and Q controller is applied and compared to a traditional amplitude and ... sinusoidal reference despite several forms of disturbance. Non-linear ... provide good waveform quality despite these disturbances. ... to the imaginary axis, around integer multiples of j2Ï·50, .... resulting from. ¡. Zg. .... The THD of current injected into.
Evaluation of repetitive control for power quality improvement of distributed generation J. Liang, T. C. Green, G. Weiss and Q.-C. Zhong Imperial College of Science, Technology and Medicine London SW7 2BT, U.K.
Abstract - Small-scale distributed generation (DG) is often not a natural 50 or 60 Hz AC source and so employs an inverter for the interface to the utility grid. Here, repetitive control is examined as a means of also using a DG inverter to improve the distortion of a local grid where a large proportion of the load is non-linear. The proposed controller can offer better waveform quality in balanced and unbalanced conditions than PI controllers in either stationary or rotating reference frames. The inverter also requires a control loop to regulate the exported power. A decoupled P and Q controller is applied and compared to a traditional amplitude and angle controller. The controllers are tested for disturbance rejection of variations in grid voltage, DC-link voltage and load power. Keywords: Distributed generation, inverter, grid connection, repetitive control, power control
1 INTRODUCTION The growing concern about availability and quality of electrical power has led to a development of distributed generation (DG) [1]. DG technologies include fuel cells, wind turbines, small gas-turbines and photovoltaic arrays. DG is relatively small and interconnected at substation, distribution feeder or customer load levels. Although some generators can be connected directly to the grid, many adopt power electronics based interfaces. Photo-voltaic arrays and fuel cells require DC to AC conversion. Variable speed wind-turbines require AC to DC to AC conversion as do high speed gas-turbine driven generators. The inverters switch at high frequency and can be designed for more than simply controlling the average power. The inverter can provide reactive power, voltage support, load balancing and harmonic mitigation. A heightened awareness of power quality issues [2] among electricity consumers means that inverter based DG may play an important role in power quality management. A voltage source inverter must be interfaced to the gird via an impedance. The impedance serves two purposes; it attenuates the switching frequency distortion of the inverter and it allows controlled fundamental frequency current flow between the two voltage sources. An inductor provides first-order filtering of the switching frequency. Because of the relatively small separation between the grid frequency and the switching frequency, it is often necessary to provide second or third order filtering. Here a second order LC filter is applied and therefore the inverter/filter output is a voltage. Local loads are connected to the filter output and a further inductor is provided for exchange of power with a grid. The filter output voltages must be controlled to follow a balanced
sinusoidal reference despite several forms of disturbance. Non-linear loads, principally rectifiers and lighting ballasts, will draw non-sinusoidal currents from the filter. Single-phase loads will create unbalanced. Transients will be caused switching loads and variation of the DC-link voltage. Proper control of the inverter and filter can provide good waveform quality despite these disturbances. In contrast, a conventional synchronous machine can not actively control the waveform and relies on a low source impedance to achieve good waveform quality. The waveform quality is, therefore, linked to the fault current contribution of the DG and a compromise must be struck. Waveform tracking is normally accomplished with a conventional controller, such as a PI controller, operating in either a stationary or rotating reference frame. Control in the rotating reference frame, through the dq transformation, of a balanced three-phase system can yield good waveform quality. For unbalanced conditions, the dq transformation is not particularly helpful and a stationary frame is commonly used. Repetitive control [3, 4, 5] offers a better alternative for voltage tracking, as it can deal with a very large number of harmonics simultaneously and even several disturbances at non-harmonic frequencies. The principle of repetitive control is that if a system is subject to periodic reference or a periodic disturbance, then error information from previous cycles, stored in a delay line, can be used to achieve better performance. This can be applied to DC to AC power converters but so far it has seen only limited use. In [6, 7] repetitive control was applied to constantvoltage, constant-frequency power sources supplying isolated single-phase loads. Once waveform quality has been addressed, the next task is to ensure control of the exported power. The power of the DG could be regulated or the export to the grid (after local loads have been supplied) could be regulated. The second approach is adopted here. The control system consists, therefore, of two nested loops. A repetitive controller is used in the inner loop to achieve good waveform tracking. Decoupling control is used in the outer loop to control active power (P) and reactive power (Q) accurately and independently. The theoretical development and a design of a repetitive controller for inverter voltage has been described in [8]. Here the controller will be applied and tested in a DG environment in combination with a power controller.
