Front. Energy DOI 10.1007/s11708-017-0459-5
RESEARCH ARTICLE
Rahul SHARMA, Sathans SUHAG
Feedback linearization based control for weak grid connected PV system under normal and abnormal conditions
© Higher Education Press and Springer-Verlag Berlin Heidelberg 2017
Abstract This paper proposes a control strategy for interface of distributed energy sources into the weak grid system with a focus on the energy and ancillary services. A novel controller has been designed and implemented to tackle the challenges of coupling terms in the LCL filter, the transient behavior under sudden changes, and the voltage support under fault condition using the feedback linearization technique. The controller proposed has been implemented on the PV system connected with the weak grid using the LCL filter and the performance of the controller has been verified using Matlab/Simulink through simulation under different conditions. The results of the controller proposed have been compared with the conventional PI dual loop controller. The simulation results obtained demonstrate the effectiveness and simplicity of the controller design strategy. Keywords PV system, grid interface, feedback linearization, inverter, LCL filter
1
Introduction
The increased penetration of distributed energy sources, primarily comprising renewable sources, in the power system is an effect of energy crisis and environmental concerns. The rapid growth of distributed sources globally makes the researchers concentrate on the topic of distributed energy and its utilization, although there are associated challenges in terms of efficiency, reliability and stability of the system [1]. Due to the unpredictable nature of the distributed energy sources, still many utilities are reluctant to establish the Received June 4, 2016; accepted September 6, 2016
✉
Rahul SHARMA ( ), Sathans SUHAG Department ofElectrical Engineering, NIT, Kurukshetra 136119, India E-mail:
[email protected]
system based on distributed sources. Thus, the research is still focused mainly on devising suitable control strategies [2,3] to increase the reliability and stability of the system with weak grid integration [4]. This paper proposes a control strategy to improve the system behavior in terms of stability and reliability under different operating conditions and reduce the adverse affects of distributed sources on the grid system under normal and fault conditions. The control strategy proposed is designed based on the feedback linearization theory, keeping in mind the objectives of the grid connected distributed sources such as keeping the DC link voltage constant, improving the behavior of current and voltage under sudden change of source input, optimal control of the reactive power injection in weak grid, and voltage support under fault condition. All objectives are achieved with the help of the control strategy proposed which is based on the feedback linearization technique with synchronous reference frame taken into account for the PV grid connected system model. The model investigated is based on the PV system connected with the weak grid with the help of an inverter and an LCL filter. All system equations are converted into d-q reference frame to design the controller [5]. The LCL filter is designed as per Ref. [6]. The controller proposed is able to eliminate the problem of decoupling term due to the LCL filter. The feedback linearization technique is used to reduce the multi-input multi-output nonlinear system into single input single output linear system which simplifies the design of the transfer function of regulators [7]. The control strategy proposed provides a robust control of the system under normal and abnormal conditions and supports the weak grid voltage as defined by the standards in grid code under grid fault conditions. The strategy has been verified with the help of simulations on the Matlab/ simulink platform and the results are compared with those of the conventional PI decoupling dual loop control strategy.
2
2
Front. Energy
After transformation, the system equations are expressed
PV grid connected system modeling as
The system considered in this paper is a PV grid connected system consisting of a PV array connected with a capacitor bank, an inverter, an LCL filter, and grid source, as shown in Fig. 1. In this system, the power generated by the PV array is Pv and the current injected into the grid is I2, while Vdc is the DC link voltage across the capacitance Cdc. The LCL filter design is implemented as per Ref. [8] to interface the inverter to the grid which helps reduce the distortion of the voltage and current. The switching frequency of the inverter is considered far more than the system frequency and resistive values of the filter components while switching losses are neglected to design the controller. In the steady state, the controlled grid current is I2abc and in the phase grid, the voltages are Vgabc. The corresponding grid voltage can be expressed as 1 0 1 0 Vm sinωt Vga C BV C B (1) @ gb A ¼ @ Vm sinðωt þ 120 ∘ Þ A, Vgc
Vm sinðωt – 120 ∘ Þ
where Vm is the amplitude and w is the angular frequency of the grid voltage. The system modeling and controller design is done based on the synchronous reference frame (d-q) by transforming the stationary frame using Park transformation as 0 1 0 1 αa αd B C B C –1 (2) @ αb A ¼ T dq0 @ αq A, αc α0 where 0 T –1
dq0
cosωt
B ¼ @ cosðωt – 120Þ cosðωt þ 120Þ
– sinωt – sinðωt – 120Þ
1
1
C 1 A:
– sinðωt þ 120Þ 1
In this modeling, the grid voltage is assumed to be sinusoidal and balanced. The transformation of the equations from stationary to synchronous reference frame is done using grid voltage angle as a reference.
