produced by parameter mismatch and other disturbances, while the implemented method allows preserving the stability properties of the IDA design. Finally, the elements of the reference vector x are chosen according to the control objectives of the system. Then, the reference DC link voltage is set constant (v dc ¼ const:) with the
Fig. 2. Proposed control strategy.
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F.M. Serra et al. / Electrical Power and Energy Systems 60 (2014) 317–324 Table 1 Controller parameters. Parameter
Value
R1 R2 R3 K1 K2
2:5 2:5 0:2 0:5 0:5
Table 2 FEC and grid parameters Parameter
Value
v dc L RL C RonðIGBTÞ
780 V 2:5 mH 1 mX 1000 uF 1 mX
fs P ea ; eb ; ec f
10 kHz 30 kW 311 V 50 Hz
Input power change Fig. 3 shows: DC link voltage, v dc , (solid line) and reference voltage, v dc , (dashed line) in subfigure (a); currents id and iq and their references, id and iq , in subfigure (b); and phase a output voltage and current, ea and ia , in subfigure (c), for a change in the input power. The results were obtained for a test where the DC link voltage is set at (v dc ¼ 780 V) constant value, with unity power factor (iq ¼ 0 A). At t ¼ 50 ms the input power is reduced from P ¼ 30 kW to P ¼ 15 kW. As it can be appreciated in Fig. 3(a), the DC link voltage (v dc ) remains constant up to t ¼ 50 ms, where it decreases until the con-
trol establish the new value of id , needed to restore the power balance. The proposed control scheme guarantees the DC voltage regulation, thus, the id current is adjusted until this voltage reaches its reference value. Besides, it can be seen (Fig. 3(b)) that the change in the quadrature current does not modify the direct current, which implies that the two variables are decoupled, as established by the election of Ja . Fig. 3(c) shows the phase a system voltage and the corresponding output current, where it is evident the reduction in the current amplitude at t ¼ 50 ms, due to the change in the input power PðtÞ. As it can be appreciated, voltage and current remain in phase, since no reactive power is exchanged with the power system (iq ¼ 0). Besides, the output current is sinusoidal, without harmonics and distortions. Reactive power change Fig. 4 shows: DC link voltage, v dc , (solid line) and reference voltage, v dc , (dashed line) in subfigure (a); currents id and iq and their references, id and iq , in subfigure (b); and phase a output voltage and current, ea and ia , in subfigure (c), corresponding to a change in the reactive power reference. In this test the reference DC voltage is set at a constant value (v dc ¼ 780 V), with constant input power (P ¼ 30 kW). At t ¼ 50 ms the reactive power set point is modified from Q ¼ 0 kV Ar to Q ¼ 7:6 kV Ar. This change in the reactive power can be implemented by changing the reference of the quadrature current component iq . In this test, the system starts with iq ¼ 0, and at t ¼ 50 ms the ref erence iq is modified according to the following trajectory,
8 t > < 0 iq ðtÞ ¼ pðtÞ 50 ms < t > : 20 55 ms < t
< 50 ms < 55 ms
with,
Fig. 3. Input power change. (a) DC link voltage, (b) currents id and iq , and (c) grid voltage and output current of phase a
Fig. 4. Reactive power change. (a) DC link voltage, (b) currents id and iq , and (c) grid voltage and output current of phase a.
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pðtÞ ¼ c3 ðt t i Þ3 þ c4 ðt ti Þ4 þ c5 ðt t i Þ5 ; where pðtÞ is a polynomial whose derivative can be analytically obtained, and coefficients c3 ; c4 and c5 can be properly calculated to achieve a soft transition of iq with bounded derivative (in this work 10 c3 ¼ 2:5 10 ; c4 ¼ 1:875 1013 and c5 ¼ 3:75 1015 ). This polynomial is used as a reference in order to avoid control signals to reach values which may saturate the actuators. As it can be appreciated in Fig. 4(a), v dc remains constant up to t ¼ 50 ms, when the change in iq from 0 A to 20 A is produced. This change in the quadrature current produces a modification in id , making the system variables to couple again, as it can be observed from (30). However, the change in v dc and id remains between the admissible values. Besides, by comparing Fig. 3(a) and 4(a) it can be seen that the reduction of v dc and id is much lower in this last test. Fig. 4(c) shows the phase a voltage and current, where it is evident the change of phase of the current at t ¼ 50 ms due to the change of iq . Finally, the same as in the previous test, the waveform of the output current is sinusoidal, without significant distortions.
Change of the power flow direction Fig. 5 shows: DC link voltage, v dc , (solid line) and voltage reference, v dc , (dashed line) in subfigure (a); currents id and iq and their references, id and iq , in subfigure (b) and the system voltage and output current corresponding to phase a; ea and ia in subfigure (c), for a test in which the power flow direction changes. In this case the system, which was working as an inverter, begins to works as a rectifier. The DC link voltage is set at a constant value (v dc ¼ 780 V), with unity power factor (iq ¼ 0 A), and at t ¼ 50 ms the input power changes from P ¼ 30 kW to P ¼ 15 kW.
Fig. 6. Change of system parameters. (a) DC link voltage and (b) currents id and iq .
