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Pole-placement control of 4-leg voltage-source inverters for standalone photovoltaic systems: Considering digital delays
Renewable Energy 36 (2011) 858e865
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Technical Note
Pole-placement control of 4-leg voltage-source inverters for standalone photovoltaic systems: Considering digital delays Reza Nasiri*, Ahmad Radan Power Electronics Laboratory, Electrical Engineering Faculty, K.N. Toosi University of Technology, Seyed Khandan Bridge, 1431714191 Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 December 2009 Accepted 4 August 2010
Three leg inverters for photovoltaic systems have a lot of disadvantages, especially when the load is unbalanced. These disadvantages are for example, small utilization of the DC link voltage, the dependency of the modulation factor of the load current and the superposition of a DC component with the output AC voltage. A solution for these problems is the four-leg inverter. Most papers dealing with 4-leg inverters ignore the effect of digital delays in control loop and suggest classic controllers, such as PI controller. However, the transient performance of the closed-loop system does not become, accurately, adjustable and acceptable, in most cases. This paper proposes a novel model, considering digital delays, for 4-leg inverters. Then, it applies pole-placement control strategy, via state feedback, to adjust the transient performance of the closed-loop system to a desired second-order system. Simulation results validate theoretical results and proposed control strategy. 2010 Elsevier Ltd. All rights reserved.
Keywords: Photovoltaic systems 4-Leg inverters State feedback Digital delays Pole-placement control strategy
1. Introduction The electrical energy gained from a photovoltaic (PV) system can be utilized in different ways. There are two kinds of the employment of solar modules, standalone and grid coupled systems. Standalone photovoltaic system [1e10] means that the system, composed of load and generator, is closed and locally limited. The energy supply of a ship or a space chattel is gained from a standalone network. Grid coupled system [10e20] means that there is a supra-regional energy supply network and the generated energy, for example, from a PV system will be injected in this network. Both operation modes must satisfy different requirements on the energy supply. In the standalone system, the problem to solve is to ensure the energy supply. As the solar energy supply works only, if there is sun radiation, the gained energy must be stored. Photovoltaic generators can only produce DC currents and therefore only DC loads can be supplied. To supply costumer AC loads an inverter unit must be applied. The inverter, which represents the interface between the photovoltaic generator and AC loads, is the main subject of this paper. Its task in context with standalone PV systems is to ensure the energy supply in threephase standalone network.
* Corresponding author. Tel.: þ98 21 84062408; fax: þ98 21 88462066. E-mail addresses: [email protected] (R. Nasiri), [email protected] (A. Radan). 0960-1481/$ e see front matter 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2010.08.002
The proposed standalone PV system is illustrated in Fig. 1. It consists of the PV generator, DC/DC1 converter, a battery energy storage, a second DC/DC2 converter, DC link capacitor bank, inverter, output filter and load. It can be seen that the proposed system is transformer less. For this reason a high DC link voltage is needed which is supplied with boost converter (DC/DC1). DC/DC2 is also a boost converter and acts as a maximum power point tracker (MPPT) and battery charger. In this way the battery will be always charged at the maximum power point. The goal of the system illustrated in Fig. 1 is to supply three as well as single phase loads of any kind with constant amplitude sinusoidal voltage and constant frequency. To realize this, all three phases must be independent of each other. For this reason the neutral point of the LC output filter and load must be connected to a neutral point. There are a lot of possibilities to realize this. One possibility, which is the four-leg inverter, is shown in Fig. 2. Because of additional neutral leg, 4-leg inverters are recommended for supplying unbalanced and/or nonlinear loads. Consequently, generation of balanced voltage, with sinusoidal waveform, is necessary for these inverters. As a result, using an appropriate load voltage controller is highly required. To apply control techniques a DC operating point is needed. Thus, the transformation T, given in Appendix, is applied to the model of 4-leg inverter in abc coordinates to get the power stage model in rotating coordinates dqo [21e25]. Most papers dealing with 4-leg inverters propose classic control techniques, such as PI controller, for the closed-loop system in rotating coordinates dqo [21e23]. But it has some drawbacks:
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2. Four-leg inverters and their large signal model
Fig. 1. Standalone photovoltaic system.
Digital implementation of the controller leads to a digital delay in control loop [25]. However, almost all of the control strategies in the literatures neglect to consider the digital delays, even papers which deal with digital control of the 4-leg inverter [23]. Most of the Classical control techniques use frequency domain factors, such as gain margin and phase margin, for designing. However, it does not lead to a precise response in time domain. As a result, classic control techniques cannot adjust transient performance of the system, such as overshoot and speed of response, accurately. However, modern control techniques can be used to this aim [25]. This paper solves all of the above problems as below: 1. It proposes a novel model considering digital delays in control loop to consider the effect of delay in model of the 4-leg inverter. 2. It proposes pole-placement control strategy via state feedback, which is a modern control technique, to make the transient performance of the system, greatly, adjustable. In fact, this paper demonstrates that by using the proposed controller the 4-leg inverter generates balanced voltage in spite of the presence RL loads. Additionally, the desired transient performance of the system is guaranteed. After this introduction, in Section 2, large signal model of the 4-leg inverter in rotating coordinates qdo is presented. Next, the proposed model considering delay, control structure and state equations is presented. Then, in Section 4, a pole-placement control strategy via state feedback is introduced and discussed. In Section 5, some simulation results of the 4-leg inverter using this control strategy are presented and discussed. Finally, in Section 6 some conclusions are pointed out. It should be noted that proposed control strategy can be used, similarly, for multilevel 4-leg inverters and other applications of the four-leg structure, such as active power filters and DVR(dynamic voltage resistor), but it is beyond the scope of this paper.