Cdc Vdc
dVdc ¼ Pv – Pinv : dt
(3)
Equation (3) represents the power balance which is derived from the DC link voltage where Pinv is the inverter output instantaneous power. But filter losses are considered zero and Vgq= 0 in d-q frame, therefore the grid active power and reactive power are expressed as 3 3 Pinv ¼ (4) Vd I1d þ Vq I1q ¼ Vgd I2d , 2 2 3 Qinv ¼ Vgd I2q : 2
(5)
Now, from Eqs. (4) and (5), Eq. (6) can be obtained. dV 3 (6) Cdc Vdc dc ¼ Pv – Vgd I2d : 2 dt With the help of the LCL filter, the system differential equations in synchronous reference (d-q) frame are dI L1 1d ¼ L1 ωI1q þ Vd – Vcd , dt dI1q ¼ – L1 ωI1d þ Vq – Vcq , L1 dt dV Cf cd ¼ – Cf ωVcq þ I1d – I2d , dt dVcq (7) ¼ Cf ωVcd þ I1q – I2q , Cf dt dI L2 2d ¼ – L2 ωI2q þ Vcd – Vgd , dt dI2q ¼ L2 ωI2d þ Vcq , L2 dt dV 3 Cdc Vdc dc ¼ Pv – Vgd I2d : dt 2 From Eq. (7), state vectors are defined as x = [I1d I1q Vcd Vcq I2d I2q Vdc]T, and the system input variables are expressed as u =[ud uq] where pffiffiffiffiffiffiffiffi Vd ¼ 3=8Vdc ud : (8) pffiffiffiffiffiffiffiffi Vq ¼ 3=8Vdc uq
Fig. 1 Schematic of PV inverter system connected to grid with an LCL filter
Rahul SHARMA et al. Feedback linearization based control for weak grid connected PV system
Equations (3)–(8) describe the PV grid connected system with the LCL filter using the inverter for grid interface. In Section 3, the feedback linearization based control strategy will be designed with the help of system model equations as described in Section 2.
3 Design of feedback linearization based control strategy 3.1
Feedback linearization theory
Feedback linearization is an approach to nonlinear control design where the central idea is to algebraically transform nonlinear system dynamics into linear (fully or partly) ones, so that linear control techniques can be applied. The detailed theory on feedback linearization can be found in Refs. [9–11]. In this paper only a brief procedure is explained before applying this theory to design the control scheme proposed for the system under study. Choose a MIMO system to be x_ ¼ f ðxÞ þ gðxÞu, y ¼ hðxÞ:
(9)
Differentiate y until u appears in one of the equations for the derivatives of y y_ €y ::: ðrÞ
y
(10)
¼ ðxÞ þ lðxÞu:
After r steps, u appears, where r is the relative degree of the system and u is the input which is related with the y(r) output after the rth differentiations. Choose u to give y(r)= V, where V is the synthetic input, u ¼ l – 1 ðxÞ½ – ðxÞ þ V :
(11)
Design a linear control law for this r-integrator linear system. 3.2
Design of control scheme proposed
The feedback linearization procedure as outlined above has been implemented to design the control scheme of the PV grid connected system using Eq. (7). The control strategy proposed is the key to simplifying the control design using feedback linearization. Rather than taking all the variables simultaneously, the scheme proposed chooses the set of variables in a cascaded manner such as taking the initial variables as the output for the first controller, which would, in turn, be the input for the next controller and so on which
3
automatically sets up the relation between the input and the output of the overall control design. The system has seven variables after d-q transformation in which the d-axis is used to control the DC link voltage of the system, while the q-axis is used to control the power factor of the system under normal condition and the reactive power (voltage) support under abnormal/faulty condition. The control scheme proposed utilizes feedback linearization in parts to reduce the complexity of the controller which is designed in parts and then connected in a cascaded manner using four d-axis variables and three q-axis variables. The d-axis variables Vdc, I1d, Vcd, and I2d, involved to control the DC link voltage, can be broken into four parts like cascaded connections. Similarly, the q-axis variables I1q, I2q, and Vcq are utilized to control the reactive power. The detailed control scheme of the system is derived in steps as follows using all the equation mentioned in Eqs. (7) and (8). To begin with, the first step is to design the close loop control using I1d and I1q as the output and Vd and Vq as the input variables. Now, the differential equation can be written as "