As it can be seen in Fig. 5(a), the DC link voltage remains constant up to the instant in which the power flow direction changes. At this time, the control acts so as to reestablish the power balance, making v dc to maintain in the reference value. Besides, it can be appreciated that the change in the direct current does not modify the quadrature current, thus the system remains decoupled. Finally, Fig. 5(c) shows the phase a system voltage and the correspondent phase output current, where it is evident the change in amplitude and the inversion of its phase at t ¼ 50 ms, due to the change in the power flow direction. It is also observed that the output current is sinusoidal, the same as in the previous tests. Change of system parameters Fig. 6 shows: DC link voltage, v dc , (solid line) and voltage reference, v dc , (dashed line) in subfigure (a) and currents id and iq with its references, id and iq in subfigure (b) when the value of the filter resistance has increased by 50% of the value considered in the controller and the value of the filter inductance is increased by 10%. In the same figure results are shown with (dark solid line) and without (gray solid line) using the proposed integral action. The results were obtained for a test where the DC link voltage is set at (v dc ¼ 780 V) constant value, with unity power factor (iq ¼ 0 A). At t ¼ 50 ms the input power is reduced from P ¼ 30 kW to P ¼ 15 kW. As it can be seen, the DC link voltage and the quadrature axis current shows a steady-state error when the integral action is not implemented (gray solid line). Using the controller with the integral action, the steady-state error in the DC link voltage and the quadrature axis current is fully eliminated. Discussion and conclusions
Fig. 5. Power flow direction change. (a) DC link voltage, (b) currents id and iq , and (c) grid voltage and output current of phase a.
A control strategy for a front end converter using the interconnection and damping assignment technique was proposed in this paper. The controller was designed in order to allow the FEC to inject all the available DC power to the grid, while controlling the exchanged reactive power. IDA design method has numerous advantages, not only from the design procedure but also regarding controller performance. Regarding design procedure, IDA is a systematic methodology, which ensures the system stability being able to obtain a decoupled control and facilitating the determination of controller
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parameters when compared with conventional strategies. The controller design, following the steps of the IDA strategy, is an organized design by matrix equations, where each matrix has a well-defined physical meaning. In this way the designer can modify the structure of the system to simplify the problem of having more control variables than control actions. Besides, in this particular case, this design allows performing a direct control of the DC link voltage, which, up to the best authors knowledge, is an innovative proposal not found in the related bibliography. An integral action is added to the proposed controller using the same design technique, which allows eliminating the steady state error produced by parameter uncertainities. Regarding controller performance, the proposed method provides a better dynamic response when compared with other well-known strategies (PI controllers in dq reference frame). The improved performance is due to IDA allows designing a nonlinear controller for this nonlinear problem, while allows decoupling d and q axes. Moreover, the obtained closed loop error dynamics results linear, thus allowing determining the controller gains in terms of the desired performance. Besides the conventional PI controller in dq coordinates, more complex strategies such as feedback linearization can be designed for this DC–AC converter. Controller design through feedback linearization allows obtaining very similar (or even the same) control laws as those obtained by IDA design [42]. However, in this case the proposed design using IDA has the advantages mentioned above regarding design procedure, when compared with feedback linearization. Four different realistic simulation tests were performed. The first one corresponds to a change in the input DC power, the second one to a change in the reactive power set point, the third one is a change in the power flow direction and the fourth one is a change of system parameters. In the first test, when the input power changes, the direct current is modified so as to guarantee the power balance, regulating the DC link voltage in its reference value. Besides, this change in the direct current does not modify significantly the quadrature current, thus meeting the decoupling objective. In the second test, it has been observed that even when the change in the quadrature axis current reference occurs again the coupling of both currents, variation of direct axis current and DC link voltage remains within acceptable limits. Also, the quadrature current follows accurately the desired reference. For the third test, when the power flow direction changes, direct axis current is modified, while DC link voltage remains in its reference value after a short transient. It is observed that the output current undergoes a change of phase of 180°, due to the system starts to consume energy. For the fourth test, when the system parameters change, the direct axis current remains with its reference value, while the steady-state error of the DC link voltage and