Improvement of the grid-connect current quality using novel proportional-integral controller for photovoltaic inverters Yuhua Cheng, Kai Chen, Libing Bai, and Jing Yang Citation: Review of Scientific Instruments 85, 025110 (2014); doi: 10.1063/1.4866023 View online: http://dx.doi.org/10.1063/1.4866023 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Grid-connected photovoltaic converters: Topology and grid interconnection J. Renewable Sustainable Energy 6, 032901 (2014); 10.1063/1.4876415 Digital control strategy for solar photovoltaic fed inverter to improve power quality J. Renewable Sustainable Energy 6, 013128 (2014); 10.1063/1.4863987 Suppression of zero-crossing distortion for single-phase grid-connected photovoltaic inverters with unipolar modulation Rev. Sci. Instrum. 84, 104705 (2013); 10.1063/1.4824708 Grid-connected photovoltaic system based on switched-inductor quasi-Z-source inverter and its low voltage ridethrough control strategy J. Renewable Sustainable Energy 5, 033120 (2013); 10.1063/1.4808262 A novel pulse width modulation for grid-connected multilevel inverter J. Renewable Sustainable Energy 1, 053102 (2009); 10.1063/1.3204460
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 025110 (2014)
Improvement of the grid-connect current quality using novel proportional-integral controller for photovoltaic inverters Yuhua Cheng,a) Kai Chen, Libing Bai, and Jing Yang School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China
(Received 7 December 2013; accepted 4 February 2014; published online 25 February 2014) Precise control of the grid-connected current is a challenge in photovoltaic inverter research. Traditional Proportional-Integral (PI) control technology cannot eliminate steady-state error when tracking the sinusoidal signal from the grid, which results in a very high total harmonic distortion in the grid-connected current. A novel PI controller has been developed in this paper, in which the sinusoidal wave is discretized into an N-step input signal that is decided by the control frequency to eliminate the steady state error of the system. The effect of periodical error caused by the dead zone of the power switch and conduction voltage drop can be avoided; the current tracking accuracy and current harmonic content can also be improved. Based on the proposed PI controller, a 700 W photovoltaic grid-connected inverter is developed and validated. The improvement has been demonstrated through experimental results. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4866023] I. INTRODUCTION
In recent years, the distribution system of renewable energy sources is gaining greater attention. As a result, new sources of renewable energy, such as solar energy and fuel cells, have become more and more critical.1 The distributed generation system comprising of photovoltaic (PV) modules, grid-connected inverter and the grid, realizes the transmission and usage of solar energy. The grid-connected inverter transforms the DC power from PV modules into AC power. The grid current total harmonic distortion (THD) affects the efficiency of system, which means the higher the efficiency, the smaller is the current harmonic distortion.2 Improper control will create significant harmonic distortion under a linear or sinusoidal load. It cannot achieve good results when the direct input signal of PI controller is sinusoidal. Besides, it also increases the THD and the tracking accuracy of the grid decreases with sinusoidal inputs.3, 4 Grid corporations give the requirements of power generation equipment for distributed grid such as the standards that deal with issues including power quality, detection of islanding operation, grounding, etc.1 For example, according to the IEEE1547 standard, the current harmonic content should be less than 5%.5 Ideally, the grid-connected current is a sine wave and the traditional control technology based on the classical feedback control is widely used. The error is obtained by subtracting the feedback current from the reference current. By eliminating this error with the current controller, the grid-collected current is close to the standard sine wave. The common control strategies include proportional-integral (PI), proportionalresonant (PR), predictive deadbeat (DB), repetitive controller (RC), and hysteresis controllers.6 However, it is difficult for PI controller to track the time-varying AC sinusoidal signal accurately and results a steady-state error. The quality of a) Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0034-6748/2014/85(2)/025110/8/$30.00
output current can be improved by PR control with a good steady state performance, but only when the controller resonant frequency is the same as the grid frequency.7 DB control is widely used to eliminate the error current, while its performance and system parameters are closely related.8, 9 Hysteresis control is easy to use and has good dynamic response characteristics. However, the sampling control frequency is not fixed, resulting in a high current ripple, which reduces the quality of the current. Besides, the output filters are difficult to design.10–12 Besides, the authors in Ref. 13 proposed a hybrid controller and a fuzzy-PI controller is introduced in Refs. 14 and 15. The feature model of the external interference signal is integrated into the controller by repetitive control in order to better suppress the interference. So, the repetitive control is widely used in high-precision servo control systems. As a result, the structure of the system is stable. In order to set the steady-state error to zero, and to ensure that the system has a good tracking ability and also eliminate interference, the dynamic model of the disturbance must be included in the controller. This interference signal dynamic model is called the internal model; the implemented internal model controller is the repetitive controller.16, 17 In recent years, a repetitive control method18–21 has been widely used in the inverter system due to its ability to suppress the periodic interference. It can eliminate harmonics and deal with a very large number of harmonics simultaneously.3, 22 Most importantly, repetitive control is one area in which the objective is mainly to remove the steady-state error due to periodic inputs.23 Ye et al.24 gave the repetitive control with Real-Time Phase-Lead FIR filter. Zhou et al.25 presented a Plug-In Dual-Mode-Structure Repetitive Control. Many interfering signals are announced to have an effect because the grid-connected inverter system tracks a reference standard sine wave with a varying amplitude and phase instead of a constant value. These interfering signals contain multiple harmonics in addition to the fundamental one
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PV Module
inverter
takes the pulse width modulated (PWM) waves as input and output the sine-wave through the LC filter. The error current is obtained by comparing the output feedback current and the expected current (Iref ), and ultimately suppressed or eliminated by the proposed controller. The Iref is obtained by Maximum Power Point Tracking method (MPPT). The voltage feed-forward can eliminate the grid disturbance, and the design of error controller is simplified after adding it. Assuming that the duty cycle of the bridge control is D , UO is the output voltage, and Udc as the input voltage, the relationship among these three variables can be derived according to the principle of equivalent areas:27
grid AC Switching
DC Switching
Local load FIG. 1. Grid-connected PV power generation system.