2. THE POWER CONVERSION SYSTEM Fig. 1 shows the DG system under study. The energy source is a high-speed AC generator with a line voltage of 1000V at 3 kHz. The generator is connected via a stepdown transformer to a rectifier and a split DC-link of ±425V. The centre point of the link forms the neutral connection of the three-phase system and is held in balance by a pair of IGBTs and a voltage controller. The inverter is switched at 10 kHz and feeds a star connected LC filter and a set of local loads. The load has a linear element (a star connected set of resistors) and a nonlinear element (an uncontrolled diode rectifier and resistor). Interface inductors connect the output of the filter to the power grid. The inductance of the transmission line is lumped into the interface inductor. The filter inductors and interface inductors are modelled with a series winding resistance and a parallel core loss resistance. Two contactors are included, Sc and Sg, which are needed in the start-up and shout-down procedures of the inverter. The principle parameters are given in table 1. DC link capacitor DC link inductor Filter inductor Filter capacitor Interface inductor Output power Local Load Reference Load Voltage
6600 F(×2) 0.75 mH 1.35 mH 50 F 0.3 mH 30 kW 25 kW 230 V
imaginary axis, leading to poor rejection of higher harmonics; too large and the system tends to be unstable. The compensator ensures exponential stability of the entire system. Detailed design of the compensator is carried out in [8]. The repetitive voltage controller, based on the internal model principle, ensures that the output voltage of each phase tracks the periodic reference. The power controller monitors the grid voltage and, based on reference values for active and reactive power, generates reference voltages for the inner loop. A PLL is used to synchronise the system. The controller also identifies the fundamental frequency voltage magnitude and generates a quadrature voltage signal. Fundamental frequency real and reactive powers are calculated from instantaneous current and direct and quadrature voltages. The powers are averaged over one cycle but updated every 2ms. An equivalent single-phase system circuit is given in Fig. 3. The active power and the reactive power exchanged with the grid will be controlled by manipulating the output voltage, vout. Assuming that the output voltages track the reference voltages accurately, and that the grid voltage, Vg is measured by the PLL (with an assumed angle of zero) the control of the power flow is achieved by adjusting the reference voltage Vref . Vref = V g +
Table 1 System design specification and parameters
3. THE CONTROL SYSTEM
Q P ⋅Zg − j ⋅Zg Vg Vg
If Zg is known, Vref can be calculated and used in an openloop, feed-forward controller. In practice, an accurate value of Zg is hard to acquire. However, the angle, Zg of Zg can be found with reasonably accuracy. The method proposed here involves forming a unit vector n in the direction of Zg. n = Z g / Z g = cos(θ zg ) + j sin(θ zg )
There are two control objectives: (i) to maintain the phase voltages close to the reference voltages and (ii) control the power injected into grid according to its reference values. The power flow to the grid is controlled by manipulating the fundamental frequency component of the three voltages on the local load through setting the reference values for these voltages. The two loop controller is shown in Fig.2. The inner loop contains the internal model, the stabilizing compensator C (computed by H∞ design) and the plant to be controlled (PWM inverter, LC filters, local loads and interface inductors). The disturbances acting on the plant are distortion current id, the positive and negative DC-link voltage VDC+ and VDC-, and grid voltage vg (which is assumed to be periodic with a frequency of 50Hz).
Two variables, VP and VQ, are used to calculate Vref. VP =
Q P ⋅ Zg ⋅ Z g , VQ = Vg Vg
From their definition, it can be seen that if Vg is known, active power P is only related to VP and reactive power Q to VQ, hence, decoupling between P and Q can be achieved. Vref = V g + V P ⋅ n − jV Q ⋅ n
Error ( Zg) between the measured (|Zg*|) and actual (|Zg|) values of impedance will lead to an error in Vref. Vref = Vg + VP* ⋅ n − jVQ* ⋅ n + ∆VP* ⋅ n − j∆VQ* ⋅ n ,
The internal model has an infinite sequence of pairs of conjugate poles of which about the first 30 are very close to the imaginary axis, around integer multiples of j2π 50, and the later ones are further to the left. The internal model is obtained from a low-pass filter with a transfer function W(s) = C / (s+ C), with C = 104 rad/sec, cascaded with a delay element with transfer function e-s , where is slightly less than the fundamental period, e.g., τ = 19.9 ms. The choice of C is based on a compromise: too small and only a few poles of the internal model will be close to the
where VP* = P ⋅ Z g* , VQ* = Q ⋅ Z g* , ∆VP* = P ⋅ ∆Z g , Vg
Vg
Vg
and ∆VQ* = Q ⋅ ∆Z g . Vg
Controllers combining feed-forward and feedback are used. The feed-forward controllers calculate VP* and VQ* directly.