the quadrature axis current are eliminated through the integral action added to the proposed controller. In the performed tests, it was observed that the proposed strategy allows to achieve the objectives of injecting all the available DC power to the grid, while controlling the reactive power exchanged with the power system. Besides, the waveform of the output current is sinusoidal, without harmonics nor distortions for all the tests. It must be mentioned that when grid voltage is distorted, (30) or (44) can be modified using the fundamental component of the positive sequence grid voltage instead of the measured grid voltage. This allows the FEC to inject sinusoidal currents according to standards. However, it must be noted that an oscillating power will be injected into the grid. In order to implement such modification into the proposed controller, PLL of Fig. 2 must be replaced by a po-
323
sitive sequence detector (see [39]) which allows obtaining frequency, amplitude and phase of the fundamental grid voltage component. Acknowledgment This work was supported by the Universidad Nacional de Río Cuarto, by FONCyT-ANPCyT and by the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). References [1] El-Kholy EE, El-Sabbe A, El-Hefnawy A, Mharous HM. Three-phase active power filter based on current controlled voltage source inverter. Int J Electr Power Energy Syst 2006;28(8):537–47. [2] Ghedamsi K, Aouzellag D. Improvement of the performances for wind energy conversions systems. Int J Electr Power Energy Syst 2010;32(9):936–45. [3] Kamatchi Kannan V, Rengarajan N. Photovoltaic based distribution static compensator for power quality improvement. Int J Electr Power Energy Syst 2012;42(1):685–92. [4] Ge B, Wang W, Bi D, Rogers CB, Peng FZ, de Almeida AT, et al. Energy storage system-based power control for grid-connected wind power farm. Int J Electr Power Energy Syst 2013;44(1):115–22. [5] Yazdani A, Iravani R. Voltage sourced converters in power systems. Wiley-IEEE Press; 2010. [6] Bahrman MP, Johansson JG, Nilsson BA. Voltage source converter transmission technologies: the right fit for the application. In: IEEE, Power engineering society general meeting, 13–17 July, 2003. [7] Alsayegh O, Alhajraf S, Albusairi H. Grid-connected renewable energy source systems: challenges and proposed management schemes. Energy Convers Manage 2010;51(8):1690–3. [8] Leão RPS, Barroso GC, Sampaio RF, Almada JB, Lima CFP, Rego MCO, et al. The future of low voltage networks: moving from passive to active. Int J Electr Power Energy Syst 2011;33(8):1506–12. [9] Iov F, Ciobotaru M, Blaabjerg F. Power electronics control of wind energy in distributed power systems. In: IEEE int. conf. on optimization of electrical and electronic equipment (OPTIM), 22–24 May 2008. p. XXIX–XLIV. [10] Rekik Mouna, Abdelkafi Achraf, Krichen Lotfi. A novel control strategy of a distributed generator operating in seven modes for ancillary services under grid faults. Int J Electr Power Energy Syst 2013;47:100–8. [11] Arboleya P, Diaz D, Guerrero JM, Garcia P, Briz F, Gonzalez-Moran C, et al. An improved control scheme based in droop characteristic for microgrid converters. Electr Power Syst Res 2010;80(10):1215–21. [12] Huang Xinhong, Wang Zhutian, Jiang Jin. Control and load-dispatching strategies for a microgrid with a DC/AC inverter of fixed frequency. Int J Electr Power Energy Syst 2012;43(1):1127–36. [13] IEEE standard for interconnecting distributed resources with electric power systems. IEEE Std 1547-2003, 2003. p. 1–16. [14] Lu Bin, Ooi Boon-Teck. Nonlinear control of voltage-source converter systems. IEEE Trans Power Electron 2007;22(4):1186–95. [15] Kim Dong-Eok, Lee Dong-Choon. Feedback linearization control of three-phase ups inverter systems. IEEE Trans Ind Electron 2010;57(3):963–8. [16] Liu F, Mei S, Lu Q, Ni Y, Wu FF, Yokoyama A. The nonlinear internal control of STATCOM: theory and application. Int J Electr Power Energy Syst 2003;25(6):421–30. [17] Shtessel Y, Baev S, Biglari H. Unity power factor control in three-phase AC/DC boost converter using sliding modes. IEEE Trans Ind Electron 2008;55(11):3874–82. [18] Lee Tzann-Shin. Lagrangian modeling and passivity-based control of threephase AC/DC voltage-source converters. IEEE Trans Ind Electron 2004;51(4):892–902. [19] Cecati C, DellAquila A, Liserre M, Monopoli VG. A passivity-based multilevel active rectifier with adaptive compensation for traction applications. IEEE Trans Ind Appl 2003;30(5):1404–13. [20] Ortega R, van der Schaft A, Maschke B, Escobar G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. AUTOMATICA 2002;38(4):585–96. [21] He B, Zhang X, Zhao X. Transient stabilization of structure preserving power systems with excitation control via energy-shaping. Int J Electr Power Energy Syst 2007;29(10):822–30. [22] Ramirez-Arredondo JM, Davalos-Marin R. TCSC control based on passivity for power system damping enhancement. Int J Electr Power Energy Syst 2001;23(2):81–90. [23] De Leon-Morales J, Espinosa-Pérez G, Macias-Cardoso I. Observer-based control of a synchronous generator: a Hamiltonian approach. Int J Electr Power Energy Syst 2002;24(8):655–63. [24] Ramirez JM, Valencia Arroyave F, Correa Gutierrez RE. Transient stability improvement by nonlinear controllers based on tracking. Int J Electr Power Energy Syst 2011;33(2):315–21. [25] Pengfei Wang, Jiuhe Wang, Zuohua Xu, Passivity-based control of three phase voltage source PWM rectifiers based on PCHD model. In: Int. conf. on electrical machines and systems (ICEMS), 17–20 October 2008. p. 1126–30.