and they are not constant. Therefore, it becomes extremely difficult to use the traditional PI control method to achieve high-precision tracking. In order to solve this problem, a novel PI controller has been developed. This method is simpler than repetitive control due to several reasons: (1) Interference signal modeling is not needed; (2) Expressions of the proposed PI method are simpler than repetitive control; (3) Less steps to achieve the purpose. In this paper, the system model is developed and analyzed. In contrast to the previous work on the traditional PI control methods, a pure delay link has been implemented in the designed system with the reference sinusoidal signal being discretized into the N discrete step input. The high-precision control is achieved through our novel PI controller approach. The rest of this paper is organized as follows: Sec. II presents the system model and transfer function with the analysis of the limitations of conventional PI controller. Section III introduces the proposed PI control algorithm and Sec. IV reports the design and experimental results of the experimental prototype. Finally, the conclusion is drawn in Sec. V.
UO = Udc × D .
Full-bridge is assumed to be a proportional component; transfer function is equal to the input; ignoring the impact of switching device voltage drop and dead zone, (1) is the ideal mathematical model. Ignoring the capacitive current of the LC filter, the voltage UL is the difference value of inverterbridge-output voltage UO and grid voltage UN . Also UL is equal to the differential of the grid-connected current IO and the voltage component on resistor of the inductor. The relationship between voltages and current can be written as UL = L
Io (s) =
As shown in Fig. 1, the grid-connected PV power generation system is formed by the PV modules, grid-connected inverter, and grid.26 Fig. 2 shows the core component of the inverter that is composed of a grid-connected inverter bridge and an LC filter. The current goes through the grid-connected inverter bridge which works as a DC-to-AC conversion, and then the gridconnected current is obtained via the LC filter. The inverter bridge works under the high-frequency switching state that
+
(3)
Grid
IO Iref +
+
U o − UN . Ls + r
Substituting it into Eq. (3), the open-loop transfer function of full-bridge and the inductance of the grid system is
Inverter LC UO Bridge Filter
PWM Generator
(2)
The full-bridge output voltage must be greater than the grid voltage in order to send the power to the grid. The output voltage is broken down into two parts: one part to offset the grid and the other part to produce an output current. Therefore, D is divided into two parts: UN + D = UN + Udc × D. (4) Uo = Udc × Udc
A. Modeling
Udc
dIO + IO r = UO − UN , dt
where L is the inductance value and r is the resistance component of L. The grid-connected current IO is integrated of the error voltage UL by the LC filter. The Laplace transform of Eq. (2) can be easily obtained as
II. PV POWER GENERATION SYSTEM AND MODEL BASED ANALYSIS
DC Power Source
(1)
Controller
-
The voltage feed-forward link
err
UN K
FIG. 2. Structure of grid-connected inverter.
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1 Udc E
Ir +
KPs+KI s
-
+ +
D
Udc
Simplified:
UN
E(s) = E (s)Ir (s) Uo +
-
1 Ls+r
Io
= =
FIG. 3. System closed-loop block diagram based on PI control.
Udc IO (s) = . (5) D Ls + r The type-0 system cannot track the sinusoidal signal according to automatic control principle. G0 (s) =
KP is the proportional coefficient and KI is the integral coefficient. Equations (5) and (6) show that after adding in the PI controller, the system changes from the type-0 system into the type-I system. There is always a steady-state error when the type-0 system tracks a step signal and this issue has been remedied in this new type-I system. The system model is simplified by setting the grid voltage feed-forward coefficient as 1/Udc so that the grid voltage is not included. When the input is a sinusoidal signal, the steady-state error of the system is analyzed as follows. Assume that the response of the linear time-invariant system (s) is (s)est , where the error transfer function is obtained as Ls 2 + rs 1 = . 1 + L(s) Ls 2 + (r + KP Udc )s + KI Udc (7) For the input sinusoidal signal, ω Ir (s) = Im 2 . s + ω2 The error response of system can be found:
+
a1 s + a2 (r+Kp Udc ) s + KiLUdc L
Im +
s2
ω + ω2
a3 s + a4 , s 2 + ω2
1 Ir (s), 1 + G(s)
(8)
(11)
(13)
where η, α, and θ are parameters. From Eq. (13), it is noted that the system with PI controller has a steady-state error in both amplitude and phase when the input is a sinusoidal signal. The reason for that is while dealing with currents with expectation as standard sinusoidal signal rather than the step function, the traditional PI control method cannot achieve high precision for tracking the sinusoidal signal. The system’s stability and control accuracy will also be affected when digital control is used to introduce phase delay. Furthermore, the actual full-bridge inverter is not a proportional component, and the grid voltage feed-forward mechanism is not a remedy for the same phase delay and error. The actual output current error is greater than the calculated results above because of the existence of the errors such as sampling, calculating, and so on. All of these errors can cumulate and lead to the final results that are not satisfying the system design requirements and are the major drive and motivation for the proposed improvement. III. THE PROPOSED PI CONTROLLER DESIGN A. Methods
