H REPETITIVE CURRENT CONTROLLER FOR

Transcription

1 H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS Tomas Hornik and Qing-Chang Zhong Dept. of Electrical Eng. & Electronics The University of Liverpool UK

2 Acknowledgement & apology T. Hornik would like to acknowledge the financial support from the EPSRC, UK under the DTA scheme and Q.-C. Zhong would like to thank the Royal Academy of Engineering and the Leverhulme Trust for awarding him a Senior Research Fellowship. Dr Zhong would like to send his sincere apology for having to cancel his trip at the last minute. T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 2/29

3 Outline Motivation Brief introduction to repetitive control Overall structure of the system Synchronisation H controller design Experimental setup and results Summary T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 3/29

4 Motivation Increasing share of renewable energy UK: 20% by 2020 EU: 22% target for the share of renewable energy sources and an 18% target for the share of CHP in electricity generation by 2010 Regulation of system frequency and voltage Threat to power system stability Power quality T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 4/29

5 Power quality improvement Power quality is an important problem for renewable energy and distributed generation. The maximum total harmonic distortion (THD) of output voltage allowed is 5% (120V 69kV ). The maximum THD allowed in current is shown below: Odd harmonics Maximum current THD < 11 th < 4% 11 th 15 th < 2% 17 th 21 th < 1.5% 23 rd 33 rd < 0.6% > 33 rd < 0.3% T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 5/29

6 Current status V DC e a e b e c L s, R s i a i b i c v c v b v a L g, R g Circuit Breaker v ga v gb v gc - C Currently, most grid-connected inverters adopt the VSI topology with a current controller to regulate the current injected into the grid by using schemes Proportional-integral (PI) controllers in the synchronously rotating (d, q) reference frame: works well with balanced systems, but cannot cope with unbalanced disturbance currents Proportional-resonant (PR) controllers in the stationary (α, β) reference frame: popular due to the capability of eliminating the steady state error, while regulating sinusoidal signals, and the possible extension to compensate multiple harmonic but difficult to cope with varying grid frequency. Hysteresis controllers in the natural (abc) frame: simple and fast but it results in high and variable sampling frequencies. T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 6/29

7 Repetitive control PI controllers are good for tracking or rejecting step signals. But for inverters, the signals are sinusoidal. In order to have good tracking performance, a pair of conjugate poles on the imaginary axis are needed. Proportional-resonant (PR) : ω s 2 ω 2 Repetitive control: 1 1 e τ d s, where τ d is close to the signal period. In order to guarantee the stability, a low-pass filter W(s) is often added so the internal model is 1 1 W(s)e τ d s. e p e -τ d s W(s) T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 7/29

8 Poles of the internal model 1 x Im * o true poles approximated poles Re T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 8/29

9 Objective of this talk To design a current controller to minimise the current THD, which is equipped with the repetitive control technique designed with the H control theory To demonstrate the performance with experimental results Also to cover other issues, such as synchronisation T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 9/29

10 Overall structure of the system Transformer DC power source Inverter bridge LC filter i a i b i c u ga u gb u gc PWM modulation u u ga u gb u gc Phase-lead low-pass filter u Internal model M and stabilizing compensator C u ga u gb u gc e i ref PLL θ dq abc I d * I q * Current controller Individual controllers are adopted for each phase in the natural abc frame. Equipped with a neutral point controller so that a balanced neutral point is available. It has a current loop including a repetitive controller so that the current injected into the grid could track the reference current i ref, which is generated from the d, q-current references Id and I q using the dq abc transformation. A phase-locked loop (PLL) is used to provide the phase information of the grid voltage, which is needed to generate i ref. T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 10/29

11 Synchronisation When the references I d and I q are all equal to 0, the generated voltage should be equal to the grid voltage, i.e., the inverter should be synchronised with the grid and the circuit breaker could be closed at any time if needed. In order to achieve this, the grid voltages (u ga, u gb and u gc ) are feed-forwarded and added to the output of the repetitive current controller via a phase-lead low-pass filter F(s) = 33(0.05s 1) (s 300)(0.002s 1), which has a gain slightly higher than 1 and a phase lead at the fundamental frequency. It is introduced to compensate the phase shift and gain attenuation caused by computational delay, PWM modulation, the inverter bridge and the LC filter. It also attenuates the harmonics in the feed-forwarded grid voltages. simple (but effective) improves the dynamics during grid voltage fluctuations does not affect the independence of each phase. T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 11/29

12 H current repetitive control w u g i ref u P plant stabilizing compensator C e W ( s) e τ s internal model To minimise the tracking error e between the current reference and the current injected to the grid. M p d T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 12/29

13 Single phase representation V DC - u PWM Inverter bridge u f S u c o i 1 i c L g i 2 R f L f R g filter inductor C f grid interface inductor u c u g grid R d neutral States: x = [ i 1 i 2 u c ] T External signals: w = [ u g i ref ] T Controlled signal: e = i ref i 2 T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 13/29

14 State-space model with A = R fr d L f R d L g ẋ = Ax B 1 w B 2 u y = e = C 1 x D 1 w D 2 u R d L f 1 L f R gr d 1 L g L g 1 C f 1 C f 0 C 1 = [ 0 1 0, B 1 = ] [, D 1 = L g ], D 2 = 0., B 2 = The corresponding plant transfer function is then P = [ D 1 D 2 ] C1 (si A) 1 [ B 1 B 2 ]. 1 L f 0 0, T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 14/29

15 Internal model M e p e -τ d s W(s) τ d = τ 1 ω c, where ω c is the cut-off frequency of the low-pass filter W(s) = ω c sω c and τ is the signal period. In order to maintain the tracking performance of the controller, a frequency adaptive mechanism could be used (not presented here). T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 15/29