Fig. 2 shows the power stage model of a three-phase 4-leg voltage-source inverter with second-order load filter. The average model in abc stationary coordinates, without considering the delay, is shown in Fig. 3. The output voltages and input current of the inverter can be represented as:
Vaf
Ip ¼
Vbf
Vcf
daf
dbf
T
¼
daf
dcf $½ ia
dbf
dcf
ib
ic T
T
$Vdc
(1) (2)
T
where [Vaf,Vbf,Vcf] is the inverter output voltages vector (lineneutral), [ia,ib,ic]T is line currents vector and, daf,dbf,dcf are line-toneutral duty ratios. The duty ratios din (i ¼ a,b,c) are controlled in a way so as to produce sinusoidal voltages at output of the filter, irrespective of the load. The system requirement can be expressed as:
3 3 2 VAG cosðutÞ 4 VBG 5 ¼ Vm 4 cosðut 120 Þ 5 cosðut þ 120 Þ VCG 2
(3)
where [VAG,VBG,VCG]T is voltage across the load and Vm is its amplitude. To produce the desired sinusoidal load voltages, the steady-state duty ratios have to be time-varying and sinusoidal in case of linear loads. But, to apply proposed control techniques we need a DC operating point. Thus, we apply the transformation T, given in Appendix, to get the power stage model in rotating coordinates dqo. Fig. 4 gives the power stage model in rotating coordinates and it is redrawn as a Signal Flow Graph in Fig. 5. Note that the load is assumed to be resistive-inductive RL. The d and q sub-circuits have coupled voltage and current sources. The steady-state output load voltages are DC quantities, in rotating coordinates and are given as:
3 3 2 Vd Vm 4 Vq 5 ¼ 4 0 5 0 Vo 2
(4)
where Vm is the rated output voltage. For balanced RL Load we have
Rd ¼ Rq ¼ Ro ¼ Ra ¼ Rb ¼ Rc ¼ R Ld ¼ Lq ¼ Lo ¼ La ¼ Lb ¼ Lc ¼ L0
(5)
where (Ra,Rb,Rc) are resistances of the load in phases (a, b, c) and (Rd,Rq,Ro) are equivalent resistances of the load in rotating coordinate dqo, R is resistance and L0 is the inductance of the load.
Fig. 2. Four-leg voltage-source inverter.
Fig. 3. Average large signal model of the inverter with LC filter.
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scheme. Finally, communication with other power processing equipment or with user interfaced equipment is most easily achieved through digital controls. While digital control may enable many simplifications for control design, it brings about the element of delay in the control loop. DSPs used in control of power electronics circuits require a finite time delay for accomplishing the functions of sampling analog data, calculation, and updating digital and analog outputs. An ideal time delay can be represented as the frequency domain function,
Hdelay ðsÞ ¼ eTs
Fig. 4. Power stage average model in rotating coordinates: The d and q sub-circuits have coupled voltage and current sources.
3. Proposed model considering digital delay 3.1. Digital delays In general, feedback control for inverters is accomplished through algorithms in digital processors. There are several reasons for this. First, control for inverters is usually performed in the rotating dqo reference frame, as mentioned in previous section. This requires coordinate transformations that would be extraordinarily difficult to accomplish in analog devices, but are fairly simple to perform in Digital Signal Processors (DSPs). Second, safety and protection functions are easily implemented in a digital control
(6)
where T is the delay time. This function has unity magnitude for all frequencies and a linearly increasing phase-lag with frequency. A delay in control loop has an obvious destabilizing effect on feedback systems [22]. Because of the destabilizing effects of delay, it becomes very important to model the delay accurately when doing control design and analysis. Fig. 6 shows the location in the control loop where the delay is usually modeled. The output sampling, subtraction from reference, and compensation are all performed within the DSP. All of the delay associated with the sampling and calculation are generally lumped into one delay block at the interface between the DSP and the continuous system plant. While the delay associated with digital control may not be a constant for various reasons, it is important to put bounds on the length of the delay, so that the control can be designed around these limits. While there may be several different strategies for digital control implementation, the simplest would be to sample output variables, calculate duty cycles, and update the Pulse Width Modulation (PWM) registers all in one switching cycle. This strategy leads to a delay of between one and two switching periods [22]. 3.2. Proposed model As discussed in previous subsection digital implementation of the controller and/or discrete operation of modulation and inverter introduce a digital delay which is time-variant. The delay results in instability of the system, in some cases [22], and must be taken into account. We assume that the delay is constant. Therefore, it is estimated by:
1 eST z 1 þ ST
(7)
where T is the delay time. Let us model the estimated delay time after the gain Vg in Fig. 5, it has an important advantage which will describe in next subsection. On the other hand, the power stage is a coupled multi-variable multi-loop system. Here, we neglect coupling and present independently control of the system. Fig. 7 shows the proposed control structure for load voltage loop control.
Fig. 5. Average model represented as a signal flow graph: the load is represented as a balanced resistive-inductive (RL).
Fig. 6. Basic system model including ideal time delay.
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Fig. 7. Average model represented as signal flow graph (couplings are neglected).
3.3. Realization of proposed model We can realize the Signal Flow Graph of the inverter, shown in Fig. 7, as Fig. 8. Consequently, we can express the state-space equations of the system, for d channel, as:
x_ ¼ A$xðtÞ þ B$uðtÞy ¼ C$xðtÞ
dq ¼ 0 do ¼ 0
(10)
Thus, from now only the state equations of d channel are expressed and a controller for this channel will be developed in the following sections.
(8) 4. Pole-placement control via state feedback
where
2
3 id 6 Vd 7 7 x ¼ 6 4 iLd 5; VTd
2
C ¼ ½0
1
0
0 ;
3 0 607 7 B ¼ 6 405
Let the controller has the linear state feedback form
u ¼ K$x þ G$r
Vg T
where r ˛ R is a new vector with 1 input and K, state feedback, and G, static lead compensator, are the unknown controller matrices with dimensions 1 3 and 1 1, respectively (Fig. 9). Substituting Eq. (11) in Eq. (8) yields the closed-loop system
And
2
0 6 1 6 C A ¼ 6 40 1 T
1 L
0
1 Ld
1 C Rd Ld
0
0
0
1 L
0 0
3 7 7 7 5
x_ ¼ ðA BKÞx þ BGry ¼ Cx
1 T
It should be noted, by using above state equations, we consider the delay as a resistant and a capacitor in the model. As a result, it introduces a new state variable, which is the capacitor voltage VTd. Fortunately, the voltage of the capacitor VTd is equal to the voltage of the filter inductance. Consequently, this state variable is accessible and known, like other state variables. This feature is so important because proposed model does not need the observer, despite adding an unreal state variable. Because of system requirements in d and q channels, we have:
Vq ¼ 0 Vo ¼ 0 Therefore:
(11)
(9)
(12)
The control problem is to determine the control law (11), to determine the controller matrices K and G, such that the closedloop system has the desired characteristics.
K ¼ ½ K11
K12
K13
K14 ;
G ¼ ½G
4.1. Pole placement via state feedback Consider the linear, time-invariant system
_ xðtÞ ¼ AxðtÞ þ BuðtÞ
(13)
where all states are accessible and known. To this system we apply a linear state feedback control law of the form
uðtÞ ¼ KxðtÞ
(14)
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Fig. 8. Average model of the four-leg inverter; neglecting coupling and realizing digital delay as a first order system with RT ¼ 1U and CT ¼ T.