As discussed in Sec. II, PI controller can improve the system to the type-I system. Although it cannot track a sinusoidal signal, the following derivations show that the steady-state error for step input signals is 0. For the same transfer function of the system shown in Eq. (7) with step input: Im . S The steady-state error can be found as Ir (S) =
(14)
s × Ism s−>0 1 + G(s)
ess = lim (9)
s × Ism × s(Ls + r) = 0. s−>0 1 + (KP S + KI )Udc + s(Ls + r)
= lim
E(s) = E (s)Ir (s) =
s2
+
Ki Udc L
ess (t) = η cos(αω + θ ),
A PI controller changes the system type and improves system dynamic performance. A feedback loop and a PI controller are designed to adjust the output current and track the desired current. The system block diagram is shown in Fig. 3. The value of transfer function for the whole bridge is Udc . Denoting IO as the output, the open-loop transfer function of the system without the controller is defined in Eq. (5). The open-loop transfer function of the system has been modified as follows: (KP ×S + KI ) × Udc . (6) L(s) = s(Ls + r)
E(s) = E (s)Ir (s) =
s2 +
(r+Kp Udc ) s L
where a1, a2, a3, a4 are parameters. The closed-loop system is stable, so the steady-state error is a3 s + a4 Ess (s) = 2 . (12) s + ω2 The steady-state error of the time-domain form can be obtained by inverse Laplace transform:
B. Traditional PI controller
E (s) =
s 2 + Lr s
Ls 2 + rs ω Im 2 . 2 Ls + (r + KP Udc )s + KI Udc s + ω2 (10)
(15)
The inverter is controlled by the pulse width modulated wave, where the period of N switching waves is equivalent to the period of the fundamental sinusoidal signal, whose amplitude in each switching period is a fixed value. So, considering
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the correction signal becomes zero and the steady-state error is eliminated: Qem = Kp × em (k) + KI × [em (k) + K × em (k − 1) +K 2 × em (k − 2) + ...... + K k × em (0)].
FIG. 4. PI controller and system compensation: open-loop Bode plots.
the same phase value of the reference signal, the input signal is a step function ignoring the errors. The output of PI controller is Qe : Qe = Kp × e (t) + KI e (t) dt, (16) where e(t) is the error. When implemented in the experiment, the Qe is Qe = Kp × err(k) + KI × [err(k) + err(k − 1) + err(k − 2) + ...... + err(0)].
(17)
The errors at different phases are calculated by comparing the corresponding angles of the output signals and the reference signals, and then the control value is calculated by the PI controller. Since the output for the N-step input can be obtained, the sinusoidal input signals are transformed into Nstep input first to eliminate steady-state error. The correction signal of the same phase in each period will accumulate as long as the error exists. When the error signal becomes zero,
(18)
Qem is the correction signal of phase point m in current cycle, em (k) represents the error signal of phase point m (m = 1, 2, . . . , N) in the current cycle, em (k − 1), em (k − 2) . . . em (0) represent the error signals of phase point m in last 1, 2, . . . , k cycles, respectively. KI and KP are the integral coefficient and proportional coefficient, respectively. K is the attenuation coefficient of integral part. With accumulation of the point m error, the part of current error will become smaller over time under the controller. Since the system has a phase delay, the current calculated value of control can only be used for the next fundamental cycle. Each control cycle has the independent amount of the error integral and it is relatively easy to implement a phase lead compensator. As the control value always has a fundamental cycle delay, the dynamic characteristics of the system become worse. As a result, the proportional component will be used to the current control cycle, so the dynamic performance can be improved. Set the zero of PI controller equal to the switching frequency, fc = 1500 Hz that fc =
KI = 1500 Hz. 2π KP
(19)
Let the crossover frequency of open-loop transfer function to be 150 Hz, which is 1/10 of the resonant frequency. The original system denoted as G0 (s) has the amplitude attenuation as 20 dB per 10 octave and 40 dB per 10 octave before and after compensation, respectively. The amplitude is 40 dB at resonant frequency and by adding 20 dB from resonant frequency to the switching frequency the attenuation coefficient is up to 60 dB. It is to be noted that the low frequency response is poor if the crossover frequency is too low, while the attenuation will be small if the crossover frequency is high.
FIG. 5. Simulation model of the proposed PI controller.
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Rev. Sci. Instrum. 85, 025110 (2014)
(a)
(b)
FIG. 6. Waveforms of grid voltage and the grid-collected current. (a) Traditional PI control method. (b) Improved PI control method.
(a)
(b)
FIG. 7. Spectra of the current sent to the grid. (a) Traditional PI control method. (b) Improved PI control method.
FIG. 8. Internal structure of the experimental photovoltaic grid-connected inverter.
FIG. 9. Experiment circuit and experimental test platform.
