Case

Proceedings of the 2005 IEEE Conference on Control Applications Toronto, Canada, August 28-31, 2005

WC5.1

An Adaptive Internal Model Control for Reactive Power Compensation M. Karimi-Ghartemani and F. Katiraei

Abstract— This paper presents an adaptive approach for reactive current extraction and adaptive shunt compensation of a three-phase system under unbalanced and time-varying frequency conditions. The reactive current component is extracted based on the concept of instantaneous symmetrical components. The extracted signals are then utilized by a shunt, PWM, voltage-sourced converter to adaptively compensate the reactive power. The converter control is based on a closed-loop adaptive internal-model strategy which dynamically adjusts itself to the variations in the system frequency. Feasibility of the proposed method is illustrated in a distribution system and computer simulations are presented. Index Terms— Adaptive internal model control, Reactive current compensation, Unbalanced conditions, Frequency variations

includes a conventional (rotating machine) distributed generation (DG) unit, a reactive power compensator, and multiple loads [7]. II. PRINCIPLES OF DETECTION ALGORITHM Assume that v(t) and i(t) denote phase-voltage and current signals of a three-phase load, respectively. Generally, these signals are distorted (by harmonics and noise) and not balanced. These signals can be decomposed into their fundamental components, harmonics (v h and ih ), and noise (nv and ni ) as follows. ⎛

⎞ ⎛ va (t) v(t) = ⎝ vb (t) ⎠ = ⎝ vc (t) ⎛ ⎞ ⎛ ia (t) i(t) = ⎝ ib (t) ⎠ = ⎝ ic (t)

I. INTRODUCTION Reactive power compensation for voltage control and/or power factor correction is widely used in electric power systems [1]. A voltage-sourced converter (VSC) based compensator requires instantaneous reactive current component [2] as the feedback signal for reactive power compensation. The existing methods for extraction of reactive current components suffer from one or more of the following drawbacks [3], [4]. • • • •

Sensitivity to the load or compensator configuration. Slow or incorrect response during transients. Sensitivity to noise. Inaccuracy (or failure to operate) when the system fundamental frequency changes as a function of time.

The existing reactive current compensation methods are primarily based on balanced and fixed-frequency system conditions. Under highly distorted waveform conditions, instead of direct control of reactive current component, a tracking controller based on a hysteresis bandwidth can be adopted. However, this approach results in high and variable switching frequency, and steady-state error [5], [6]. The objective of this paper is to propose an alternative adaptive control method that can properly perform under unbalanced and frequency-varying conditions. The latter enables the compensator to accurately perform during transients and operating scenarios where frequency changes are inevitable, e.g. transients leading to islanding and autonomous operation of a micro-grid [7]. Performance of the proposed method is evaluated in a utility distribution system which The authors are with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, Ontario, Canada, E-mail Addresses: [email protected],

[email protected]

0-7803-9354-6/05/$20.00 ©2005 IEEE

⎞ Va sin(φva ) + vah (t) Vb sin(φvb ) + vbh (t) ⎠ + nv (t), Vc sin(φvc ) + vch (t) ⎞ Ia sin(φia ) + iha (t) Ib sin(φib ) + ihb (t) ⎠ + ni (t). Ic sin(φic ) + ihc (t)

Assume that the instantaneous positive-sequence components of voltage and current waveforms are described by

⎞ ⎞ ⎛ + I sin(ωt + θ) V + sin(ωt) =⎝ V + sin(ωt − 120) ⎠ , i+ =⎝ I + sin(ωt − 120 + θ) ⎠ , I + sin(ωt + 120 + θ) V + sin(ωt + 120) ⎛

v+

where θ denotes the phase-displacement between the corresponding positive-sequence voltage and current components. The current positive-sequence component can be decomposed into instantaneous real and reactive components as: ⎞ I + cos θ sin(ωt) ⎝ I + cos θ sin(ωt − 120) ⎠ , i+ a (t) = I + cos θ sin(ωt + 120) ⎞ ⎛ + I sin θ cos(ωt) ⎝ I + sin θ cos(ωt − 120) ⎠ . i+ r (t) = I + sin θ cos(ωt + 120) ⎛