2
0 6 60 h i Fc ¼ B«AB«A2 B«A3 B ¼ 6 60 6 4
Then, the closed-loop system is given by the homogeneous equation (15)
_ xðtÞ ¼ ðA BFÞxðtÞ
(15)
Vg T
Now, the design problem is to find the appropriate controller matrix so as to improve the performance of the closed-loop system Eq. (15). One such method of improving the performance of (15) is that of pole placement. The pole-placement method consists in finding a particular matrix K, such that the poles of the closed-loop (15) take on desirable preassigned values. Using this method, the behavior of the open-loop system may be improved significantly. For example, the method can stabilize an unstable system, increase or decrease the speed of response etc. For this reasons, improving the system performance via the pole-placement method is widely recommended. The pole-placement or eigenvalue assignment problem can be defined as follows: let l1,l2,l3,l4 be the eigenvalues of the matrix A l 1 ; bl 2 ; bl 3 ; bl 4 be the desired of the open-loop system (8) and b eigenvalues of the matrix A BK of the closed-loop system (15), where all complex eigenvalues exist in complex conjugate pairs. Also, let p(s) and b p ðsÞ be the respective characteristic polynomials. i.e., let us find a matrix K so that Eq. (17) is satisfied.
pðsÞ ¼
4 Y
b p ðsÞ ¼
4 Y
li sb
2
1 60 j ¼ 6 40 0 a2
(19)
a3 1 0 0
a2 a3 1 0
a1
a0 ;
3 a1 a2 7 7; a3 5 1
a ¼ ½ a3 a2 a1 a0 ;
K ¼ ½ K11
K12
K13
K14
By substituting A,B and Fc in (19), state feedback matrix K can be expressed as:
LT Vg
0
LC LCTa3 Vg Vg Ld ðTþC ðða23 a2 ÞLVg La3 þ1ÞÞ Vg
LCT Vg LCLd Vg
0
0
0
3T
7 0 7 7 7 LCLd 7 5 0
(20)
4.2. Input reference tracking via static lead compensator
¼ jsI A þ BFj
¼ s þ a3 s þ a2 s þ a1 s þ a0 3
ð2$TLÞ L$T 3
where Fc is the controllability matrix, defined in Eq. (18), and
a ¼ ½ a3
i¼1 4
0 Vg Vg T 3 L$T 2
K ¼ ða aÞj1 F1 c
(16)
K ¼ ða
V T 2g
It is evident that rank(Fc) ¼ 4, for Vg s 0, L s T. Therefore, the system (A,B) is controllable, a fact which guarantees that there exists a matrix K which satisfies the pole-placement problem. Several methods have been proposed for determining K. One of the most popular pole-placement methods is the Base-Gura Formula which gives the following simple solution:
ðs li Þ ¼ jsI Aj ¼ s4 þ a3 s3 þ a2 s2 þ a1 s þ a0
LTa3 L Vg Vg 2 6 LCT ða3 a2 Þ 3 6 þ CLCa 6 Vg Vg aÞ6 3 2 6 Ld ðTa3 þða3 2a3 a2 þC ÞLCVg LCa3 þCa3 þLCa2 1Þ 4 Vg T Vg
0 0
ðT 2 þC$TL$C Þ 3 L2 $C$T 2 7 1 7 L$C$T 7 7 1 7 L$C$Ld 5 (18)
i¼1
2
Vg L$T 2 Vg L$C$T
Vg L$T
2
(17)
The controllability matrix Fc of system (18) can be expressed as:
The poles of system can be assigned in every place by using state feedback method, presented in previous subsection. However, the system does not track the reference input. To modify the performance of the system a static lead compensator, i.e. the matrix G, can
R. Nasiri, A. Radan / Renewable Energy 36 (2011) 858e865
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Fig. 9. Pole-placement control strategy via state feedback and static lead compensator.
be used. To guarantee the tracking performance of system (12) we have
lim yðtÞ ¼ r
(21)
t/N
On the other hand when t / N, (12) is given by
_ xðNÞ ¼ 0 ¼ ðA BKÞxðNÞ þ BGryðNÞ ¼ CxðNÞ
(22)
By substituting (21) in (22), G can be expressed as:
G ¼ CðA BKÞ1 B
(23)
We can get G for our case, by substituting A,B,C and K, as follow
G ¼
LCT a0 Ld Vg R d
(24)
Fig. 10 shows diagram of the pole-placement control strategy. This system has poles, assigned by K vector, and the output load voltage tracks the reference input.
5. Simulation results and discussion In this section, the performance of the proposed control strategy is investigated by a computer simulation using SIMULINK-MATLAB software. The system parameters specifications used in the simulation are as follows:
L ¼ 333 mH C ¼ 100 mH Rd ¼ Rq ¼ Ro ¼ 100U Ld ¼ Lq ¼ Lo ¼ 200 mH Vg ¼ 800 V Vm ¼ 400 V T ¼ 100 ms
Fig. 11. Time response of the proposed second-order system and desired fourth-order systems when the input is unit step.
5.1. The respective characteristic polynomial of the open-loop system By substituting system’s parameters in (16), the respective characteristic polynomial of the desired system can be expressed as:
pðsÞ ¼ s4 þ a3 s3 þ a2 s2 þ a1 s þ a0 where
a3 a2 a1 a0
¼ ¼ ¼ ¼
510; 000 5:11006 109 3:08303 1013 1:516516 1017
Therefore, the open-loop system’s eigenvalues are given by
l1 l2 l3 l4
¼ ¼ ¼ ¼
1354 þ 6261j 1354 6261j 7391 50; 000
5.2. The desired characteristic polynomial A desired respective characteristic polynomial for proposed model has 4 poles, or 4 eigenvalues as below:
And, according to [21], the best PI controller parameters are given as below:
Kp ¼ 0:07;
Ki ¼ 1:2 105
Fig. 10. Structure of proposed pole-placement control strategy via state feedback and static lead compensator.
Fig. 12. Time response of the closed-loop system with proposed control strategy.