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TABLE I. Experimental setup parameters. Parameters
Symbol
Value
Normal power DC voltage Decoupling capacitor Interface inductance Interface capacitance Low-frequency transformer Grid voltage Fundamental frequency Switching/sampling frequency
PN Ud C1 L1, L2 C2 KT UN fr fc/fs
700 W 200 V 3000 μF 1.2 mH 2.2 μF 1.5 120 V 60 Hz 21.6 kHz
By assuming the loop gain at crossover is 1, then (KP s + KI ) × Udc = 1. s(Ls + r) s=j 2π×150
The duty cycle of the PWM wave is determined by the ratio of the grid voltage and the DC voltage; and the amount that reference current and feedback current error signal through the controller. The results in Figs. 6 and 7 are the comparison between the proposed method and the traditional PI controller with the same KP and KI values, which clearly demonstrated the improvements. It is to be noted that in the model of improved algorithm, N = 360, f = 18000 Hz, a load with r = 20, the reference current is 20 A. KP = 0.0012, KI = 11.06. Figs. 6(a) and 6(b) show the waveforms of grid voltage and the grid-collected current with the PI controller and improved PI controller, respectively. Figs. 7(a) and 7(b) show the average THDs of the grid-collected current, which are 0.04% and 22.32%, respectively.
(20)
Using Eqs. (19) and (20), we can easily obtain that KP = 0.0012, KI = 11.06. Substituting them into (6) to get the system open-loop transfer function: 0.24s + 2212 . (21) 0.0024s 2 + 0.13s Fig. 4 is the corresponding Bode plots for an open-loop transfer function of the compensation system, attenuation up to 62 dB at 21 kHz, and the phase margin is positive, which implies that the system is stable. G(s) =
B. Simulation results
Our methods have been tested and validated through simulation studies and the MATLAB simulation model of improved algorithm is shown in Fig. 5. The solar panel input is equivalent to DC voltage and the high frequency PWM waves are obtained after the inverter and sent to the grid after passing through filters. The phase of the grid-connected current and the grid voltage must be the same so the grid will be harmfully affected by the current.
IV. EXPERIMENTAL VALIDATION
Experimental validation was also carefully designed and conducted in our study. A PV grid-connected inverter designed using power frequency isolation topology of fullbridge inverter is shown in Fig. 8, where C1 is the energy storage capacitor; L1, L2, and C2 form a filter. The transformer adjusts the input voltage range and isolates PV modules from the grid. A. Experimental platform
The experiment setup is shown in Fig. 9, where the DC voltage is supplied by the DC voltage stabilizer that is in series with a resistance, so its power can be adjusted in order to simulate the internal resistance of the solar PV modules. Our setup uses a low-frequency isolation PV grid-connected inverter with a maximum power of 700 W. The system parameters are shown in Table I. The inverter’s output is connected to the public power grid. The resistance is adjustable to test different power points.
FIG. 10. ((a) and (b)) Experimental results with the improved controller.
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TABLE II. Harmonic content and output power. Output power (W) THD (%)
140 5.3
210 4.5
280 4.0
350 3.0
420 2.9
ACKNOWLEDGMENTS 490 2.9
560 2.7
630 2.2
700 2.1
This work is partially supported by the National Natural Science Foundation of China (Grant No. 61102141) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20100185120005).
B. Experimental results and discussion
As the circuit diagram shown in Fig. 8, by adjusting the variable resistor R the output current waveform at two different power points can be obtained. Fig. 10(a) shows the waveforms of the grid current and voltage at the output power of 440 W and Fig. 10(b) shows the similar waveforms at the output power of 700 W. The grid experiment waveform harmonic and power factor shown in Table II are measured using a power quality analyzer. By using a proposed PI controller method that eliminates the steady-state error, the grid current harmonic content is significantly reduced and shown in Table II in which the THD values are measured at different power points. Table II shows that the harmonic content is reduced as the power increases: 3.0% at 50% of the rated power (350 W); at full power, the lowest is the 2.1%, which meets the requirements of less than 3% of the index. From Fig. 10(b), the harmonic still exists in the current waveform under full power operation. This is because the output current accuracy is affected by sampling. The proportion of inherent noise of the sampling circuit is high, which leads to high harmonic content. As the power increases, the current increases, the effect of noise decreases, and the harmonic content is reduced significantly.
V. CONCLUSIONS
The conventional PI controller reduces the steady-state error to zero with the step input, but cannot track time-varying sinusoidal signal. There is a steady-state error, which leads to system instability. At the moment these systems connecting to the grid, it will distort the grid signal. The PI controller can eliminate the steady-state error and improve the quality of the output current in the case of sinusoidal signal discretized into multiple step input, because the value of fixed phase in each cycle is equivalent to a straight flow which is a step function. It has been implemented in this paper, called the improved PI controller method. This method can eliminate not only the steady-state error in the system, but also the periodic error caused by switching dead or voltage drop of power tube. The existing PI controller has greatly improved, which cannot eliminate the system steady-state error caused by the sinusoidal input. The effectiveness and practicality of this control method have been verified by experiments. Because of the limitations of the hardware, for example, the current sensor’s accuracy is not high enough in the sample circuit. The final current THD cannot reach 0. It can be 1.1% with further improvement. In the following work, we will attempt to improve the hardware circuit so that current THD can be improved significantly.
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compensation of 6l±1 harmonic components,” Ind. Electron. 55(8), 3150– 3158 (2008). 21 S. Hara, Y. Yamamoto, T. Omata, and M. Nakano, “Repetitive control system: A new type servo system for periodic exogenous signals,” IEEE Trans. Autom. Control 33(7), 659–668 (1988). 22 S. Chen, Y. M. Lai, S.-C. Tan, and C. K. Tse, “Analysis and design of repetitive controller for harmonic elimination in PWM voltage source inverter systems,” Power Electron., IET 1, 497–506 (2008). 23 W.-S. Yao and M.-C. Tsai, “Analysis and estimation of tracking errors of plug-in type repetitive control systems,” IEEE Trans. Autom. Control 50(8), 1190–1195 (2005).