0.0013 DC link 5
3
ic_a
ig_a
30.5
1
2 G5
0.0003
Vout_a 50.0
0.053
7.0 Sc
2 G3
2 G1
A
0.0013
0.0003
Vout_b 50.0
0.053
ic_b
ig_b
B
30.5 2
6
Vg_b 0.1
7.0 C
4
2 G2
Vg_a 0.1
2 G6
2 G4
0.0013 ic_c
30.5
DC to AC inverter
0.0003
Vout_c 50.0
0.053
ig_c 0.0003 0.0003 0.0003
LC filters
Vg_c 0.1
7.0
0.00075 interface inductors
47000.06600.0
7
D
D
D
2 G7 0.01 0.5
A
A #1
#2
A
D
D
0.01
B
B
B C
30.0
47000.06600.0 C
8 2 G8
D
neutural leg
D
D
0.64
C
1.0
B
Sg
A R=0
C 100.0 100.0 100.0
D
D
rectifier bridge
D
high frequency
nonlinear
AC generator
local load
C
D
B
A
grid
Fig. 1 Circuit diagram of complete distributed generation system model (note capacitances are in F)
power controller
FF
Pref Qref
+
+
+ +VP
PI
_
+
PI
_
VQ +
formation of reference voltage
vref
repetitive voltage controller +
_
e+ +
vout
vave id v u
C internal model
W (s ) ⋅ e −τs
FF
plant
ic
sin( t), cos( t), amplitude of vg
Q
P
sampling and calculation of P and Q
vquad
PLL
v ig
Fig. 2 Block diagram of the two-level control system
u
IGBT bridge
iS
Zg
ig
vg
P, Q
Filter inductance
Vdc−
Vdc+
vout
Local load
Grid
neutral
Fig. 3 An equivalent single-phase system circuit
PI feedback controllers are used to compensate errors resulting from Zg. Pref K ⋅ Z g* + K PP + IP ( Pref − P) VP = s Vg
VQ =
Qref Vg
⋅ Z + K PQ + * g
K IQ s
(Qref − Q)
The outputs from the PI controllers, VP and VQ, are converted to instantaneous references for the inner loop
voltage controllers by multiplying by signals from the PLL. Good transient and steady-state performance has been achieved using: KPP = KPQ = 0.2, KIP = KIQ = 2. The inverter switches use sampled PWM at 10 kHz with a triangular carrier waveform. The inverter voltage can be found from the duty-cycle value ( ) or the depth of modulation (m). In turn, m is determined from the desired plant input as m = u/VDCref.
+ − V Inv = δ ⋅ VDC + (1 − δ ) ⋅ V DC 1 2
=u
(1 + m) ⋅ VDC+
+
1 2
(1 − m) ⋅ V DC−
4
+ − + − V DC V DC − VDC + VDC + Re f 2 2VDC
− + Re f then, VInv = u. If the DC-link If V DC = −V DC = VDC voltages are in error, compensation can be applied by defining the modulating signal as:
(
+ − 2u − VDC + VDC m′ = + − VDC − VDC
THD (%)
=
3 2 1 0 0
)
0.2
0.4
0.6 0.8 1 Time (s)
1.2
1.4
1.6
Fig.5 Total harmonic distortion of the load voltage
4. EVALUATION OF THE CONTROL SYSTEM PI controllers in both stationary and rotating reference frames were used for comparison; their schematic diagrams are given in Appendix A. Trial and error was used to select PI gains that avoid instability but maintain good waveform quality. The proposed power controller was compared to a conventional magnitude-angle power controller and an open-loop feed-forward controller. 4.1 Comparison of voltage controllers with balanced loads In each simulation the load is balanced but nonlinear. Initially SC is open and Sg is closed so that the local load is supplied from the grid. At 0.2s SC is closed and the inverter is connected to the local load and starts to influence the voltage waveform. The power references are set so that the power flows are undisturbed.