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Fig. 13. Comparison of PI controller and proposed controller (a) Vd (b) voltage of phase a Va.
b l1 b l2 b l3 b l 4
¼ ¼ ¼ ¼
3836 3913j 3836 þ 3913j 50; 000 50; 000
l 1 and bl 2 ). Two assigned poles are assumed to be complex (b Therefore, we can consider them, together, as a second-order system. To have an acceptable transient performance (reasonable overshoot, and very high speed of response), a second-order system with z ¼ 0.69 and un ¼ 5480, (25), is ideal. G2 ¼
u2n 54802 ¼ 2 s2 þ 2zun þ u2n s þ 2 0:69 5480 þ 54802
¼
54802 l l sb sb 1
(25)
2
l 3 and bl 4 ) are assumed to be real and Two other assigned poles (b be located far from the second-order system’s poles. Therefore, we can ignore their effect on the final, fourth-order, system. In other words, two other assigned poles, which are located ins ¼ 50,000, guarantee that the behavior of the resulted, fourth-order, system (26), is matched to the behavior of the second-order system (25). In fact, they are non-dominant poles of system (26). G4 ¼
¼
G4 ¼
l3 sb
b l 3 bl 4 u2n l 4 s2 þ 2 z un þ u2n sb
50; 0002 54802 ðs 50; 000Þ s2 þ 2 :69 5480 þ 54802
(26)
50; 0002 54802 s4 þ a3 s3 þ a2 s2 þ a1 s þ a0
(27)
2
(28)
where
a3 a2 a1 a0
¼ ¼ ¼ ¼
5.3. Time response of the closed-loop system with proposed controller for step unit input Fig. 12 depicts the time response of the closed-loop system with proposed control strategy. As expected, the output voltage tracks the reference input (Vd-reference ¼ Vm ¼ 400 V) in steady-state conditions. Additionally, the transient performance of the system, such as overshoot and speed of response, is matched with the desired fourth-order system (27) or second-order system (25), shown in Fig. 11, presented in previous subsection. Consequently, this simulation result validates the theoretical results. 5.4. Comparison between the performance of the proposed controller and PI controller Fig. 13(a) depicts the response of the system with proposed pole-placement control strategy and with Conventional PI controller in a picture, for comparison. The PI controller has a very slow response. It tracks the input after 3 cycles (0.09 s) while it has the best parameters(Kp ¼ 0.07, Ki ¼ 1.2 105). However, the system with state feedback tracks the input less than 0.0015 s. Fig. 13(b) shows the equivalent load voltage in phase a. 6. Conclusion
Thus, the respective characteristic polynomial of the desired system can be expressed as:
b p ðsÞ ¼ s4 þ a3 s3 þ a2 s2 þ a1 s þ a0
step input. It can be seen that the time response of the fourth-order system is matched with the second one. It should be noted that the systems, (25) and (26), have an overshoot not larger than 5% and a time of response less than 0.0015 s. This system (26) behaves as our desired (designed) system. Thus, its respective characteristic polynomial (28) is chosen as the vector a, see (17), in designing procedure.
1007672 257702756150 1:9482160658648 1015 7:5090094293545 1018
Fig. 11 shows the time response of the proposed second-order system (25) and the desired fourth-order system (26) for the unit
Modern control theory, which uses state equations for controller design, offers precise control in time domain. However, it has not yet been noticed for 4-leg inverter’s controller design issue. This paper proposes it as a suitable approach in the 4-leg inverter case. Moreover, it demonstrates unique advantages of this approach. In conclusion, the following can be made. This paper proposes a novel model, considering delay of the digital implementation, for 4-Leg voltage-source inverters. It uses state feedback matrix for pole placement of the system. Consequently, it can stabilize the system, if the system is unstable. Moreover, we can adjust the overshoot and the speed of response to any desired values. In addition, it guarantees the tracking of the reference input by using static lead compensator.
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It should be noted that this paper considers the digital delay as a first order system with a resistant and a capacitor in the model. As a result, it introduces a new state variable, which is the capacitor voltage. Fortunately, the voltage of the capacitor is equal to the voltage of the output filter inductance. Consequently, this state variable is accessible and known, like other state variables. This feature is very valuable because proposed model does not need the observer, despite adding an unreal state variable. As simulation results show, this approach has unique responses in both transient and steady states conditions. In comparison with other control strategies, which have been proposed for 4-Leg inverters, like PI controllers, this approach is very suitable for implementation. This approach provides a precise control in time domain, like the speed and overshoot of the response, which cannot be achieved by designing in frequency domain. Appendix Transformation matrix T from abc to dqo
2
Vd
3
2
Va
3
6 7 6 7 4 Vq 5 ¼ T 4 Vb 5 Vo
2
Vc CosðutÞ
6 T ¼ 234 SinðutÞ 1 2
Cosðut 120 Þ Sinðut 120 Þ 1 2
Cosðut þ 120 Þ
3
Sinðut þ 120 Þ 7 5 1 2
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[6] Kolhe M, Agbossou K, Hamelin J, Bose TK. Analytical model for predicting the performance of photovoltaic array coupled with a wind turbine in a standalone renewable energy system based on hydrogen. Renewable Energy 2003;28:727e42. [7] Hrayshat ES. Brackish water desalination by a stand alone reverse osmosis desalination unit powered by photovoltaic solar energy. Renewable Energy 2008;33:1784e90. [8] Celik AN. Effect of different load profiles on the loss-of-load probability of stand-alone photovoltaic systems. Renewable Energy 2007;32:2096e115. [9] Mellit A, Kalogirou SA, Shaari S, Salhi H, Arab AH. Methodology for predicting sequences of mean monthly clearness index and daily solar radiation data in remote areas: application for sizing a stand-alone PV system. Renewable Energy 2008;33:1570e90. [10] Li C-H, Zhu X-J, Cao G-Y, Sui S, Hu M-R. Dynamic modeling and sizing optimization of stand-alone photovoltaic power systems using hybrid energy storage technology. Renewable Energy 2009;34:815e26. [11] Notton G, Lazarov V, Stoyanov L. Optimal sizing of a grid-connected PV system for various PV module technologies and inclinations inverter efficiency characteristics and locations. Renewable Energy 2010;35:541e54. [12] Nofuentes G, Almonacid G. An approach to the selection of the inverter for architecturally integrated photovoltaic grid-connected systems. Renewable Energy 1998;15:487e90. [13] Rahim NA, Selvaraj J, Krismadinata C. Five-level inverter with dual reference modulation technique for grid-connected PV system. Renewable Energy 2010;35:712e20. [14] Hassaine L, Olias E, Quintero J, Haddadi M. Digital power factor control and reactive power