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Ye, K. Zhou, B. Zhang, D. Wang, and J. Wang, “High-performance repetitive control of PWM DC-AC converters with real-time phase-lead FIR filter,” IEEE Trans. Circuits Syst. 53(8), 768–772 (2006). 25 K. Zhou, D. Wang, B. Zhang, and Y. Wang, “Plug-in dual-mode-structure repetitive controller for CVCF PWM inverters,” IEEE Trans. Ind. Electron. 56(3), 784–791 (2009). 26 S. B. Kjaer, J. K. Pedersen, and F. Blaabjerg, “A review of single-phase grid-connected inverters for photovoltaic modules,” IEEE Trans. Ind. Appl. 41(5), 1292–1306 (2005). 27 A. Sakamoto and N. Hori, “New PWM schemes based on the principle of equivalent areas,” in Proceedings of the 2002 IEEE International Symposium on Industrial Electronics (IEEE-ISIE, 2002), Vol. 2, pp. 505–510.
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 025110 (2014)
Improvement of the grid-connect current quality using novel proportional-integral controller for photovoltaic inverters Yuhua Cheng,a) Kai Chen, Libing Bai, and Jing Yang School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China
(Received 7 December 2013; accepted 4 February 2014; published online 25 February 2014) Precise control of the grid-connected current is a challenge in photovoltaic inverter research. Traditional Proportional-Integral (PI) control technology cannot eliminate steady-state error when tracking the sinusoidal signal from the grid, which results in a very high total harmonic distortion in the grid-connected current. A novel PI controller has been developed in this paper, in which the sinusoidal wave is discretized into an N-step input signal that is decided by the control frequency to eliminate the steady state error of the system. The effect of periodical error caused by the dead zone of the power switch and conduction voltage drop can be avoided; the current tracking accuracy and current harmonic content can also be improved. Based on the proposed PI controller, a 700 W photovoltaic grid-connected inverter is developed and validated. The improvement has been demonstrated through experimental results. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4866023] I. INTRODUCTION
In recent years, the distribution system of renewable energy sources is gaining greater attention. As a result, new sources of renewable energy, such as solar energy and fuel cells, have become more and more critical.1 The distributed generation system comprising of photovoltaic (PV) modules, grid-connected inverter and the grid, realizes the transmission and usage of solar energy. The grid-connected inverter transforms the DC power from PV modules into AC power. The grid current total harmonic distortion (THD) affects the efficiency of system, which means the higher the efficiency, the smaller is the current harmonic distortion.2 Improper control will create significant harmonic distortion under a linear or sinusoidal load. It cannot achieve good results when the direct input signal of PI controller is sinusoidal. Besides, it also increases the THD and the tracking accuracy of the grid decreases with sinusoidal inputs.3, 4 Grid corporations give the requirements of power generation equipment for distributed grid such as the standards that deal with issues including power quality, detection of islanding operation, grounding, etc.1 For example, according to the IEEE1547 standard, the current harmonic content should be less than 5%.5 Ideally, the grid-connected current is a sine wave and the traditional control technology based on the classical feedback control is widely used. The error is obtained by subtracting the feedback current from the reference current. By eliminating this error with the current controller, the grid-collected current is close to the standard sine wave. The common control strategies include proportional-integral (PI), proportionalresonant (PR), predictive deadbeat (DB), repetitive controller (RC), and hysteresis controllers.6 However, it is difficult for PI controller to track the time-varying AC sinusoidal signal accurately and results a steady-state error. The quality of a) Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0034-6748/2014/85(2)/025110/8/$30.00
output current can be improved by PR control with a good steady state performance, but only when the controller resonant frequency is the same as the grid frequency.7 DB control is widely used to eliminate the error current, while its performance and system parameters are closely related.8, 9 Hysteresis control is easy to use and has good dynamic response characteristics. However, the sampling control frequency is not fixed, resulting in a high current ripple, which reduces the quality of the current. Besides, the output filters are difficult to design.10–12 Besides, the authors in Ref. 13 proposed a hybrid controller and a fuzzy-PI controller is introduced in Refs. 14 and 15. The feature model of the external interference signal is integrated into the controller by repetitive control in order to better suppress the interference. So, the repetitive control is widely used in high-precision servo control systems. As a result, the structure of the system is stable. In order to set the steady-state error to zero, and to ensure that the system has a good tracking ability and also eliminate interference, the dynamic model of the disturbance must be included in the controller. This interference signal dynamic model is called the internal model; the implemented internal model controller is the repetitive controller.16, 17 In recent years, a repetitive control method18–21 has been widely used in the inverter system due to its ability to suppress the periodic interference. It can eliminate harmonics and deal with a very large number of harmonics simultaneously.3, 22 Most importantly, repetitive control is one area in which the objective is mainly to remove the steady-state error due to periodic inputs.23 Ye et al.24 gave the repetitive control with Real-Time Phase-Lead FIR filter. Zhou et al.25 presented a Plug-In Dual-Mode-Structure Repetitive Control. Many interfering signals are announced to have an effect because the grid-connected inverter system tracks a reference standard sine wave with a varying amplitude and phase instead of a constant value. These interfering signals contain multiple harmonics in addition to the fundamental one
85, 025110-1
© 2014 AIP Publishing LLC
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Rev. Sci. Instrum. 85, 025110 (2014)
PV Module
inverter
takes the pulse width modulated (PWM) waves as input and output the sine-wave through the LC filter. The error current is obtained by comparing the output feedback current and the expected current (Iref ), and ultimately suppressed or eliminated by the proposed controller. The Iref is obtained by Maximum Power Point Tracking method (MPPT). The voltage feed-forward can eliminate the grid disturbance, and the design of error controller is simplified after adding it. Assuming that the duty cycle of the bridge control is D , UO is the output voltage, and Udc as the input voltage, the relationship among these three variables can be derived according to the principle of equivalent areas:27
grid AC Switching
DC Switching
Local load FIG. 1. Grid-connected PV power generation system.