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Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Interconnection and damping assignment control of a three-phase front end converter Federico M. Serra ⇑, Cristian H. De Angelo, Daniel G. Forchetti Grupo de Electrónica Aplicada (GEA), UNRC, Río Cuarto, Córdoba, Argentina
a r t i c l e
i n f o
Article history: Received 19 October 2012 Received in revised form 24 November 2013 Accepted 20 March 2014 Available online 16 April 2014 Keywords: Front end converter Nonlinear control Interconnection and damping assignment Passivity based control
a b s t r a c t A new nonlinear control strategy for a three-phase front end converter used to connect renewable energy sources to the grid is proposed in this paper. The controller is designed in order to inject all the generated power into the grid, while the reactive power can be controlled to meet the power system requirements. The system is represented through its port controlled Hamiltonian model, and the controller is designed by interconnection and damping assignment. This design method allows an intuitive way to remove the undesired couplings between system dynamics while assigning the damping required to achieve the expected convergence rate. The proposed controller allows a direct control of the DC link voltage by proper selection of the controller parameters. Moreover, an integral action is added to the proposed controller in order to eliminate the steady-state error in the system variables. The proposal is validated through simulation tests performed using a realistic converter model. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction In the lasts years, electronic power converters have become important components in electrical power systems, mainly due to issues related to power quality [1–4]. Even when different converter configurations have been proposed, Voltage Source Converters (VSC) are the most used in energy conversion systems due to its bidirectional power flow capability and its low harmonic distortion [5,6]. Such power converters have allowed the increase of renewable energy sources integration into the grid [7,8]. In those applications, the VSC can be used as a Front End Converter (FEC) for wind energy or photo voltaic generation systems, where the operating mode depend on the application [9,3,10]. In the case of grid connected generation units in strong power systems, voltage and frequency are imposed by the grid, so the FEC must be synchronized and inject all the available DC power to the grid. Besides, it must be able to control the reactive power exchanged with the power system. For weak power systems or micro-grids, droop control is usually used to control the active and reactive powers that the FEC exchange with the grid, thus contributing to grid stability [11,12,10]. In order to fulfill this requirements, it is necessary that the FEC be able to control the waveform of the injected currents. Due to the actual requirements to connect power converters to the grid [13], precise control strategies are needed. Such strategies ⇑ Corresponding author. Tel.: +54 265715543445. E-mail address: [email protected] (F.M. Serra). http://dx.doi.org/10.1016/j.ijepes.2014.03.033 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
require, in several cases, a complex design including nonlinear control techniques [14]. The most used design strategies are Feedback Linearization (FL) [15,16], Sliding Mode Control (SMC) [17], and Passivity Based Control (PBC) [18,19], among others. Interconnection and Damping Assignment (IDA) is a PBC technique [20] which has been widely used to design controllers for electric power systems stabilization and control [21–24]. More specifically, for controlling VSC this technique has been utilized in rectifiers and inverters. An IDA controller for a three-phase rectifier is presented in [25] in order to obtain unity power factor, sinusoidal phase currents and constant DC output voltage. Simulation results show that the controller is stable and robust against load variations. The controller proposed in [26] is designed with similar objectives, but an integral control loop is added in order to eliminate the steady state error in the DC output. In [27] an IDA controller is designed to control the DC output even in cases of variable load. It is also shown that the controller is robust against input voltage changes. An IDA controller for a three-phase rectifier with LCL input filter is designed in [28], which is also robust to grid impedance variations. For single-phase rectifiers, an IDA controller is designed in [29], which allows controlling the DC output while obtaining unity power factor, with a reduced number of sensors. IDA is used in [30] to design a controller for a bidirectional power flow single-phase rectifier. In [31] this strategy is used to control the DC voltage in case of complex loads, considering bidirectional power flow, too.
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Proposals to control VSC used in single-phase inverters using IDA can be found in [32,33]. In [32] the controller is designed to control a grid connected inverter, allowing to solve the instability problems introduced by the output filters. In [33] the system robustness against load changes is explored through experimental results. For three-phase inverters, in [34] a controller for a VSC with a pure resistive load is designed using IDA, whereas [35] presents the design of a passivity-based controller for a three-phase inverter for the connection of a photo voltaic system to the grid. The control objectives are to maintain unity power factor and control of active power to fast track the maximum power of photo voltaic array. A new control strategy for grid connected FEC based on IDA is proposed in this paper. This strategy is aimed to inject all the available DC power into the grid, while controlling the reactive power exchanged with the power system. To this aim, the system is first represented as a Port-Controlled Hamiltonian (PCH) system. In this way, the coupling between the direct and quadrature currents, and the DC voltage dynamics can be clearly seen in the so-called interconnection matrix, while the damping matrix represents the natural damping of the system. The IDA technique was chosen because it is a method based on energy, which gives a physical interpretation to the controller and it also has the advantage of showing clearly the structure of the system and the couplings between the system dynamics. Then, interconnection matrix represents the energy flow inside the system, while damping matrix represents the dissipated energy. Thus, control objective can be expressed not only in terms of system stability, but also in terms of controller performance. The design of this strategy is based on the selection of an energy function for the closed-loop system so that it allows ensuring the system stability. In this way, the control laws for the system are obtained by solving the differential equation that results from the correct choice of interconnection and damping assignments. Different from previous proposals, the particular design proposed in this paper allows a direct control of the DC link voltage dynamics using the same approach (IDA). In addition, a dynamic extension of the system through a controller with integral action is proposed, which allows eliminating the steady state error produced by parameter variation. The performance of the proposed controller is validated through simulations performed using a realistic model of the system.
Front end converter model In this section, the FEC model in Park coordinates is first presented using the power invariant transformation. Then, this model is written in the PCH form. As it can be seen in Fig. 1, the FEC is composed by an electronic power converter with IGBT (Isolated Gate Bipolar Transitors), ðS1 . . . S6 Þ, and an output RL filter. The filter inductance value is selected with the aim of achieving the adequate attenuation of the switching frequencies in the output current. Therefore, in this work we adopted L ¼ 2:5 mH. Current is comes from a slowly variant power source, which models the generation system. This current can be obtained through the quotient between the power of this source and the DC bus voltage (v dc ). The electric grid is modeled using sinusoidal voltage sources ea ; eb and ec . Model in Park coordinates The FEC model in Park coordinates (dq) can be written as follows [36],
Fig. 1. Front end converter – FEC.