(1)

Based on (1), to calculate the reactive component of the current, one needs the magnitude of the positive-sequence currents (I + ), the phase-displacement between the voltage and current positive-sequence components (θ), and the 90degree phase-shifted version of the voltage positive-sequence component (cos ωt). The instantaneous positive-sequence components can be calculated by a linear transformation described by matrix P + as following: ⎛ 1 1⎝ P = −1/2 3 −1/2 +

√0 −√ 3/2 3/2

−1/2 1 −1/2

√

3/2 √0 − 3/2

−1/2 −1/2 1

√ ⎞ −√ 3/2 3/2 ⎠ . 0 (2)

The transformation operates on the six-dimensional vector comprising of the fundamental and their corresponding 90-degree phase-shifted components. For the voltage signals, for example, this vector is [Va sinφva , Va cosφva , Vb sinφvb , Vb cosφvb , Vc sinφvc , Vc cosφvc ]T .

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III. SHUNT COMPENSATION OF REACTIVE CURRENTS A. Study System

Fig. 1.

Proposed system for extracting reactive current components.

The above fact is observed by noting that the instantaneous positive-sequence component of a set of three-phase voltages (or currents) is defined as: ⎞ ⎛ va+ (t) 1 1 + ⎝ v (t) ⎠ = ⎝ β 2 b 3 vc+ (t) β ⎛

β 1 β2

⎞ ⎞⎛ f va (t) β2 f β ⎠ ⎝ vb (t) ⎠ 1 vcf (t)

(3)

where β stands for the 120-degree phase-shift operator in the time domain and the superscript f refers to the fundamental component. Rewriting (3) in terms of the 90-degree phaseshift operator S90 yields va+ (t) = 13 vaf (t) − 16 (vbf (t) + vcf (t)) − vb+ (t) = −va+ (t) − vc+ (t) vc+ (t) = 13 vcf (t) − 16 (vaf (t) + vbf (t)) −

1 √ S (v f (t) 2 3 90 b

− vcf (t))

1 √ S (v f (t) 2 3 90 a

− vbf (t)). (4)

In (4), S90 represents the 90-degree phase-shift operator. Thus S90 sin(θ) = sin(θ + 90) = cos(θ). Equation (4) provides the linear transformation described by (2). The enhanced phase-locked loop (EPLL) of [8] is used to extract the fundamental component and its 90-degree phaseshifted version. It receives the input signal u(t) and extracts its fundamental component y(t) and its 90-degree phaseshifted version y
(t). The EPLL also estimates amplitude A, phase-angle φ and frequency ω = ωo + ∆ω of the fundamental component. Mathematical properties of the EPLL and its applications to power systems are available in [8] and [9], respectively. The proposed system for extraction of reactive current components is shown in Fig. 1. The system receives the three-phase voltage and current waveforms. Two set of EPLL units (each comprising of three EPLLs) extract the fundamental components and their 90-degree phase-shifted versions. These signals are then forwarded to the symmetrical component (SC) extraction blocks. The SC blocks implement the linear transformation of (2) to calculate the instantaneous positive-sequence components of the voltage and current signals. Two single EPLL units are used in the next stage to extract the magnitude of the current positive-sequence (I + ), its phase-angle (ωt + θ), and the phase-angle of the voltage positive-sequence (ωt). The last block, i.e. R, uses these three pieces of information to calculate the reactive currents based on (1). This method of detection has three distinct properties as follows. It is robust with regard to unbalanced conditions in both voltage and current signals, it tolerates variations in the frequency, and it is highly immune to noise and distortions.