and they are not constant. Therefore, it becomes extremely difficult to use the traditional PI control method to achieve high-precision tracking. In order to solve this problem, a novel PI controller has been developed. This method is simpler than repetitive control due to several reasons: (1) Interference signal modeling is not needed; (2) Expressions of the proposed PI method are simpler than repetitive control; (3) Less steps to achieve the purpose. In this paper, the system model is developed and analyzed. In contrast to the previous work on the traditional PI control methods, a pure delay link has been implemented in the designed system with the reference sinusoidal signal being discretized into the N discrete step input. The high-precision control is achieved through our novel PI controller approach. The rest of this paper is organized as follows: Sec. II presents the system model and transfer function with the analysis of the limitations of conventional PI controller. Section III introduces the proposed PI control algorithm and Sec. IV reports the design and experimental results of the experimental prototype. Finally, the conclusion is drawn in Sec. V.
UO = Udc × D .
Full-bridge is assumed to be a proportional component; transfer function is equal to the input; ignoring the impact of switching device voltage drop and dead zone, (1) is the ideal mathematical model. Ignoring the capacitive current of the LC filter, the voltage UL is the difference value of inverterbridge-output voltage UO and grid voltage UN . Also UL is equal to the differential of the grid-connected current IO and the voltage component on resistor of the inductor. The relationship between voltages and current can be written as UL = L
Io (s) =
As shown in Fig. 1, the grid-connected PV power generation system is formed by the PV modules, grid-connected inverter, and grid.26 Fig. 2 shows the core component of the inverter that is composed of a grid-connected inverter bridge and an LC filter. The current goes through the grid-connected inverter bridge which works as a DC-to-AC conversion, and then the gridconnected current is obtained via the LC filter. The inverter bridge works under the high-frequency switching state that
+
(3)
Grid
IO Iref +
+
U o − UN . Ls + r
Substituting it into Eq. (3), the open-loop transfer function of full-bridge and the inductance of the grid system is
Inverter LC UO Bridge Filter
PWM Generator
(2)
The full-bridge output voltage must be greater than the grid voltage in order to send the power to the grid. The output voltage is broken down into two parts: one part to offset the grid and the other part to produce an output current. Therefore, D is divided into two parts: UN + D = UN + Udc × D. (4) Uo = Udc × Udc
A. Modeling
Udc
dIO + IO r = UO − UN , dt
where L is the inductance value and r is the resistance component of L. The grid-connected current IO is integrated of the error voltage UL by the LC filter. The Laplace transform of Eq. (2) can be easily obtained as
II. PV POWER GENERATION SYSTEM AND MODEL BASED ANALYSIS
DC Power Source
(1)
Controller
-
The voltage feed-forward link
err
UN K
FIG. 2. Structure of grid-connected inverter.
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1 Udc E
Ir +
KPs+KI s
-
+ +
D
Udc
Simplified:
UN
E(s) = E (s)Ir (s) Uo +
-
1 Ls+r
Io
= =
FIG. 3. System closed-loop block diagram based on PI control.
Udc IO (s) = . (5) D Ls + r The type-0 system cannot track the sinusoidal signal according to automatic control principle. G0 (s) =
KP is the proportional coefficient and KI is the integral coefficient. Equations (5) and (6) show that after adding in the PI controller, the system changes from the type-0 system into the type-I system. There is always a steady-state error when the type-0 system tracks a step signal and this issue has been remedied in this new type-I system. The system model is simplified by setting the grid voltage feed-forward coefficient as 1/Udc so that the grid voltage is not included. When the input is a sinusoidal signal, the steady-state error of the system is analyzed as follows. Assume that the response of the linear time-invariant system (s) is (s)est , where the error transfer function is obtained as Ls 2 + rs 1 = . 1 + L(s) Ls 2 + (r + KP Udc )s + KI Udc (7) For the input sinusoidal signal, ω Ir (s) = Im 2 . s + ω2 The error response of system can be found:
+
a1 s + a2 (r+Kp Udc ) s + KiLUdc L
Im +
s2
ω + ω2
a3 s + a4 , s 2 + ω2
1 Ir (s), 1 + G(s)
(8)
(11)
(13)
where η, α, and θ are parameters. From Eq. (13), it is noted that the system with PI controller has a steady-state error in both amplitude and phase when the input is a sinusoidal signal. The reason for that is while dealing with currents with expectation as standard sinusoidal signal rather than the step function, the traditional PI control method cannot achieve high precision for tracking the sinusoidal signal. The system’s stability and control accuracy will also be affected when digital control is used to introduce phase delay. Furthermore, the actual full-bridge inverter is not a proportional component, and the grid voltage feed-forward mechanism is not a remedy for the same phase delay and error. The actual output current error is greater than the calculated results above because of the existence of the errors such as sampling, calculating, and so on. All of these errors can cumulate and lead to the final results that are not satisfying the system design requirements and are the major drive and motivation for the proposed improvement. III. THE PROPOSED PI CONTROLLER DESIGN A. Methods
As discussed in Sec. II, PI controller can improve the system to the type-I system. Although it cannot track a sinusoidal signal, the following derivations show that the steady-state error for step input signals is 0. For the same transfer function of the system shown in Eq. (7) with step input: Im . S The steady-state error can be found as Ir (S) =
(14)
s × Ism s−>0 1 + G(s)
ess = lim (9)