regulation for grid-connected photovoltaic inverter. Renewable Energy 2009;34:315e21. [15] Salas V, Alonso-Abella M, Chenlo F, Olias E. Analysis of the maximum power point tracking in the photovoltaic grid inverters of 5 kW. Renewable Energy 2009;34:2366e72. [16] Calais M, Agelidis VG, Dymond MS. A cascaded inverter for transformer less single-phase grid-connected photovoltaic systems. Renewable Energy 2001;22:255e62. [17] Gounden NA, Peter SA, Nallandula H, Krithiga S. Fuzzy logic controller with MPPT using line-commutated inverter for three-phase grid-connected photovoltaic systems. Renewable Energy 2009;34:909e15. [18] Islam S, Woyte A, Belmans R, Heskes PJM, Rooij PM. Investigating performance reliability and safety parameters of photovoltaic module inverter: test results and compliances with the standards. Renewable Energy 2006;31:1157e81. [19] Hamrouni N, Jraidi M, Cherif A. New control strategy for 2-stage grid-connected photovoltaic power system. Renewable Energy 2008;33:2212e21. [20] Mondol JD, Yohanis YG, Norton B. The impact of array inclination and orientation on the performance of a grid-connected photovoltaic system. Renewable Energy 2007;32:118e40. [21] Thandi GS, Zhang R, Zhing K, Lee FC, Boroyevich D. Modeling, control and stability analysis of a PEBB based DC DPS. IEEE Transactions on Power Delivery 1999;14:497e505. [22] Gannet RA. Control strategies for high power four-leg voltage-source inverters. Dissertation, Virginia Tech; 2001. [23] Elbarbari S, Hofman W. Digital control of a four-leg inverter for standalone photovoltaic systems with unbalanced load. In: Proc. IEEE-IECON ’00 conf.; 2000. p. 729e34. [24] Nasiri R, Radan A. Pole -placement control strategy for 4 leg voltage source inverters. In: Proc. IEEE-PEDSTC ’10 conf.; 2010. p. 74e9. [25] Hiti S, Boroyevich D. Small-signal modeling of three-phase PWM modulators. In: Proc. IEEE PES conf.; 1996. p. 550e5.
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Technical Note
Pole-placement control of 4-leg voltage-source inverters for standalone photovoltaic systems: Considering digital delays Reza Nasiri*, Ahmad Radan Power Electronics Laboratory, Electrical Engineering Faculty, K.N. Toosi University of Technology, Seyed Khandan Bridge, 1431714191 Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 December 2009 Accepted 4 August 2010
Three leg inverters for photovoltaic systems have a lot of disadvantages, especially when the load is unbalanced. These disadvantages are for example, small utilization of the DC link voltage, the dependency of the modulation factor of the load current and the superposition of a DC component with the output AC voltage. A solution for these problems is the four-leg inverter. Most papers dealing with 4-leg inverters ignore the effect of digital delays in control loop and suggest classic controllers, such as PI controller. However, the transient performance of the closed-loop system does not become, accurately, adjustable and acceptable, in most cases. This paper proposes a novel model, considering digital delays, for 4-leg inverters. Then, it applies pole-placement control strategy, via state feedback, to adjust the transient performance of the closed-loop system to a desired second-order system. Simulation results validate theoretical results and proposed control strategy. 2010 Elsevier Ltd. All rights reserved.
Keywords: Photovoltaic systems 4-Leg inverters State feedback Digital delays Pole-placement control strategy
1. Introduction The electrical energy gained from a photovoltaic (PV) system can be utilized in different ways. There are two kinds of the employment of solar modules, standalone and grid coupled systems. Standalone photovoltaic system [1e10] means that the system, composed of load and generator, is closed and locally limited. The energy supply of a ship or a space chattel is gained from a standalone network. Grid coupled system [10e20] means that there is a supra-regional energy supply network and the generated energy, for example, from a PV system will be injected in this network. Both operation modes must satisfy different requirements on the energy supply. In the standalone system, the problem to solve is to ensure the energy supply. As the solar energy supply works only, if there is sun radiation, the gained energy must be stored. Photovoltaic generators can only produce DC currents and therefore only DC loads can be supplied. To supply costumer AC loads an inverter unit must be applied. The inverter, which represents the interface between the photovoltaic generator and AC loads, is the main subject of this paper. Its task in context with standalone PV systems is to ensure the energy supply in threephase standalone network.
* Corresponding author. Tel.: þ98 21 84062408; fax: þ98 21 88462066. E-mail addresses: [email protected] (R. Nasiri), [email protected] (A. Radan). 0960-1481/$ e see front matter 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2010.08.002
The proposed standalone PV system is illustrated in Fig. 1. It consists of the PV generator, DC/DC1 converter, a battery energy storage, a second DC/DC2 converter, DC link capacitor bank, inverter, output filter and load. It can be seen that the proposed system is transformer less. For this reason a high DC link voltage is needed which is supplied with boost converter (DC/DC1). DC/DC2 is also a boost converter and acts as a maximum power point tracker (MPPT) and battery charger. In this way the battery will be always charged at the maximum power point. The goal of the system illustrated in Fig. 1 is to supply three as well as single phase loads of any kind with constant amplitude sinusoidal voltage and constant frequency. To realize this, all three phases must be independent of each other. For this reason the neutral point of the LC output filter and load must be connected to a neutral point. There are a lot of possibilities to realize this. One possibility, which is the four-leg inverter, is shown in Fig. 2. Because of additional neutral leg, 4-leg inverters are recommended for supplying unbalanced and/or nonlinear loads. Consequently, generation of balanced voltage, with sinusoidal waveform, is necessary for these inverters. As a result, using an appropriate load voltage controller is highly required. To apply control techniques a DC operating point is needed. Thus, the transformation T, given in Appendix, is applied to the model of 4-leg inverter in abc coordinates to get the power stage model in rotating coordinates dqo [21e25]. Most papers dealing with 4-leg inverters propose classic control techniques, such as PI controller, for the closed-loop system in rotating coordinates dqo [21e23]. But it has some drawbacks:
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2. Four-leg inverters and their large signal model
Fig. 1. Standalone photovoltaic system.