Fig. 2 shows a single-line diagram of the system used to investigate performance of the detection and control algorithms. The system is composed of a 13.8-kV, radial, threefeeder distribution substation which is connected to the utility main grid through a 69-kV radial line. The utility system is represented as a 69-kV, 1000-MVA short-circuit capacity bus. The system parameters are given in [7]. The system includes a distributed generation unit DG1 (5-MVA) and a dedicated static compensator DSC (2.5-MVA) on feeders 1 and 3 respectively. DG1 is a synchronous rotating machine equipped with excitation and governor control systems. It represents either a diesel-generator or a gas-turbine-generator unit. DSC employs a multilevel voltage-sourced converter [10]. The load on feeder 3 is assumed to be a sensitive industrial load. Hence, the main objective of DSC is to provide the power quality requirements of the load in terms of fast reactive power compensation and voltage regulation. The system of Fig. 2 operates in both the grid-connected mode and the off-grid (islanded) mode. This is a departure from the conventional utility practice in which only the grid connected mode is permitted. Transition to the islanded mode and autonomous operation of the system is accompanied by frequency deviations and voltage variations, and there is a need for a robust control algorithm to maintain voltage/angle stability and power quality. A schematic representation of DSC is given in Fig. 3 which shows the DSC Controller and the DSC power circuit including the converter and the interface apparatus. B. Power Circuit of DSC The three-phase DSC is composed of 7 cascaded cells of H-bridge converter in each phase which presents 15 discrete levels of voltage injections [11]. The three phases are starconnected and the neutral point is connected to the system neutral. Thus, DSC can inject three independent current components for reactive compensation and voltage control. Fig. 4 illustrates the fundamental-frequency equivalent model of the power circuit of DSC. The power convertor is represented by three separate current-controlled voltage sources, generating three-phase sinusoidal voltages at the fundamental frequency. Each phase also includes a series connected R and L branch. The dynamic model of the power circuit, in the abc frame, are obtained from the ordinary differential equations (ODEs) of the three phases. s vabc = Risabc + L

d s t i + vabc dt abc

(5)

s t where vabc , vabc and isabc are vectors of instantaneous values of converter output voltages, bus voltages and converter output currents respectively. The power circuit model (5), as the plant model, is used to design the controllers of the system in time-domain.

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Fig. 2.

Single-line diagram of the study system.

Fig. 4. verter.

Fig. 3.

Schematic diagram of DSC of Fig. 2.

C. Control System of DSC The building blocks for the control system of DSC are shown in Fig. 5 which include (i) a signal monitoring block, (ii) an EPLL block, (iii) a reactive current extraction block, (iv) an adaptive current controller and (v) a PWM pulse generator. The instantaneous values of the bus voltage (v t (t)), the load current (il (t)), and the DSC output current (is (t)) are measured by the signal monitoring block. The EPLL block uses the bus voltage to determine the system angular frequency (ωo ), and track the direction of the voltage space vector. The reactive current extraction block estimates the instantaneous reactive current component of the load current based on the algorithm outlined in Section II. The threephase reactive current components are then multiplied by the

Fundamental-frequency equivalent model of the multilevel con-

calculated compensation factor Q to generate the reference currents corresponding to the three-phase output currents of DSC. The reference currents are used by the adaptive current controller to determine reference values for the converter output voltages, v r . The compensation factor Q is identified by the pre-set value of the compensation power factor (pfc ) and the load power factor (pfl ), and varies in the range of zero to one. Inputs to the current controller are the reference currents for compensation, the measured values of the DSC currents, and the estimated frequency of the system. The current controller generates the reference voltages corresponding to the output of the DSC converter by which DSC injects the same amount of reactive currents that is needed for compensation of the load reactive power, Fig. 5. The adaptive current controller is discussed in next section. IV. PROPOSED DSC CONTROL STRATEGY The adopted control strategy for the compensator is based on the concept of internal-model control [12]. The internalmodel control addresses the problem of tracking/rejecting families of exogenous inputs. These inputs can be considered

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Fig. 6. Block diagram representation of the adaptive current controller and the plant model. Fig. 5.