s × Ism × s(Ls + r) = 0. s−>0 1 + (KP S + KI )Udc + s(Ls + r)
= lim
E(s) = E (s)Ir (s) =
s2
+
Ki Udc L
ess (t) = η cos(αω + θ ),
A PI controller changes the system type and improves system dynamic performance. A feedback loop and a PI controller are designed to adjust the output current and track the desired current. The system block diagram is shown in Fig. 3. The value of transfer function for the whole bridge is Udc . Denoting IO as the output, the open-loop transfer function of the system without the controller is defined in Eq. (5). The open-loop transfer function of the system has been modified as follows: (KP ×S + KI ) × Udc . (6) L(s) = s(Ls + r)
E(s) = E (s)Ir (s) =
s2 +
(r+Kp Udc ) s L
where a1, a2, a3, a4 are parameters. The closed-loop system is stable, so the steady-state error is a3 s + a4 Ess (s) = 2 . (12) s + ω2 The steady-state error of the time-domain form can be obtained by inverse Laplace transform:
B. Traditional PI controller
E (s) =
s 2 + Lr s
Ls 2 + rs ω Im 2 . 2 Ls + (r + KP Udc )s + KI Udc s + ω2 (10)
(15)
The inverter is controlled by the pulse width modulated wave, where the period of N switching waves is equivalent to the period of the fundamental sinusoidal signal, whose amplitude in each switching period is a fixed value. So, considering
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the correction signal becomes zero and the steady-state error is eliminated: Qem = Kp × em (k) + KI × [em (k) + K × em (k − 1) +K 2 × em (k − 2) + ...... + K k × em (0)].
FIG. 4. PI controller and system compensation: open-loop Bode plots.
the same phase value of the reference signal, the input signal is a step function ignoring the errors. The output of PI controller is Qe : Qe = Kp × e (t) + KI e (t) dt, (16) where e(t) is the error. When implemented in the experiment, the Qe is Qe = Kp × err(k) + KI × [err(k) + err(k − 1) + err(k − 2) + ...... + err(0)].
(17)
The errors at different phases are calculated by comparing the corresponding angles of the output signals and the reference signals, and then the control value is calculated by the PI controller. Since the output for the N-step input can be obtained, the sinusoidal input signals are transformed into Nstep input first to eliminate steady-state error. The correction signal of the same phase in each period will accumulate as long as the error exists. When the error signal becomes zero,
(18)
Qem is the correction signal of phase point m in current cycle, em (k) represents the error signal of phase point m (m = 1, 2, . . . , N) in the current cycle, em (k − 1), em (k − 2) . . . em (0) represent the error signals of phase point m in last 1, 2, . . . , k cycles, respectively. KI and KP are the integral coefficient and proportional coefficient, respectively. K is the attenuation coefficient of integral part. With accumulation of the point m error, the part of current error will become smaller over time under the controller. Since the system has a phase delay, the current calculated value of control can only be used for the next fundamental cycle. Each control cycle has the independent amount of the error integral and it is relatively easy to implement a phase lead compensator. As the control value always has a fundamental cycle delay, the dynamic characteristics of the system become worse. As a result, the proportional component will be used to the current control cycle, so the dynamic performance can be improved. Set the zero of PI controller equal to the switching frequency, fc = 1500 Hz that fc =
KI = 1500 Hz. 2π KP
(19)
Let the crossover frequency of open-loop transfer function to be 150 Hz, which is 1/10 of the resonant frequency. The original system denoted as G0 (s) has the amplitude attenuation as 20 dB per 10 octave and 40 dB per 10 octave before and after compensation, respectively. The amplitude is 40 dB at resonant frequency and by adding 20 dB from resonant frequency to the switching frequency the attenuation coefficient is up to 60 dB. It is to be noted that the low frequency response is poor if the crossover frequency is too low, while the attenuation will be small if the crossover frequency is high.
FIG. 5. Simulation model of the proposed PI controller.
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Rev. Sci. Instrum. 85, 025110 (2014)
(a)
(b)
FIG. 6. Waveforms of grid voltage and the grid-collected current. (a) Traditional PI control method. (b) Improved PI control method.
(a)
(b)
FIG. 7. Spectra of the current sent to the grid. (a) Traditional PI control method. (b) Improved PI control method.
FIG. 8. Internal structure of the experimental photovoltaic grid-connected inverter.
FIG. 9. Experiment circuit and experimental test platform.
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TABLE I. Experimental setup parameters. Parameters
Symbol
Value
Normal power DC voltage Decoupling capacitor Interface inductance Interface capacitance Low-frequency transformer Grid voltage Fundamental frequency Switching/sampling frequency
PN Ud C1 L1, L2 C2 KT UN fr fc/fs
700 W 200 V 3000 μF 1.2 mH 2.2 μF 1.5 120 V 60 Hz 21.6 kHz
By assuming the loop gain at crossover is 1, then (KP s + KI ) × Udc = 1. s(Ls + r) s=j 2π×150
The duty cycle of the PWM wave is determined by the ratio of the grid voltage and the DC voltage; and the amount that reference current and feedback current error signal through the controller. The results in Figs. 6 and 7 are the comparison between the proposed method and the traditional PI controller with the same KP and KI values, which clearly demonstrated the improvements. It is to be noted that in the model of improved algorithm, N = 360, f = 18000 Hz, a load with r = 20, the reference current is 20 A. KP = 0.0012, KI = 11.06. Figs. 6(a) and 6(b) show the waveforms of grid voltage and the grid-collected current with the PI controller and improved PI controller, respectively. Figs. 7(a) and 7(b) show the average THDs of the grid-collected current, which are 0.04% and 22.32%, respectively.