Grid Supply Only PI control in stationary frame PI control in synchronous frame Repetitive control
Voltage THD (%) 2.75 0.95 0.76 0.43
Table 2 Voltage control performance
4.2 Comparison of voltage controllers with unbalanced loads The load was unbalanced by adding a 10 resistive load in parallel with phase-A. Table 3 summarises the performance of the various controllers and demonstrates that the repetitive controller maintains its good performance. It also reveals that the PI controller in the synchronous reference frame is not able to adequately regulate the voltage amplitude. Phase
Fig. 4 shows that quality of current flowing in the grid after connection improves when the repetitive controller starts to operate (at 0.2s). It takes about 0.1s (5 cycles) to correct the distorted voltage and hence the current. The THD (total harmonic distortion) of load voltage (vout) is decreased from 2.75% to 0.428% with the repetitive voltage control, Fig. 5. The THD of current injected into grid is decreased from 24.9% to 1.45%.
Amplitude Error (V) n/a 3.0 2.7 0.2
Grid Supply Only PI control in stationary frame PI control in synchronous frame Repetitive control
A B&C A B&C A B&C A B&C
THD (%) 2.64 2.73 0.98 1.07 0.72 0.94 0.40 0.41
Amplitude Error (V) n/a n/a 2.7 2.9 7.2 5.3 0.2 0.2
Table 3 Voltage control performance
Table 2 compares the various controllers by THD and error in the amplitude of the fundamental component output voltage. It can be seen that the repetitive voltage controller gives a significant improvement in waveform quality and voltage amplitude.
Current (A)
80 40 0
-40 -80 0.16
0.2
0.24
Time (s)
Fig. 4 Waveform of current injected into grid
0.28
0.32
4.3 Response of Voltage Control to Disturbances Fig. 6 shows the response of the repetitive controller to a change in the rectifier resistance from 12 to 24 at 1.4s. There is an initial deterioration in voltage quality because the repetitive controller is basing its output on predisturbance data. It takes about 0.1s for the controller to converge and gives good disturbance rejection once more. The THD has reduced from 0.4% to 0.22% because the distortion current has reduced. Fig. 7 shows the THD following a change in grid voltage from 325V to 335V over 0.01s from 1.4s. Again the repetitive controller takes about 0.1s to re-converge.
For comparison, Fig 9 shows the response of a magnitudephase angle power controller with disturbances of Q=13 kVAr and P=3 kW respectively.
THD (%)
1 0.8 0.6 0.4 0.2
Fig. 10 shows that the feed-forward power controller produces a well-damped response but Fig. 11 shows that if there is an error in Zg ( Zg = 42° in the plant but 50° was used for controller design) then an error in the active and reactive powers results. In contrast, an error in the assumed Zg has little effect on the proposed decoupling controller, Fig. 12.
0 1.3
1.4
1.5
1.6
1.7
1.8
Time (s)
Fig. 6 Response of the load voltage THD to load changes
10 5
THD (%)
1
0 -5 0
0.8 0.6
0.2
0.4
0.6
0.8
1
1.2
-10 -15
0.4
1.4
1.6
Qg (kVar)
-20
0.2
Pg (kW)
-25
0 1.3
1.4
1.5
1.6
1.7
1.8
-30 Time (s)
Time (s) Fig. 7 Response of the load voltage THD to grid voltage changes
Fig. 10 P and Q response with the feed forward control 10
4.4 Comparison of Power Controllers
5
Fig.8 shows the response of the power controller to a step in P from -25 kW to +5kW at 0.6s and a step in Q from -3.5 kVAr to +1.5 kVAr at 1.0s. The responses are slightly under-damped and show some residual coupling between the two terms. There are disturbances of Q=1.5 kVAr and P=0.3kW respectively.
0 -5 0
0.2
0.4
0.6
0.8
1
1.2
-10
Qg (kVar)
-15
Pg (kW)
1.4
1.6
1.4
1.6
-20 -25 -30 Time (s)
10 5
Fig. 11 Feed forward control with error in
0 -5 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Zg
1.6
-10
10
-15
Qg (kVar)
-20
P g (kW)
5 0 -5 0
-25 -30
0.2
0.4
0.6
0.8
1
1.2
-10
Time (s)
Fig. 8 P and Q response with decoupling control
-15
Qg (kVar)
-20
Pg (kW)
-25
10 5 0 -5 -10 -15 -20 -25 -30
-30 Time (s)
Fig. 12 Power curves with the decoupling control Qg ( kVar ) Pg ( kW)
0
0.2
0.4
0.6
0.8 1 Time (s)
1.2
1.4
Fig. 9 P and Q response with magnitude-phase control
4.5 Response to Power Control to Disturbances
1.6
Fig. 13 shows the response of the power controller to the same disturbance of grid voltage as used in Section 4.3.