I_ 1d I_ 1q
#
" ¼
ωI1q – Vcd =L1
# "
1=L1
0
#"
þ – ωI1d – Vcq =L1 0 1=L1 " # I1d ηd y1 þ ¼ , ηq y2 I1q
# τd , þ τq Vq
Vd
(12) where τ= 0.5sin(0.2t) and η = 0.5e are considered the disturbance and noise in the system to verify the satisfactory response of the system under more realistic operating conditions. Using Park transformation, the three phase τ and η are transformed into the d and q-axis. With the help of feedback linearization, Eq. (12) can be rewritten as –0.2t
y_ 1 ¼ ωI1q – Vcd =L1 þ Vd =L1 þ τ d y_ 2 ¼ – ωI1d – Vcq =L1 þ Vq =L1 þ τ q where the relative degree is {1,1}. Choose u =[Vd Vq] to give " # y_ 1 ¼ ½ V1 V2 T , y_ 2
(13)
(14)
where V1 and V2 are the synthetic input and Eq. (14) can be written in the standard form of Eq. (11). " # " #" # ð – ωI1q þ Vcd =L1 – τ d =L1 Þ þ V1 Vd L1 0 ¼ , Vq ðωI1d þ Vcq =L1 – τ q =L1 Þ þ V2 0 L1 (15)
4
Front. Energy
" l – 1 ðxÞ ¼
where
L1
0
# G2 ðsÞ ¼
,
0 L1 1 ðxÞ ¼ ωI1q – Vcd =L1 þ τ d =L1 , 2 ðxÞ ¼ – ωI1d – Vcq =L1 þ τ q =L1 :
With the help of Eqs. (13) and (15), the decoupled linearized control law is obtained in Eq. (14) which provides the transfer function and closed loop control to achieve the desired output which are expressed as – k1 ðI1 – I *1 Þ ¼ 0,
(16)
k1 e ¼ 0: Therefore, the transfer function V ¼ – k1 ðI1 – I *1 Þ,
(17)
V ðsÞ ¼ syðsÞ:
Taking Laplace transformation of Eq. (17), Eq. (18) can be obtained. G1 ðsÞ ¼
þ
#"
1=Cf
0
0
1=Cf
I1d I1q
After applying the feedback linearization technique on Eq. (23), Eq. (24) can be obtained. " # " #" # Vcd L2 0 ðωI2q – Vgd =L2 Þ þ V5 ¼ , (24) Vcq – ωI2d þ V6 0 L2 "
(18) –1
where e is the error and I*1d and I*1q are the desired outputs for the d and q axes respectively. Subsequently, in the next step, the feedback linearization technique is applied for closed loop control of Vcd and Vcq as the output variables whereas I1d and I1q as the input variables which are assumed to be equal to the reference values (desired outputs). The differential equations can be expressed as " # " # V_ cd – ωVcq – I2d =Cf ¼ ωVcd – I2q =Cf V_ cq # :
(19)
Similar to the first step, using the feedback linearization, the closed loop control is written as " # " #" # ðωVcq – I2d =Cf Þ þ V3 I1d Cf 0 ¼ , (20) I1q – ðωVcd – I2q =Cf Þ þ V4 0 Cf
l ðxÞ ¼
" l – 1 ðxÞ ¼
Cf
0
# ,
0 Cf 1 ðxÞ ¼ ωVcq – I2d =Cf , 2 ðxÞ ¼ – ωVcd þ I2q =Cf , and V3 and V4 are the synthetic inputs. Thus, the transfer function can be derived as
(21)
L2
0
#
, 0 L2 1 ðxÞ ¼ ωI2q – Vgd =L2 ,
(25)
2 ðxÞ ¼ – ωI2d : V5 and V6 are the synthetic inputs. Therefore, the transfer function can be expressed as k3 , ðs þ k3 Þ sk3 þ kqi : G3q ðsÞ ¼ 2 s þ sk3 þ kqi G3d ðsÞ ¼
(26)
It is evident here that the q-axis control loop is the outer loop which is the proportional as well as the integral controller to eliminate the error in the presence of the inner loop, but the d-axis control loop is not the outer loop which is only the proportional controller. In the last step, the PI controller is implemented to control the DC link voltage using power balancing Eq. (6) and the transfer function can be accordingly represented as G4 ðsÞ ¼
where
(22)
Further, in the third step, grid side variables Vcd and Vcq are taken as the inputs with I2d and I2q acting as the outputs to obtain the closed loop control. Similarly, in this step, the differential equations can be expressed as " # " # I_ 2d – ωI2q – Vgd =L2 ¼ ωI2d I_ 2q " #" # 1=L2 0 Vcd þ : (23) Vcq 0 1=L2
where
I1 ðsÞ k1 ¼ , * ðs þ k1 Þ I 1 ðsÞ
"
Vc ðsÞ k2 ¼ : * ðs þ k2 Þ V c ðsÞ
s2
sk4 þ kdi : þ sk4 þ kdi
(27)