Schematic diagram of the DSC controller. 800

600

400

200

Imag Axis

as a set of signals generated by a fixed autonomous finitedimensional dynamical system (the initial condition of which identifies the actual, unknown, exogenous input affecting the plant). Thus, the internal-model control approach can be considered as the generalized classical control method in which integral-control based schemes deal with constant but unknown disturbances. The main limitation of the internal-method control is that a precise model of the system that generates all exogenous inputs must be available to be replicated in the control law. This limitation does not exist in the set-point-control case where the uncertain exogenous input is constant and thus obeys a trivial, parameter independent, differential equation. However, it becomes evident for the case of rejecting/tracking, e.g. a sinusoidal waveform of unknown amplitude, phaseangle and frequency. An internal-model based controller can cope with uncertainties of amplitude and phase-angle of the exogenous sinusoid, but the frequency at which the internal-model oscillates must match the frequency of the exogenous sinusoid. A mismatch in the frequencies results in a nonzero steady-state error. This requires an adaptive controller which adjusts its frequency. The proposed control algorithm uses the estimated frequency (provided by the detection algorithm) to adjust its parameter as follows. For a three-phase system, three identical controllers are used to track the sinusoidal inputs with the variable frequency of ωo . Fig. 6 shows a generic controller to track the desired sinusoidal reactive current component to be injected by the compensator [13]. Note that this state-space realization guarantees the applicability of the internal model control for time-varying frequency conditions. Input to the controller is the reference current signal determined by the detection algorithm. The output signal of each controller is the reference signal corresponding to the terminal voltage of the phase where the controller is applied to. The controller 1 transfer function is C(s) = Ks22s+K +ωo2 . The controller performance is governed by K1 and K2 while ωo is adaptively provided by the detection algorithm. An optimum design of the controller parameters K1 and K2 is achieved based on the pole placement and the root-locus methods as follows. 1 where The power circuit is represented by P (s) = Ls+R R and L identify the branch parameters connecting the converter to the corresponding network bus. The open-loop

k=0.03 0

O

−200

−400

−600

−800 −25

−20

Fig. 7.

−15

−10 Real Axis

−5

0

5

Root locus of the control system

transfer function is C(s)P (s) =

1 + αs K2 s + K 1 =k (Ls + R)(s2 + ωo2 ) (Ls + R)(1 + s2 /ωo2 )

K2 1 where k = K ωo2 is the root-locus gain and α = K1 is the zero of the system. The open-loop system has three poles at ±jωo and −R/L. For the study system of Fig. 2, −R/L is −25. A root-locus test verifies that the zero of the system must be between 0 and −25. Based on trial and error, −0.5 is a desirable value for the zero. The root-locus of the system for this zero and the gain of k ∈ [0, .04] is shown in Fig. 7. For a value of k around 0.03, all three poles have almost the same real value of approximately -8. Smaller k’s correspond to conjugate poles closer to the imaginary axis. Larger k’s correspond to a real pole closer to the imaginary axis. Timedomain simulation studies show that a desirable behavior is achieved when all three poles have almost the same real value and k = 0.03.

V. CASE STUDIES

Some case studies are conducted on the system of Fig. 2 to examine performance of the compensator that is equipped with the detection algorithm and the proposed time-domain controller. The grid-connected mode and the island mode, where the system frequency is constant and subject to variations respectively, are considered. Case studies illustrate (i) the steady-state response to the changes in the reference signal of the controllers of DSC, and (ii) the dynamic response when the system undergoes transients, such as a three-phase to ground fault on the grid 69-kV line. The 1671

a) Ph−a

2

a) Reactive Power

3

t

va

0 c

il

−1

1

1 0

ia

a

−2

1.05

1.1

1.15

−1 0.8

1.2

0.9

1

1.1

1.2

t

1.4

1.5

1.3

1.4

1.5

1.3

1.4

1.5

1 Compensated pf ( pf ) c

Lag

0 −1

Load pf ( pf ) l

ic

l

ib 1

1.3

b) Power factor

va

1

p.u

Grid

b) Ph−b

2

−2

DSC

2

Q (MVAr)

p.u

1

b

1.05

1.1

1.15

0.8

1.2

0.9

1

1.1

2

1.2

c) Bus voltage

c) Ph−c vt

p.u

Vrms (p.u)

a

1 0 −1

ic c

−2

1

l

ic 0.8

1

1.05

1.1

1.15

0.9

1

1.1

1.2

1.2

Time (s) Time (s)

Fig. 8. Case(A)- A step change in the controller set-point to pfc = 1 at t=1 s, bus-voltages, feeder currents and load currents.