Digital implementation of the controller leads to a digital delay in control loop [25]. However, almost all of the control strategies in the literatures neglect to consider the digital delays, even papers which deal with digital control of the 4-leg inverter [23]. Most of the Classical control techniques use frequency domain factors, such as gain margin and phase margin, for designing. However, it does not lead to a precise response in time domain. As a result, classic control techniques cannot adjust transient performance of the system, such as overshoot and speed of response, accurately. However, modern control techniques can be used to this aim [25]. This paper solves all of the above problems as below: 1. It proposes a novel model considering digital delays in control loop to consider the effect of delay in model of the 4-leg inverter. 2. It proposes pole-placement control strategy via state feedback, which is a modern control technique, to make the transient performance of the system, greatly, adjustable. In fact, this paper demonstrates that by using the proposed controller the 4-leg inverter generates balanced voltage in spite of the presence RL loads. Additionally, the desired transient performance of the system is guaranteed. After this introduction, in Section 2, large signal model of the 4-leg inverter in rotating coordinates qdo is presented. Next, the proposed model considering delay, control structure and state equations is presented. Then, in Section 4, a pole-placement control strategy via state feedback is introduced and discussed. In Section 5, some simulation results of the 4-leg inverter using this control strategy are presented and discussed. Finally, in Section 6 some conclusions are pointed out. It should be noted that proposed control strategy can be used, similarly, for multilevel 4-leg inverters and other applications of the four-leg structure, such as active power filters and DVR(dynamic voltage resistor), but it is beyond the scope of this paper.
Fig. 2 shows the power stage model of a three-phase 4-leg voltage-source inverter with second-order load filter. The average model in abc stationary coordinates, without considering the delay, is shown in Fig. 3. The output voltages and input current of the inverter can be represented as:
Vaf
Ip ¼
Vbf
Vcf
daf
dbf
T
¼
daf
dcf $½ ia
dbf
dcf
ib
ic T
T
$Vdc
(1) (2)
T
where [Vaf,Vbf,Vcf] is the inverter output voltages vector (lineneutral), [ia,ib,ic]T is line currents vector and, daf,dbf,dcf are line-toneutral duty ratios. The duty ratios din (i ¼ a,b,c) are controlled in a way so as to produce sinusoidal voltages at output of the filter, irrespective of the load. The system requirement can be expressed as:
3 3 2 VAG cosðutÞ 4 VBG 5 ¼ Vm 4 cosðut 120 Þ 5 cosðut þ 120 Þ VCG 2
(3)
where [VAG,VBG,VCG]T is voltage across the load and Vm is its amplitude. To produce the desired sinusoidal load voltages, the steady-state duty ratios have to be time-varying and sinusoidal in case of linear loads. But, to apply proposed control techniques we need a DC operating point. Thus, we apply the transformation T, given in Appendix, to get the power stage model in rotating coordinates dqo. Fig. 4 gives the power stage model in rotating coordinates and it is redrawn as a Signal Flow Graph in Fig. 5. Note that the load is assumed to be resistive-inductive RL. The d and q sub-circuits have coupled voltage and current sources. The steady-state output load voltages are DC quantities, in rotating coordinates and are given as:
3 3 2 Vd Vm 4 Vq 5 ¼ 4 0 5 0 Vo 2
(4)
where Vm is the rated output voltage. For balanced RL Load we have
Rd ¼ Rq ¼ Ro ¼ Ra ¼ Rb ¼ Rc ¼ R Ld ¼ Lq ¼ Lo ¼ La ¼ Lb ¼ Lc ¼ L0
(5)
where (Ra,Rb,Rc) are resistances of the load in phases (a, b, c) and (Rd,Rq,Ro) are equivalent resistances of the load in rotating coordinate dqo, R is resistance and L0 is the inductance of the load.
Fig. 2. Four-leg voltage-source inverter.
Fig. 3. Average large signal model of the inverter with LC filter.
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scheme. Finally, communication with other power processing equipment or with user interfaced equipment is most easily achieved through digital controls. While digital control may enable many simplifications for control design, it brings about the element of delay in the control loop. DSPs used in control of power electronics circuits require a finite time delay for accomplishing the functions of sampling analog data, calculation, and updating digital and analog outputs. An ideal time delay can be represented as the frequency domain function,
Hdelay ðsÞ ¼ eTs
Fig. 4. Power stage average model in rotating coordinates: The d and q sub-circuits have coupled voltage and current sources.
3. Proposed model considering digital delay 3.1. Digital delays In general, feedback control for inverters is accomplished through algorithms in digital processors. There are several reasons for this. First, control for inverters is usually performed in the rotating dqo reference frame, as mentioned in previous section. This requires coordinate transformations that would be extraordinarily difficult to accomplish in analog devices, but are fairly simple to perform in Digital Signal Processors (DSPs). Second, safety and protection functions are easily implemented in a digital control
(6)
where T is the delay time. This function has unity magnitude for all frequencies and a linearly increasing phase-lag with frequency. A delay in control loop has an obvious destabilizing effect on feedback systems [22]. Because of the destabilizing effects of delay, it becomes very important to model the delay accurately when doing control design and analysis. Fig. 6 shows the location in the control loop where the delay is usually modeled. The output sampling, subtraction from reference, and compensation are all performed within the DSP. All of the delay associated with the sampling and calculation are generally lumped into one delay block at the interface between the DSP and the continuous system plant. While the delay associated with digital control may not be a constant for various reasons, it is important to put bounds on the length of the delay, so that the control can be designed around these limits. While there may be several different strategies for digital control implementation, the simplest would be to sample output variables, calculate duty cycles, and update the Pulse Width Modulation (PWM) registers all in one switching cycle. This strategy leads to a delay of between one and two switching periods [22]. 3.2. Proposed model As discussed in previous subsection digital implementation of the controller and/or discrete operation of modulation and inverter introduce a digital delay which is time-variant. The delay results in instability of the system, in some cases [22], and must be taken into account. We assume that the delay is constant. Therefore, it is estimated by:
1 eST z 1 þ ST
(7)
where T is the delay time. Let us model the estimated delay time after the gain Vg in Fig. 5, it has an important advantage which will describe in next subsection. On the other hand, the power stage is a coupled multi-variable multi-loop system. Here, we neglect coupling and present independently control of the system. Fig. 7 shows the proposed control structure for load voltage loop control.
Fig. 5. Average model represented as a signal flow graph: the load is represented as a balanced resistive-inductive (RL).
Fig. 6. Basic system model including ideal time delay.
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Fig. 7. Average model represented as signal flow graph (couplings are neglected).
3.3. Realization of proposed model We can realize the Signal Flow Graph of the inverter, shown in Fig. 7, as Fig. 8. Consequently, we can express the state-space equations of the system, for d channel, as:
x_ ¼ A$xðtÞ þ B$uðtÞy ¼ C$xðtÞ
dq ¼ 0 do ¼ 0
(10)
Thus, from now only the state equations of d channel are expressed and a controller for this channel will be developed in the following sections.