Li_d ¼ RL id xdq Liq þ md v dc ed ; Li_q ¼ RL iq þ xdq Lid þ mq v dc eq ; C v_ dc ¼ is md id mq iq ;
ð1Þ ð2Þ ð3Þ
where xdq is the angular speed of the dq reference frame, which is considered equal to the grid frequency in this work; id and iq are the currents in the selected reference frame, which are obtained through the transformation of the ia ; ib and ic currents; ed and eq are the grid voltages, obtained from transformations of ea ; eb and ec ; md and mq are the modulation indexes; L and RL are the filter inductance and resistance respectively, and C is the DC link capacitor. The system (1)–(3) can be written in matrix form as follows,
2
3 2 32 3 2 3 RL xdq L md ed id Li_d 6 _ 7 6 76 7 6 7 RL mq 54 iq 5 þ 4 eq 5: 4 Liq 5 ¼ 4 xdq L md mq 0 is v dc C v_ dc
ð4Þ
In order to design the controller for this system using the IDA technique, the system must be represented through its PCH model [20]. PCH model The PCH model of a dynamic system can be written as,
x_ ¼ ½JðxÞ RðxÞ
@HðxÞ þ gðxÞu þ f; @x
ð5Þ
where x is the state vector, u is the input vector, JðxÞ is the interconnection matrix, RðxÞ is the damping matrix, HðxÞ is the energy function of the system, gðxÞ is the input matrix and f is an external disturbance. In this work, the state vector is defined as follows:
x ¼ ½ x1
T x3 ¼ Lid
x2
Liq
C v dc
T
;
ð6Þ
and the input vector is,
mq T :
u ¼ ½ md
ð7Þ
The interconnection and damping matrices are defined from (4) as follows:
2
xdq L 0
0
6 J ¼ 4 xdq L 0 2
RL
0
3
0
7 0 5;
0
0
0
ð8Þ
3
6 R¼40
RL
7 0 5;
0
0
0
ð9Þ
where J is anti-symmetric and R is symmetric positive semi-definite, that is,
J ¼ JT
and R ¼ RT P 0:
ð10Þ
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Besides, for this system, matrices J and R do not depend on the states or the inputs. The input matrix is,
2
v dc
v dc 75:
ð11Þ
iq
The system energy function HðxÞ can be obtained as the sum of the energy stored in the filter inductances and in the DC link capacitor,
HðxÞ ¼
2 2 Liq C v 2dc 1 x21 x22 x23 Li ¼ dþ þ þ þ ; 2 L L C 2 2 2
ð12Þ
@HðxÞ ¼ id @x
iq
v dc
T
j13 7 j23 5; 0
0 j23
R1
0
6 Ra ¼ RTa ¼ 4 0
R2
0
0
0
ð20Þ
3
7 0 5: R3
ð21Þ
In order to ensure that the system states (x) tend to its reference value (x ) asymptotically, we may choose Hd ðxÞ as,
Hd ðxÞ ¼
1 T 1 ð P Þ; 2
ð22Þ
2
:
ð13Þ
3 L 0 0 6 7 P ¼ 4 0 L 0 5; 0 0 C
ð14Þ
with ¼ x x . The time derivative of Hd ðxÞ is,
Finally, the external disturbance f is given by
eq
is
T
;
where ed and eq are the measured grid voltages and is is the DC current which comes from the generation system. Controller design The FEC operating mode considered in this paper shows the main objective of injecting all the power available in the DC side (the generated power) to the grid, while controlling the reactive power exchanged with the power system. One way to perform this power control is by controlling the currents id and iq ; this is achieved by modifying the modulation indexes md and mq . The objective of injecting all the available DC power into the grid can be achieved by maintaining the DC link voltage constant. Furthermore, this voltage control can be performed with the id current control through the design method. Moreover, the reactive power control can be made by directly controlling the iq current. Therefore, the controller design has to ensure that the state variables of the system (5) reach the reference vector,
x ¼ Lid
j12
j13 2
3
where,
then,
f ¼ ed
0 6 Ja ¼ JTa ¼ 4 j12
3
0
6 gðxÞ ¼ 4 0 id
2
C v dc
T
H_d ðxÞ ¼ T P1 Rd P1 < 0;
2
0
½J R
0
7 0 5:
0
0
@Ha ðxÞ @Hd ðxÞ ¼ ½Ja Ra þ gðxÞu þ f; @x @x
Ha ðxÞ ¼ Hd ðxÞ HðxÞ:
ð16Þ
ð27Þ
RL id þ xdq Liq R1 ðid id Þ þ ed
v dc
;
ð28Þ
:
ð29Þ
RL iq xdq Lid R2 ðiq iq Þ þ eq
v dc
In addition, id can be obtained from (26) and (3) considering id ¼ id ; iq ¼ iq and eq ¼ 0,
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 1 4 ed ed D 2 ¼ þ 4iq 5; 2 RL RL RL
id
so that x be a minimum of Hd ðxÞ,
@ 2 Hd ðxÞ @x2
ð26Þ
The solution of (26) results in the control laws for md and mq ,
@Hd ðxÞ ; @x
ð25Þ
where,
mq ¼
@Hd ðxÞ ¼ 0; @x x¼x
3
Using (5) and (16), the following differential equation can be obtained,
with the desired dynamic response, while ensuring closed-loop stability. We propose to design a control law u, such that the dynamics of closed-loop system can be described by a PCH system with the form [20],
x_ ¼ ½Jd ðxÞ Rd ðxÞ
xdq L 0
0
6 Ja ¼ 4 xdq L
md ¼
:
ð24Þ