Fig. 9. Case(A)- A step change in the controller set-point to pfc = 1.0 at t=1 s. a) Feeder currents 1

i(t) p.u

0.5 0 −0.5 −1 1.45

1.5

1.55

1.65

1.7

DSC

1.5

Q (MVAr)

1.6 b) Reactive power

2

1 0.5 Grid 0 1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.7

1.75

1.8

c) Power factor 1

Compensated pf (pf ) c

Lag

latter leads to the islanding of the distribution system. For the presented studies, the power system is simulated in the PSCAD/EMTDC environment and the detection and control sub-systems are simulated in the MATLAB/Simulink. The two software environments are interfaced to allow interactive simulation of the overall system. Case (A)- Power Factor Variations: The load connected on Bus-3, Fig. 2, is set at 0.9-MW/0.6-MVAr with the lagging power factor of 0.74. The DSC controller is to adjust the power factor to a pre-specified power factor of pfc . Figs. 8 and 9 show the controller response to a change in its reference signal. At t=1 s, the DSC control is activated to compensate the reactive power of the load by setting Q from zero to one. As a result, the compensated grid currents (icabc ), t and the three-phase bus voltages (vabc ) in Fig. 8 become inphase within 5 to 6 cycles after the compensator is activated. The reactive power delivery by the grid is also reduced to zero, Fig. 9(a), and the compensated load power factor approaches unity, Fig. 9(b). Consequently, the rms value of the bus voltage is increased, Fig. 9(c), due to the decrease in the feeder current. Fig. 10 illustrates performance of the control system in response to a step change in the reference signal of the power factor. The system load and the operating conditions are the same as the previous case. However, it is assumed that DSC initially compensates the total reactive power of the load, and pfc is unity, Fig. 10(c), and the grid does not supply any reactive power to the load, Fig. 10(b). At t=1.5 s the set-point of the controller, corresponding to the targeted compensated power factor of the load, is changed to 0.9-lag by changing Q from unity to 0.68. DSC responds to the new set-point by decreasing the injected reactive power, leading to an increase in the grid reactive power supply, Fig. 10(b), and an increase in the feeder currents, Fig. 10(a). The reactive power supply from the grid to Feeder 3 is consequently changed to the difference between the reactive power requirements of the

0.9 0.8 0.7 1.45

Load pf (pfl) 1.5

1.55

1.6

1.65 Time (s)

Fig. 10. Case(A)- A step change in the power factor set-point from pfc = 1.0 to pfc = 0.9 at t=1.5 s.

load and the reactive power injection of DSC by which the compensated power factor of pfc = 0.9 is achieved, Fig. 10(c). Case (B)- Three-Phase Line to Ground Fault: The objective of this case study is to verify robustness of the detection algorithm and the adaptive controller to frequency variations. It is assumed that DSC initially compensates the reactive power of the 0.9-MW/0.6-MVAr load to unity. A threephase line-to-ground fault occurs on the 69-kV utility line, Fig. 2, at t=0.8 s and causes a voltage drop, Fig. 11(a). The under-voltage protection of the system detects the fault and disconnects the distribution system from the grid at t=0.85 s and forms an island. The islanded distribution system undergoes transient frequency variations and transient power imbalance. However, increase in the power generated by

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a) Bus voltage

Vrms (p.u)

1

0.5

0

1

1.5

2

2.5

3

2.5

3

2.5

3

b) Reactiev Power

4

Q (MVAr)

3 2 DSC

Grid

1 0 1

1.5

2 c) Power factor

1

Lag

0.5

Load pf ( pf ) l

Compensated pf ( pf ) c

0 −0.5 −1 1

1.5

2 d) Frequency

1.005

f (p.u)

1 0.995 0.99 0.985

Fig. 11.