(8) 4. Pole-placement control via state feedback
where
2
3 id 6 Vd 7 7 x ¼ 6 4 iLd 5; VTd
2
C ¼ ½0
1
0
0 ;
3 0 607 7 B ¼ 6 405
Let the controller has the linear state feedback form
u ¼ K$x þ G$r
Vg T
where r ˛ R is a new vector with 1 input and K, state feedback, and G, static lead compensator, are the unknown controller matrices with dimensions 1 3 and 1 1, respectively (Fig. 9). Substituting Eq. (11) in Eq. (8) yields the closed-loop system
And
2
0 6 1 6 C A ¼ 6 40 1 T
1 L
0
1 Ld
1 C Rd Ld
0
0
0
1 L
0 0
3 7 7 7 5
x_ ¼ ðA BKÞx þ BGry ¼ Cx
1 T
It should be noted, by using above state equations, we consider the delay as a resistant and a capacitor in the model. As a result, it introduces a new state variable, which is the capacitor voltage VTd. Fortunately, the voltage of the capacitor VTd is equal to the voltage of the filter inductance. Consequently, this state variable is accessible and known, like other state variables. This feature is so important because proposed model does not need the observer, despite adding an unreal state variable. Because of system requirements in d and q channels, we have:
Vq ¼ 0 Vo ¼ 0 Therefore:
(11)
(9)
(12)
The control problem is to determine the control law (11), to determine the controller matrices K and G, such that the closedloop system has the desired characteristics.
K ¼ ½ K11
K12
K13
K14 ;
G ¼ ½G
4.1. Pole placement via state feedback Consider the linear, time-invariant system
_ xðtÞ ¼ AxðtÞ þ BuðtÞ
(13)
where all states are accessible and known. To this system we apply a linear state feedback control law of the form
uðtÞ ¼ KxðtÞ
(14)
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Fig. 8. Average model of the four-leg inverter; neglecting coupling and realizing digital delay as a first order system with RT ¼ 1U and CT ¼ T.
2
0 6 60 h i Fc ¼ B«AB«A2 B«A3 B ¼ 6 60 6 4
Then, the closed-loop system is given by the homogeneous equation (15)
_ xðtÞ ¼ ðA BFÞxðtÞ
(15)
Vg T
Now, the design problem is to find the appropriate controller matrix so as to improve the performance of the closed-loop system Eq. (15). One such method of improving the performance of (15) is that of pole placement. The pole-placement method consists in finding a particular matrix K, such that the poles of the closed-loop (15) take on desirable preassigned values. Using this method, the behavior of the open-loop system may be improved significantly. For example, the method can stabilize an unstable system, increase or decrease the speed of response etc. For this reasons, improving the system performance via the pole-placement method is widely recommended. The pole-placement or eigenvalue assignment problem can be defined as follows: let l1,l2,l3,l4 be the eigenvalues of the matrix A l 1 ; bl 2 ; bl 3 ; bl 4 be the desired of the open-loop system (8) and b eigenvalues of the matrix A BK of the closed-loop system (15), where all complex eigenvalues exist in complex conjugate pairs. Also, let p(s) and b p ðsÞ be the respective characteristic polynomials. i.e., let us find a matrix K so that Eq. (17) is satisfied.
pðsÞ ¼
4 Y
b p ðsÞ ¼
4 Y
li sb
2
1 60 j ¼ 6 40 0 a2
(19)
a3 1 0 0
a2 a3 1 0
a1
a0 ;
3 a1 a2 7 7; a3 5 1
a ¼ ½ a3 a2 a1 a0 ;
K ¼ ½ K11
K12
K13
K14
By substituting A,B and Fc in (19), state feedback matrix K can be expressed as:
LT Vg
0
LC LCTa3 Vg Vg Ld ðTþC ðða23 a2 ÞLVg La3 þ1ÞÞ Vg
LCT Vg LCLd Vg
0
0
0
3T
7 0 7 7 7 LCLd 7 5 0
(20)
4.2. Input reference tracking via static lead compensator
¼ jsI A þ BFj
¼ s þ a3 s þ a2 s þ a1 s þ a0 3
ð2$TLÞ L$T 3
where Fc is the controllability matrix, defined in Eq. (18), and
a ¼ ½ a3
i¼1 4
0 Vg Vg T 3 L$T 2
K ¼ ða aÞj1 F1 c
(16)
K ¼ ða
V T 2g
It is evident that rank(Fc) ¼ 4, for Vg s 0, L s T. Therefore, the system (A,B) is controllable, a fact which guarantees that there exists a matrix K which satisfies the pole-placement problem. Several methods have been proposed for determining K. One of the most popular pole-placement methods is the Base-Gura Formula which gives the following simple solution:
ðs li Þ ¼ jsI Aj ¼ s4 þ a3 s3 þ a2 s2 þ a1 s þ a0
LTa3 L Vg Vg 2 6 LCT ða3 a2 Þ 3 6 þ CLCa 6 Vg Vg aÞ6 3 2 6 Ld ðTa3 þða3 2a3 a2 þC ÞLCVg LCa3 þCa3 þLCa2 1Þ 4 Vg T Vg
0 0
ðT 2 þC$TL$C Þ 3 L2 $C$T 2 7 1 7 L$C$T 7 7 1 7 L$C$Ld 5 (18)
i¼1
2
Vg L$T 2 Vg L$C$T
Vg L$T
2
(17)
The controllability matrix Fc of system (18) can be expressed as:
The poles of system can be assigned in every place by using state feedback method, presented in previous subsection. However, the system does not track the reference input. To modify the performance of the system a static lead compensator, i.e. the matrix G, can
R. Nasiri, A. Radan / Renewable Energy 36 (2011) 858e865
863
Fig. 9. Pole-placement control strategy via state feedback and static lead compensator.
be used. To guarantee the tracking performance of system (12) we have
lim yðtÞ ¼ r
(21)
t/N
On the other hand when t / N, (12) is given by
_ xðNÞ ¼ 0 ¼ ðA BKÞxðNÞ þ BGryðNÞ ¼ CxðNÞ
(22)
By substituting (21) in (22), G can be expressed as:
G ¼ CðA BKÞ1 B
(23)
We can get G for our case, by substituting A,B,C and K, as follow
G ¼
LCT a0 Ld Vg R d
(24)
Fig. 10 shows diagram of the pole-placement control strategy. This system has poles, assigned by K vector, and the output load voltage tracks the reference input.