8
REFERENCES
6
Qg (kVar)
4
Pg (kW)
2 0 -2 1.3
1.4
1.5
1.6
1.7
1.8
-4 Time (s) Fig. 13 Response of powers to grid voltage changes
So far it has been assumed that the DC-link voltage is constant but it may be subject to disturbances if the original energy source is disturbed. A step change from 1000 V to 1100 V was introduced at 1.4s. Fig. 14 shows that very little disturbance is caused to the output voltage THD and Fig. 15 shows that the exported power is also little disturbed. Without the compensation scheme described in section 3 there were perturbations of approximately 10 kW and 10 kVAr in P and Q.
[1] N. Jenkins, R. Allan, P. Crossley, D. Kirschen and G. Strbac, Embedded Generation, IEE Power and Energy Series, London, 2000. [2] J. Stones and A. Collinson, “Power quality,” Power Engineering Journal, Vol. 15 Issue: 2, pp 58 –64. , 2001. [3] Y. Yamamoto, “Learning control and related problems in infinitedimensional systems,” in Essays on control: perspectives in the theory and its applications, H. Trentelman and J. Willems, Eds., pp.191-222, Boston: Birkhauser, 1993. [4] G. Weiss and M. Hafele, “Repetitive control of MIMO systems using H Design,” Automatica, vol.35, no.7, pp.1185-1199, 1999. [5] G. Weiss, “Repetitive control systems: Old and new ideas,” in Systems and control in the twenty-first century, C. Byrnes, B. Datta, D. Gilliam, and C. Martin, Eds., pp. 389-404, Birkauser, 1997. [6] Y.-Y. Tzou, R.-S. Ou, S.-L. Jung and M.Y. Chang, “Highperformance programmable AC power source with low harmonic distortion using DSP-based repetitive control technique,” IEEE Trans PE, Vol. 12, No. 5, pp. 715-725, 1997. [7] K. Zhou, D. Wang and K.-S. Low, “Periodic error elimination in CVCF PWM DC/AC converter systems: repetitive control approach,” IEE Proc. Control Theory Appl., Vol. 147, No. 6, pp. 694-700, 2000. [8] Q.C. Zhong, T. Green, J. Liang and G. Weiss, “Robust repetitive control of grid-connected DC-AC converters,” submitted to IEEE CDC 2002.
THD (%)
ACKNOWLEDGEMENT 1 0.8 0.6 0.4 0.2 0
This work was supported by the EPSRC (www.epsrc.ac.uk) on grant number
1.3
1.4
1.5
1.6
1.7
1.8
APPENDIX A Voltage controllers and power controllers for comparison to proposed system are shown in the following diagrams vref u + P plant _
Time (s)
vout Kp = 20; KI = 1,000
Fig. 14 Response of the output voltage THD to DC voltage changes
A1. PI voltage controller in stationary frame
6 5 4 3 2 1 0
vref
ABC to DQ
Qg (kVar) Pg (kW)
vrefd vrefq
+
+
vout 1.3
1.4
1.5 1.6 Time (s)
1.7
ABC to DQ
_
PI
_
PI
voutd voutq
DQ to ABC
u plant
Kp = 20; KI = 1,000
1.8
Fig. 15 Response of powers to DC voltage changes
A2. PI voltage controller in rotating frame
Pref
+ _
5. CONCLUSIONS A control system with a repetitive voltage controller and a decoupling power controller has been designed and applied to a distributed generation system based on an inverter. The repetitive voltage controller was found to provide lower THD and more accurate magnitude than conventional PI controllers in stationary or synchronous reference frame, however, it does require approximately 5 cycles to re-converge following a disturbance. The decoupled power controller provides reasonable independence of P and Q despite parameter inaccuracies.
δ V
Qref +
PI
_
Q
PI
Form reference voltage
vr Voltage control
u plant
P
A3. Magnitude-angle power controller
Pref FF
Qref
VP VQ
FF
Form reference voltage
A4. Feed forward power controller
vref
Voltage control
u plant