and Rd must be positive definite matrix. This can be achieved by properly choosing the elements of Ra . The elements of the matrix Ja can be chosen to cancel the existing coupling between the state variables,
ð15Þ
Liq
ð23Þ
ð30Þ
with,
> 0;
ð17Þ
x¼x
where Hd ðxÞ is the desired energy function for the system, and Jd ðxÞ and Rd ðxÞ are the desired interconnection and damping matrices, which can be written as,
Jd ðxÞ ¼ JðxÞ þ Ja ðxÞ;
ð18Þ
Rd ðxÞ ¼ RðxÞ þ Ra ðxÞ:
ð19Þ
Matrices Ja ðxÞ and Ra ðxÞ are used to synthesize the proposed control strategy. For this particular case these matrices can be written in generic form as follows:
D ¼ 4v dc is þ R3 v dc v dc :
ð31Þ
Then, the equations for the error dynamics are,
RL þ R1 id ; L RL þ R2 _ iq ¼ iq ; L R _ v dc ¼ 3 v dc : C
_ i
d
¼
ð32Þ ð33Þ ð34Þ
From which R1 ; R2 and R3 can be calculated so that to obtain the desired convergence speed.
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Integral action The control laws (28) and (29) are dependent on the parameters of the system. Therefore a steady-state error in the state variables may appear due to parameter variation. To cope with this problem an integral action is added to the proposed controller, by extending the system as proposed in [37,38]. The PCH model of the system with the output equation can be written as,
x_ ¼ ½JðxÞ RðxÞ
@HðxÞ þ gðxÞu þ f; @x
ð35Þ
@HðxÞ : @x
y ¼ gT ðxÞ
ð36Þ
The PCH model of the additional controller is,
u_ ¼ uca ;
ð37Þ
@Hca ðuÞ @Hca ðuÞ yca ¼ g ðuÞ ¼ ; @u @u
ð38Þ
T
where u is the state variable of the additional controller, uca is the input vector, yca is the output vector and Hca ðuÞ is the energy function of the integral controller. The energy function of the closed loop system with the additional controller is,
HT ðx; uÞ ¼ Hd ðxÞ þ Hca ðuÞ;
ð39Þ
and,
Hca ðuÞ ¼
1 T 1 u P u: 2
ð40Þ
The closed-loop system with the additional controller is a PCH system of the form,
2 3
@HT ðx;uÞ ½Jd ðxÞ Rd ðxÞ gðxÞ x_ @ðxÞ 4 5: ¼ @HT ðx;uÞ gT ðxÞ 0 u_ @ðuÞ
ð41Þ
Solving (41), the new control laws are obtained,
md ¼
RL id þ xdq Liq R1 ðid id Þ þ ed
v dc
mq ¼
id
ud ;
ð42Þ
uq :
ð43Þ
RL iq xdq Lid R2 ðiq iq Þ þ eq
v dc
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 1 4 ed ed D 2 ¼ þ 4iq 5; 2 RL RL RL
ð44Þ
objective of guaranteeing that all the available power on the DC link can be injected into the grid, id is calculated using (44) and iq is defined in terms of the reactive power to be injected into the grid. To achieve the control objectives mentioned above, the reference frame must be oriented on the d axis grid voltage, such that eq ¼ 0. In this work we used a phase locked loop (PLL) based on a generalized second-order integrator to achieve such synchronization with the grid voltage [39]. Fig. 2 shows a block diagram for the proposed control strategy. As it can be appreciated, the proposed control scheme requires measurement of the complete state vector and the disturbance vector. Even when those variables are can be easly measured, state observers [40,41] could be combined with the proposed controller in order to reduce the number of sensors, if desired.
Simulation results With the aim of evaluating the FEC behavior under different operating conditions, numerical simulations where performed using a complete converter model of the SimPowerSystems package, in Matlab™. This model considers the actual behaviour of the converter, including conduction and switching losses, modeled through the equivalent resistances and capacitances of the switches (IGBTs). In the simulated system, the input DC current, is , is obtained from a generation systems which extracts the energy from a – usually – slowly variant source. Then, the generation system is considered a power source PðtÞ such that,
PðtÞ ¼ is v dc ;
ð48Þ
from which is possible to obtain is from the DC link voltage (v dc ) and the generated power PðtÞ. The controller parameters used in the simulations are shown in Table 1 and the specifications of the FEC and the grid, are listed in Table 2. In Section ‘Input power change’, the system behavior under changes in the input power is shown. Section ‘Reactive power change’ shows the proposed control performance when a change in the reactive power reference is applied, by changing the quadra ture current component (iq ). Section ‘Change of the power flow direction’ shows the system performance in the face of a change in the power flow direction. Finally, Section ‘Change of system parameters’ shows the control performance under changes in the system parameters.