5. Simulation results and discussion In this section, the performance of the proposed control strategy is investigated by a computer simulation using SIMULINK-MATLAB software. The system parameters specifications used in the simulation are as follows:
L ¼ 333 mH C ¼ 100 mH Rd ¼ Rq ¼ Ro ¼ 100U Ld ¼ Lq ¼ Lo ¼ 200 mH Vg ¼ 800 V Vm ¼ 400 V T ¼ 100 ms
Fig. 11. Time response of the proposed second-order system and desired fourth-order systems when the input is unit step.
5.1. The respective characteristic polynomial of the open-loop system By substituting system’s parameters in (16), the respective characteristic polynomial of the desired system can be expressed as:
pðsÞ ¼ s4 þ a3 s3 þ a2 s2 þ a1 s þ a0 where
a3 a2 a1 a0
¼ ¼ ¼ ¼
510; 000 5:11006 109 3:08303 1013 1:516516 1017
Therefore, the open-loop system’s eigenvalues are given by
l1 l2 l3 l4
¼ ¼ ¼ ¼
1354 þ 6261j 1354 6261j 7391 50; 000
5.2. The desired characteristic polynomial A desired respective characteristic polynomial for proposed model has 4 poles, or 4 eigenvalues as below:
And, according to [21], the best PI controller parameters are given as below:
Kp ¼ 0:07;
Ki ¼ 1:2 105
Fig. 10. Structure of proposed pole-placement control strategy via state feedback and static lead compensator.
Fig. 12. Time response of the closed-loop system with proposed control strategy.
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Fig. 13. Comparison of PI controller and proposed controller (a) Vd (b) voltage of phase a Va.
b l1 b l2 b l3 b l 4
¼ ¼ ¼ ¼
3836 3913j 3836 þ 3913j 50; 000 50; 000
l 1 and bl 2 ). Two assigned poles are assumed to be complex (b Therefore, we can consider them, together, as a second-order system. To have an acceptable transient performance (reasonable overshoot, and very high speed of response), a second-order system with z ¼ 0.69 and un ¼ 5480, (25), is ideal. G2 ¼
u2n 54802 ¼ 2 s2 þ 2zun þ u2n s þ 2 0:69 5480 þ 54802
¼
54802 l l sb sb 1
(25)
2
l 3 and bl 4 ) are assumed to be real and Two other assigned poles (b be located far from the second-order system’s poles. Therefore, we can ignore their effect on the final, fourth-order, system. In other words, two other assigned poles, which are located ins ¼ 50,000, guarantee that the behavior of the resulted, fourth-order, system (26), is matched to the behavior of the second-order system (25). In fact, they are non-dominant poles of system (26). G4 ¼
¼
G4 ¼
l3 sb
b l 3 bl 4 u2n l 4 s2 þ 2 z un þ u2n sb
50; 0002 54802 ðs 50; 000Þ s2 þ 2 :69 5480 þ 54802
(26)
50; 0002 54802 s4 þ a3 s3 þ a2 s2 þ a1 s þ a0
(27)
2
(28)
where
a3 a2 a1 a0
¼ ¼ ¼ ¼
5.3. Time response of the closed-loop system with proposed controller for step unit input Fig. 12 depicts the time response of the closed-loop system with proposed control strategy. As expected, the output voltage tracks the reference input (Vd-reference ¼ Vm ¼ 400 V) in steady-state conditions. Additionally, the transient performance of the system, such as overshoot and speed of response, is matched with the desired fourth-order system (27) or second-order system (25), shown in Fig. 11, presented in previous subsection. Consequently, this simulation result validates the theoretical results. 5.4. Comparison between the performance of the proposed controller and PI controller Fig. 13(a) depicts the response of the system with proposed pole-placement control strategy and with Conventional PI controller in a picture, for comparison. The PI controller has a very slow response. It tracks the input after 3 cycles (0.09 s) while it has the best parameters(Kp ¼ 0.07, Ki ¼ 1.2 105). However, the system with state feedback tracks the input less than 0.0015 s. Fig. 13(b) shows the equivalent load voltage in phase a. 6. Conclusion
Thus, the respective characteristic polynomial of the desired system can be expressed as:
b p ðsÞ ¼ s4 þ a3 s3 þ a2 s2 þ a1 s þ a0
step input. It can be seen that the time response of the fourth-order system is matched with the second one. It should be noted that the systems, (25) and (26), have an overshoot not larger than 5% and a time of response less than 0.0015 s. This system (26) behaves as our desired (designed) system. Thus, its respective characteristic polynomial (28) is chosen as the vector a, see (17), in designing procedure.
1007672 257702756150 1:9482160658648 1015 7:5090094293545 1018
Fig. 11 shows the time response of the proposed second-order system (25) and the desired fourth-order system (26) for the unit
Modern control theory, which uses state equations for controller design, offers precise control in time domain. However, it has not yet been noticed for 4-leg inverter’s controller design issue. This paper proposes it as a suitable approach in the 4-leg inverter case. Moreover, it demonstrates unique advantages of this approach. In conclusion, the following can be made. This paper proposes a novel model, considering delay of the digital implementation, for 4-Leg voltage-source inverters. It uses state feedback matrix for pole placement of the system. Consequently, it can stabilize the system, if the system is unstable. Moreover, we can adjust the overshoot and the speed of response to any desired values. In addition, it guarantees the tracking of the reference input by using static lead compensator.
R. Nasiri, A. Radan / Renewable Energy 36 (2011) 858e865
It should be noted that this paper considers the digital delay as a first order system with a resistant and a capacitor in the model. As a result, it introduces a new state variable, which is the capacitor voltage. Fortunately, the voltage of the capacitor is equal to the voltage of the output filter inductance. Consequently, this state variable is accessible and known, like other state variables. This feature is very valuable because proposed model does not need the observer, despite adding an unreal state variable. As simulation results show, this approach has unique responses in both transient and steady states conditions. In comparison with other control strategies, which have been proposed for 4-Leg inverters, like PI controllers, this approach is very suitable for implementation. This approach provides a precise control in time domain, like the speed and overshoot of the response, which cannot be achieved by designing in frequency domain. Appendix Transformation matrix T from abc to dqo
2
Vd
3
2
Va
3
6 7 6 7 4 Vq 5 ¼ T 4 Vb 5 Vo
2
Vc CosðutÞ
6 T ¼ 234 SinðutÞ 1 2
Cosðut 120 Þ Sinðut 120 Þ 1 2
Cosðut þ 120 Þ
3
Sinðut þ 120 Þ 7 5 1 2
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