The SPWM technique is employed to control the switching of the two level inverter using the complete control scheme output ud and uq. Figure 2 illustrates the complete block diagram of the control scheme for the PV based grid connected inverter using the LCL filter with the introduction of the disturbance and noise in the system on the AC side to analyze more realistic performance of the control system.
Rahul SHARMA et al. Feedback linearization based control for weak grid connected PV system
5
Fig. 2 Control structure of grid connected PV system
4
Results and discussion
The control scheme proposed is verified on the PV based grid connected inverter using the LCL filter model with the help of Matlab/Simulink for two different cases. Under normal condition, only the irradiance variation is considered while under abnormal condition, the L-G fault is taken at 1.3 s and cleared at 1.5 s. The parameters are given in Table 1. Table 1 Simulation parameters Parameters
Value
Grid line voltage(rms)/V
270
Grid frequency /Hz
50
Grid rating /kVA
500
Inductance L1/H
0.45
Inductance L2/H
0.15
DC link capacitance Cdc/F
5000
Filter capacitance Cf/F
170
The irradiance variation, as depicted in Fig. 3, is considered as the input of the PV panel to verify the control design and its improvement by the results at transient and steady state conditions. The 100 kW PV system has been modeled to verify the control design. Figure 4 shows the power variation from
Fig. 3
100 kW to 53.5 kW of the system with the change in irradiance from 550 to 300. Figure 5 demonstrates the voltage of the weak grid having a capacity of 500 kVA which is constant throughout the period and as can be seen, there is no effect of change in irradiance on the voltage profile of the grid. This demonstrates the robustness and improved performance of the control design for the weak grid or low level microgrid for distributed energy. The current waveform in Fig. 6 validates the power injection into the grid and the change in the current with the change in active power according to the irradiance variation of the model. The transient behavior of the system is improved due to the control design proposed at 1s where the irradiance changes from 550 to 300. The system efficiency is also improved due to the unity power factor of the system by way of exercising the control over reactive power with the help of the control scheme proposed. The same has been verified with the help of the results shown in Fig.7. The DC link voltage of the inverter should be constant for the purpose of grid interface of the system. This objective has been achieved as shown in Fig. 8 which verifies the working of the control design for the grid connected PV system and shows better transient behavior of the DC link voltage. The results of Id and Iq current waveforms, shown in Fig. 9 along with the reference current waveforms of Id* and Iq*,
Irradiance waveform
6
Front. Energy
Fig. 4
Active power waveform of the system
Fig. 5
Grid voltage of the system
Fig. 6 Grid current of the system
verify the accuracy of tracking, by Id and Iq, of the respective reference values to control the active and reactive power where Iq* is zero to obtain unity power factor. The control design proposed is verified for the system with the help of the above results and improvement in performance, as compared to the conventional PI dual loop control design, has been validated as observed and shown as the combined results in Figs. 10 and 11 respectively for
the control design proposed and the conventional control designs at transient as well as steady state conditions. The control design proposed has reduced the harmonics considerably as compared to the conventional one which is validated with the help of the FFT results as shown in Figs. 12 and 13 respectively. Therefore, the control design proposed thus helps the system to inject power with reduced harmonic level which, in turn, improves the power quality of the weak grid.