16 Formulation of the H problem ~ P b z~ w ~ v ξ a W µ w P e ~ y u C To minimise the H norm of T z w = F l( P,C) from w = [ v w ] T to z = [ z 1 z 2 ] T, after opening the local positive feedback loop of the internal model and introducing weighting parameters ξ and µ. T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 16/29

17 The closed-loop system can be represented as z ỹ = P w u, u = Cỹ, The extended plant P consists of the original plant P together with the lowpass filter W and weighting parameters ξ and µ. The additional parameters ξ and µ are added to provide more freedom in design. P = A 0 0 B 1 B 2 B w C 1 A w B w ξ B w D 1 B w D 2 0 C w µ C 1 0 ξ D 1 D 2 The stabilising controller C can be calculated using the well-known results on H controller design for the extended plant P.. T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 17/29

18 Stability evaluation Assume that the state-space realisation of the controller is [ ] Ac B C = c. C c D c The closed-loop system is exponentially stable if the closed-loop system designed above is stable and the transfer function from a to b, T ba = satisfies T ba < 1. A B 2 D c C 1 B 2 C c B 2 D c C w 0 B c C 1 A c B c C w A w B w C 1 0 C w 0, T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 18/29

19 Experimental setup It consists of an inverter board, a three-phase LC filter, a three-phase grid interface inductor, a board consisting of voltage and current sensors, a step-up transformer, a dspace DS1104 R&D controller board with ControlDesk software, and MATLAB Simulink/SimPower software package. The inverter board consists of two independent three-phase inverters and has the capability to generate PWM voltages from a constant 42V DC voltage source. The generated three-phase voltage is connected to the grid via a controlled circuit breaker and a step-up transformer. The grid voltage and the current injected into the grid are measured for control purposes. The sampling frequency of the controller is 5 khz and the PWM switching frequency is 20 khz. T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 19/29

20 Block diagram of the system Measure 1 PCB Measure 2 Transformer DC power source Inverter bridge LC filter d a d b d c Circuit breaker i u g dspace 1104 Inverter parameters Parameter Value Parameter Value L f 150µH R f 0.045Ω L g 450µH R g 0.135Ω C f 22µF R d 1Ω T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 20/29

21 H controller design The low-pass filter W is chosen as, for f = 50Hz, [ ] W =. 1 0 The weighting parameters are chosen to be ξ = 100 and µ = Using the MATLAB hinfsyn algorithm, the H controller C which nearly minimises the H norm of the transfer matrix from w to z is obtained as C(s) = (s 300.8)(s s ) (s )(s 2550)(s s ). T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 21/29

22 Controller reduction To replace s with 0 for very high-frequency modes To cancel the poles and zeros that are close to each other. The reduced controller is C(s) = (s 300.8) s 2550 = W(s)C PD (s) with (s 300.8) C PD (s) = The resulting T ba is and, hence, the closedloop system is stable. T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 22/29

23 Comparison of the controllers Magnitude (db) Phase (deg) Original Reduced Frequency (rad/sec) There is little difference at low frequencies. The Bode plots in the discrete time domain are almost identical, for the sampling frequency of 5kHz used for implementation. T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 23/29

24 The designed controller P e τ d s w u g i ref u plant e W (s) internal model M (s) C PD It is interesting to see that C PD (s) can actually be regarded as an inductor that converts the output (current) signal from the internal model to a voltage signal u. Using MATLAB c2d (ZOH) algorithm, the discretised controller can be obtained as C(z) = C (z ) z T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 24/29.

25 Experimental results Synchronisation process Steady-state responses Transient responses T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 25/29

26 Synchronisation process As explained before, grid voltages (u ga, u gb and u gc ) are feed-forwarded through a phase-lead low-pass filter and added to the control signal for the inverter to synchronise with the grid. The inverter synchronisation process was started at around t = second and, immediately, it is synchronised and ready to be connected to the grid. 20 u g 20 Voltage [V] #1:2 u A #1:1 Voltage error [A] #1: Time [sec] Time [sec] (a) output voltage u A and grid voltage u g (b) u A -u g T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 26/29

27 Steady-state responses Current [A] #1:1 #1:2 i A iref Current error [A] #1: Time [sec] Time [sec] (a) current output i A and its reference i ref (b) the current error e The current reference Id was set at 3A. This corresponds to 76.4W active power generated by the inverter. The reactive power was set at 0VAR (Iq = 0). This corresponds to the unity power factor. Since there is no local load included in the experiment, all generated active power was injected into the grid via a step-up transformer. The recorded current THD was 0.99%, while the grid voltage THD was 2.21%. T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 27/29

28 Transient responses Current [A] #1:1 #1:2 i ref i A Current error [A] #1: Time [sec] Time [sec] (a) current output i A and its reference i ref (b) the current error e A step change in the current reference Id from 2A to 3A was applied (while keeping Iq = 0). The inverter responded to the current step change in about 5 cycles. T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 28/29

29 Summary The H repetitive control strategy has been applied to the design of a current controller for grid-connected inverters. The resulting controller is simple and consists of an internal model and a proportional-derivative controller. It has shown that advanced control theories can be applied to design implementable controllers for practical applications and can offer insightful understanding to real problems. A simple and effective synchronisation mechanism has also been introduced for the proposed control strategy to quickly synchronise the inverter with the grid. Experimental results have shown that the proposed H repetitive current controller offers excellent performance with a recorded current THD less than 1%. T. HORNIK & Q.-C. ZHONG: H REPETITIVE CURRENT CONTROLLER FOR GRID-CONNECTED INVERTERS p. 29/29