(20)
Using Eqs. (19) and (20), we can easily obtain that KP = 0.0012, KI = 11.06. Substituting them into (6) to get the system open-loop transfer function: 0.24s + 2212 . (21) 0.0024s 2 + 0.13s Fig. 4 is the corresponding Bode plots for an open-loop transfer function of the compensation system, attenuation up to 62 dB at 21 kHz, and the phase margin is positive, which implies that the system is stable. G(s) =
B. Simulation results
Our methods have been tested and validated through simulation studies and the MATLAB simulation model of improved algorithm is shown in Fig. 5. The solar panel input is equivalent to DC voltage and the high frequency PWM waves are obtained after the inverter and sent to the grid after passing through filters. The phase of the grid-connected current and the grid voltage must be the same so the grid will be harmfully affected by the current.
IV. EXPERIMENTAL VALIDATION
Experimental validation was also carefully designed and conducted in our study. A PV grid-connected inverter designed using power frequency isolation topology of fullbridge inverter is shown in Fig. 8, where C1 is the energy storage capacitor; L1, L2, and C2 form a filter. The transformer adjusts the input voltage range and isolates PV modules from the grid. A. Experimental platform
The experiment setup is shown in Fig. 9, where the DC voltage is supplied by the DC voltage stabilizer that is in series with a resistance, so its power can be adjusted in order to simulate the internal resistance of the solar PV modules. Our setup uses a low-frequency isolation PV grid-connected inverter with a maximum power of 700 W. The system parameters are shown in Table I. The inverter’s output is connected to the public power grid. The resistance is adjustable to test different power points.
FIG. 10. ((a) and (b)) Experimental results with the improved controller.
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Rev. Sci. Instrum. 85, 025110 (2014)
TABLE II. Harmonic content and output power. Output power (W) THD (%)
140 5.3
210 4.5
280 4.0
350 3.0
420 2.9
ACKNOWLEDGMENTS 490 2.9
560 2.7
630 2.2
700 2.1
This work is partially supported by the National Natural Science Foundation of China (Grant No. 61102141) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20100185120005).
B. Experimental results and discussion
As the circuit diagram shown in Fig. 8, by adjusting the variable resistor R the output current waveform at two different power points can be obtained. Fig. 10(a) shows the waveforms of the grid current and voltage at the output power of 440 W and Fig. 10(b) shows the similar waveforms at the output power of 700 W. The grid experiment waveform harmonic and power factor shown in Table II are measured using a power quality analyzer. By using a proposed PI controller method that eliminates the steady-state error, the grid current harmonic content is significantly reduced and shown in Table II in which the THD values are measured at different power points. Table II shows that the harmonic content is reduced as the power increases: 3.0% at 50% of the rated power (350 W); at full power, the lowest is the 2.1%, which meets the requirements of less than 3% of the index. From Fig. 10(b), the harmonic still exists in the current waveform under full power operation. This is because the output current accuracy is affected by sampling. The proportion of inherent noise of the sampling circuit is high, which leads to high harmonic content. As the power increases, the current increases, the effect of noise decreases, and the harmonic content is reduced significantly.
V. CONCLUSIONS
The conventional PI controller reduces the steady-state error to zero with the step input, but cannot track time-varying sinusoidal signal. There is a steady-state error, which leads to system instability. At the moment these systems connecting to the grid, it will distort the grid signal. The PI controller can eliminate the steady-state error and improve the quality of the output current in the case of sinusoidal signal discretized into multiple step input, because the value of fixed phase in each cycle is equivalent to a straight flow which is a step function. It has been implemented in this paper, called the improved PI controller method. This method can eliminate not only the steady-state error in the system, but also the periodic error caused by switching dead or voltage drop of power tube. The existing PI controller has greatly improved, which cannot eliminate the system steady-state error caused by the sinusoidal input. The effectiveness and practicality of this control method have been verified by experiments. Because of the limitations of the hardware, for example, the current sensor’s accuracy is not high enough in the sample circuit. The final current THD cannot reach 0. It can be 1.1% with further improvement. In the following work, we will attempt to improve the hardware circuit so that current THD can be improved significantly.
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compensation of 6l±1 harmonic components,” Ind. Electron. 55(8), 3150– 3158 (2008). 21 S. Hara, Y. Yamamoto, T. Omata, and M. Nakano, “Repetitive control system: A new type servo system for periodic exogenous signals,” IEEE Trans. Autom. Control 33(7), 659–668 (1988). 22 S. Chen, Y. M. Lai, S.-C. Tan, and C. K. Tse, “Analysis and design of repetitive controller for harmonic elimination in PWM voltage source inverter systems,” Power Electron., IET 1, 497–506 (2008). 23 W.-S. Yao and M.-C. Tsai, “Analysis and estimation of tracking errors of plug-in type repetitive control systems,” IEEE Trans. Autom. Control 50(8), 1190–1195 (2005).
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