Rahul SHARMA et al. Feedback linearization based control for weak grid connected PV system
7
Fig. 7 Phase voltage and current waveform
Fig. 8 DC link voltage of the system without disturbance and noise
Fig. 9 Id, Id* and Iq, Iq*current waveforms of the system
The control design proposed is tested under abnormal condition of an L-G fault at phase A and as can be seen from Figs.14 and 15, the control design proposed produces better results under fault condition as well. The dip in the voltage profile of the system shows improvement with the help of reactive power control while the current waveform shows better transient behavior at the time of fault clearance with the controller proposed as compared to the conventional one. The control design proposed shows a satisfactory performance under disturbance and noise which can also be seen in Fig. 16.
A small distortion is introduced in the DC link voltage waveform with disturbance and noise but still the control scheme is regulating the DC link voltage at the reference point. This shows the disturbance rejection capability and robustness of the control strategy.
5
Conclusions
This paper proposed and implemented the design of a novel control scheme based on the feedback linearization theory for grid connected PV system using an LCL filter.
8
Front. Energy
Fig. 10
Results waveform using the control design proposed
Fig. 11 Results waveform using the conventional control design
Fig. 12
FFT graph using conventional control design
Rahul SHARMA et al. Feedback linearization based control for weak grid connected PV system
Fig. 13
Fig. 14
Fig. 15
9
FFT graph using the control design proposed
Waveforms under fault condition using controller proposed
Waveforms under fault condition using conventional controller
The controller design proposed shows improved performance in terms of better transient behavior, reduction in harmonics, better tracking of active and reactive power at steady as well as in the event of sudden change in irradiance level to maintain the quality of the power injected with minimum disturbance into the weak grid
according to the grid code. The results illustrate the working performance of the controller proposed with a significant improvement under disturbance and noise condition as compared to the conventional dual loop PI controller. The control design proposed is tested under fault
10
Front. Energy
Fig. 16
DC link voltage of the system with disturbance and noise
condition as well and is verified from the simulation results. Not only has the condition of collapse in grid voltage been improved significantly due to the control design proposed by regulating the reactive power of the system, but also the fault severity of the system is reduced.
5.
6.
References 1. Eid B M, Rahim N A, Selvaraj J, El Khateb A H. Control methods and objectives for electronically coupled distributed energy resources in microgrids: a review. IEEE Systems Journal, 2014, 99: 1–13 2. Nehrir M H, Wang C, Strunz K, Aki H, Ramakumar R, Bing J, Zhixhin M, Salameh Z. A review of hybrid renewable/alternative energy systems for electric power generation: configurations, control and applications. IEEE Transactions on Sustainable Energy, 2011, 2(4): 392–403 3. Karabiber A, Keles C, Kaygusuz A, Alagoz B B. An approach for the integration of renewable distributed generation in hybrid DC/ AC microgrids. Renewable Energy, 2013, 52(2): 251–259 4. Delfino F, Pampararo F, Procopio R, Rossi M. A feedback linearization control scheme for the integration of wind energy
7.
8.
9. 10. 11.
conversion systems into distribution grids. IEEE Systems Journal, 2012, 6(1): 85–93 Masrur M A. Studies on the effect of filtering, digitization, and computation algorithm on the ABC-DQ current transformation in PWM inverter drive system. IEEE Transactions on Vehicular Technology, 1995, 44(2): 356–365 Bina M T, Pashajavid E. An efficient procedure to design passive LCL-filters for active power filters. Electric Power Systems Research, 2009, 79(4): 606–614 Bao X, Zhuo F, Tian Y, Tan P. Simplified feedback linearization control of three-phase photovoltaic inverter with an LCL filter. IEEE Transactions on Power Electronics, 2013, 28(6): 2739–2752 Figueres E, Garcera G, Sandia J, Gonzalez-Espin F, Rubio J C. Sensitivity study of the dynamics of three phase photovoltaic inverters with an LCL grid filter. IEEE Transactions on Industrial Electronics, 2009, 56 (3): 706–717 Megretski A. Lecture 13: feedback linearization. Department of Electrical Engineering and Computer Science, Cambridge, 2003 Slotine J J E, Li W. Applied Nonlinear Control. Englewood Liffs: Prentice Hall,1991 Hedrick J K, Girard A. Control of Nonlinear Dynamic Systems: Theory and Applications. Berkeley: University of California, 2010, 151–181