with,
D ¼ 4v dc is þ R3 v dc v dc ;
ð45Þ
where,
ud ¼ K 1 uq ¼ K 2
Z Z
v dc ðid id Þdt þ K 1 v dc ðiq iq Þdt þ K 2
Z Z
id ðv dc v dc Þdt;
ð46Þ
iq ðv dc v dc Þdt;
ð47Þ
and K 1 ; K 2 are the integral gains. In this way, the added integral action allows eliminating the steady state error produced by parameter mismatch and other disturbances, while the implemented method allows preserving the stability properties of the IDA design. Finally, the elements of the reference vector x are chosen according to the control objectives of the system. Then, the reference DC link voltage is set constant (v dc ¼ const:) with the
Fig. 2. Proposed control strategy.
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F.M. Serra et al. / Electrical Power and Energy Systems 60 (2014) 317–324 Table 1 Controller parameters. Parameter
Value
R1 R2 R3 K1 K2
2:5 2:5 0:2 0:5 0:5
Table 2 FEC and grid parameters Parameter
Value
v dc L RL C RonðIGBTÞ
780 V 2:5 mH 1 mX 1000 uF 1 mX
fs P ea ; eb ; ec f
10 kHz 30 kW 311 V 50 Hz
Input power change Fig. 3 shows: DC link voltage, v dc , (solid line) and reference voltage, v dc , (dashed line) in subfigure (a); currents id and iq and their references, id and iq , in subfigure (b); and phase a output voltage and current, ea and ia , in subfigure (c), for a change in the input power. The results were obtained for a test where the DC link voltage is set at (v dc ¼ 780 V) constant value, with unity power factor (iq ¼ 0 A). At t ¼ 50 ms the input power is reduced from P ¼ 30 kW to P ¼ 15 kW. As it can be appreciated in Fig. 3(a), the DC link voltage (v dc ) remains constant up to t ¼ 50 ms, where it decreases until the con-
trol establish the new value of id , needed to restore the power balance. The proposed control scheme guarantees the DC voltage regulation, thus, the id current is adjusted until this voltage reaches its reference value. Besides, it can be seen (Fig. 3(b)) that the change in the quadrature current does not modify the direct current, which implies that the two variables are decoupled, as established by the election of Ja . Fig. 3(c) shows the phase a system voltage and the corresponding output current, where it is evident the reduction in the current amplitude at t ¼ 50 ms, due to the change in the input power PðtÞ. As it can be appreciated, voltage and current remain in phase, since no reactive power is exchanged with the power system (iq ¼ 0). Besides, the output current is sinusoidal, without harmonics and distortions. Reactive power change Fig. 4 shows: DC link voltage, v dc , (solid line) and reference voltage, v dc , (dashed line) in subfigure (a); currents id and iq and their references, id and iq , in subfigure (b); and phase a output voltage and current, ea and ia , in subfigure (c), corresponding to a change in the reactive power reference. In this test the reference DC voltage is set at a constant value (v dc ¼ 780 V), with constant input power (P ¼ 30 kW). At t ¼ 50 ms the reactive power set point is modified from Q ¼ 0 kV Ar to Q ¼ 7:6 kV Ar. This change in the reactive power can be implemented by changing the reference of the quadrature current component iq . In this test, the system starts with iq ¼ 0, and at t ¼ 50 ms the ref erence iq is modified according to the following trajectory,
8 t > < 0 iq ðtÞ ¼ pðtÞ 50 ms < t > : 20 55 ms < t
< 50 ms < 55 ms
with,
Fig. 3. Input power change. (a) DC link voltage, (b) currents id and iq , and (c) grid voltage and output current of phase a
Fig. 4. Reactive power change. (a) DC link voltage, (b) currents id and iq , and (c) grid voltage and output current of phase a.
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pðtÞ ¼ c3 ðt t i Þ3 þ c4 ðt ti Þ4 þ c5 ðt t i Þ5 ; where pðtÞ is a polynomial whose derivative can be analytically obtained, and coefficients c3 ; c4 and c5 can be properly calculated to achieve a soft transition of iq with bounded derivative (in this work 10 c3 ¼ 2:5 10 ; c4 ¼ 1:875 1013 and c5 ¼ 3:75 1015 ). This polynomial is used as a reference in order to avoid control signals to reach values which may saturate the actuators. As it can be appreciated in Fig. 4(a), v dc remains constant up to t ¼ 50 ms, when the change in iq from 0 A to 20 A is produced. This change in the quadrature current produces a modification in id , making the system variables to couple again, as it can be observed from (30). However, the change in v dc and id remains between the admissible values. Besides, by comparing Fig. 3(a) and 4(a) it can be seen that the reduction of v dc and id is much lower in this last test. Fig. 4(c) shows the phase a voltage and current, where it is evident the change of phase of the current at t ¼ 50 ms due to the change of iq . Finally, the same as in the previous test, the waveform of the output current is sinusoidal, without significant distortions.
Change of the power flow direction Fig. 5 shows: DC link voltage, v dc , (solid line) and voltage reference, v dc , (dashed line) in subfigure (a); currents id and iq and their references, id and iq , in subfigure (b) and the system voltage and output current corresponding to phase a; ea and ia in subfigure (c), for a test in which the power flow direction changes. In this case the system, which was working as an inverter, begins to works as a rectifier. The DC link voltage is set at a constant value (v dc ¼ 780 V), with unity power factor (iq ¼ 0 A), and at t ¼ 50 ms the input power changes from P ¼ 30 kW to P ¼ 15 kW.
Fig. 6. Change of system parameters. (a) DC link voltage and (b) currents id and iq .
As it can be seen in Fig. 5(a), the DC link voltage remains constant up to the instant in which the power flow direction changes. At this time, the control acts so as to reestablish the power balance, making v dc to maintain in the reference value. Besides, it can be appreciated that the change in the direct current does not modify the quadrature current, thus the system remains decoupled. Finally, Fig. 5(c) shows the phase a system voltage and the correspondent phase output current, where it is evident the change in amplitude and the inversion of its phase at t ¼ 50 ms, due to the change in the power flow direction. It is also observed that the output current is sinusoidal, the same as in the previous tests. Change of system parameters Fig. 6 shows: DC link voltage, v dc , (solid line) and voltage reference, v dc , (dashed line) in subfigure (a) and currents id and iq with its references, id and iq in subfigure (b) when the value of the filter resistance has increased by 50% of the value considered in the controller and the value of the filter inductance is increased by 10%. In the same figure results are shown with (dark solid line) and without (gray solid line) using the proposed integral action. The results were obtained for a test where the DC link voltage is set at (v dc ¼ 780 V) constant value, with unity power factor (iq ¼ 0 A). At t ¼ 50 ms the input power is reduced from P ¼ 30 kW to P ¼ 15 kW. As it can be seen, the DC link voltage and the quadrature axis current shows a steady-state error when the integral action is not implemented (gray solid line). Using the controller with the integral action, the steady-state error in the DC link voltage and the quadrature axis current is fully eliminated. Discussion and conclusions
Fig. 5. Power flow direction change. (a) DC link voltage, (b) currents id and iq , and (c) grid voltage and output current of phase a.
A control strategy for a front end converter using the interconnection and damping assignment technique was proposed in this paper. The controller was designed in order to allow the FEC to inject all the available DC power to the grid, while controlling the exchanged reactive power. IDA design method has numerous advantages, not only from the design procedure but also regarding controller performance. Regarding design procedure, IDA is a systematic methodology, which ensures the system stability being able to obtain a decoupled control and facilitating the determination of controller
F.M. Serra et al. / Electrical Power and Energy Systems 60 (2014) 317–324
parameters when compared with conventional strategies. The controller design, following the steps of the IDA strategy, is an organized design by matrix equations, where each matrix has a well-defined physical meaning. In this way the designer can modify the structure of the system to simplify the problem of having more control variables than control actions. Besides, in this particular case, this design allows performing a direct control of the DC link voltage, which, up to the best authors knowledge, is an innovative proposal not found in the related bibliography. An integral action is added to the proposed controller using the same design technique, which allows eliminating the steady state error produced by parameter uncertainities. Regarding controller performance, the proposed method provides a better dynamic response when compared with other well-known strategies (PI controllers in dq reference frame). The improved performance is due to IDA allows designing a nonlinear controller for this nonlinear problem, while allows decoupling d and q axes. Moreover, the obtained closed loop error dynamics results linear, thus allowing determining the controller gains in terms of the desired performance. Besides the conventional PI controller in dq coordinates, more complex strategies such as feedback linearization can be designed for this DC–AC converter. Controller design through feedback linearization allows obtaining very similar (or even the same) control laws as those obtained by IDA design [42]. However, in this case the proposed design using IDA has the advantages mentioned above regarding design procedure, when compared with feedback linearization. Four different realistic simulation tests were performed. The first one corresponds to a change in the input DC power, the second one to a change in the reactive power set point, the third one is a change in the power flow direction and the fourth one is a change of system parameters. In the first test, when the input power changes, the direct current is modified so as to guarantee the power balance, regulating the DC link voltage in its reference value. Besides, this change in the direct current does not modify significantly the quadrature current, thus meeting the decoupling objective. In the second test, it has been observed that even when the change in the quadrature axis current reference occurs again the coupling of both currents, variation of direct axis current and DC link voltage remains within acceptable limits. Also, the quadrature current follows accurately the desired reference. For the third test, when the power flow direction changes, direct axis current is modified, while DC link voltage remains in its reference value after a short transient. It is observed that the output current undergoes a change of phase of 180°, due to the system starts to consume energy. For the fourth test, when the system parameters change, the direct axis current remains with its reference value, while the steady-state error of the DC link voltage and the quadrature axis current are eliminated through the integral action added to the proposed controller. In the performed tests, it was observed that the proposed strategy allows to achieve the objectives of injecting all the available DC power to the grid, while controlling the reactive power exchanged with the power system. Besides, the waveform of the output current is sinusoidal, without harmonics nor distortions for all the tests. It must be mentioned that when grid voltage is distorted, (30) or (44) can be modified using the fundamental component of the positive sequence grid voltage instead of the measured grid voltage. This allows the FEC to inject sinusoidal currents according to standards. However, it must be noted that an oscillating power will be injected into the grid. In order to implement such modification into the proposed controller, PLL of Fig. 2 must be replaced by a po-
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