1

1.5

2 Time (s)

2.5

3

3.5

Case(B)- 3L-G fault at 0.8 s, disconnection at 0.85 s.

DG1, based on the load demand, and fast reactive power compensation provided by DSC guarantee continuation of the system operation in the autonomous mode. In the islanding mode, DG1 supplies the system real power demands leading to frequency restoration after bounded oscillations, Fig. 11(d), while DSC injects the required reactive power to compensate the load on Bus3, Fig. 11(b). The large power factor variations shown in Fig. 11(c) during the fault condition are due to indistinguishable input voltage signals to the detection system.

[2] H. Akagi, Y. Kanazawa, and A. Nabae , “Instantaneous reactive power compensators comprising switching devices without energy storage components,” IEEE Trans. Ind. Applicat., vol. 20, no. 3, pp. 625–630, 1984. [3] J. L. Willems, “A new interpretation of the Akagi-Nabae power components for nonsinusoidal three phase situations,” IEEE Trans. Instrum. Meas., vol. 41, no. 4, pp. 523–529, 1992. [4] A. Ferrero and G. Supeti-Furga, “A new approach to the definition of power components in three-phase systems under nonsinusoidal conditions,” IEEE Trans. Instrum. Meas., vol. 40, no. 3, pp. 568–577, 1991. [5] M. K. Mishra, A. Joshi, and A. Ghosh, “Unified shunt compensator algorithm based on generalized instantaneous reactive power theory,” in IEE Proc. Generation, Transmission and Distribution, vol. 148, no. 6, 2001, pp. 583–589. [6] M. Ardes, J. Hafner, and K. Heumann, “Three-phase four-wire shunt active filter control strategies,” IEEE Trans. Power Electron., vol. 12, no. 2, pp. 311–318, 1997. [7] F. Katiraei, M. R. Iravani, and P. W. Lehn, “Micro-grid autonomous operation during and subsequent to islanding process,” IEEE Trans. Power Delivery, vol. 20, no. 1, pp. 248–257, Jan. 2005. [8] M. Karimi-Ghartemani and M. R. Iravani, “A nonlinear adaptive filter for on-line signal analysis in power systems: applications,” IEEE Trans. Power Delivery, vol. 17, no. 1, pp. 617–622, 2002. [9] M. Karimi-Ghartemani and A. K. Ziarani, “Periodic orbit analysis of two dynamical systems for electrical engineering applications,” Journal of Engineering Mathematics, Kluwer Academic Publishers, vol. 45, no. 2, pp. 135–154, Feb. 2003. [10] D. Soto, T. Green, and A. Coonick, “Multi-level converters and large power inverters,” in Sixth International Conference on Power Electronics and Variable Speed Drive, no. 429, 1996, pp. 354 – 359. [11] F. Peng and J. Lai, “Dynamic performance and control of a static var generator using cascaded multilevel inverters,” IEEE Trans. Ind. Applicat., vol. 33, no. 3, pp. 748–754, May/June 1997. [12] A. Datta and J. Ochoa, “Adaptive internal model control: Design and stability analysis,” in Journal of Automatica, vol. 32, no. 2, 1996, pp. 261–266. [13] M. Bodson, “Equivalence between adaptive cancellation algorithms and linear time-varying compensators,” in 43rd IEEE Conference on Decision and Control, Bahamas, Dec. 2004, pp. 638 – 643.

VI. CONCLUSION An adaptive control method for reactive power compensation is proposed and its performance is evaluated. The compensation method is an adaptive internal-model control which ensures tracking of sinusoidal reference signals with varying frequency. Performance of the proposed method is investigated based on computer simulation studies of an islanded micro-grid. The main features of the proposed extraction/control method are as follows. Performance of the system is robust regardless of distortions in voltage and current signals; including harmonics, inter-harmonics, transient disturbances and unbalanced conditions. The control algorithm adapts itself to variations in the frequency of the power system. Robustness and frequency adaptivity of the proposed compensation method are particularly advantages for distribution systems, including distributed resources, which may have to operate in both grid-connected and autonomous modes, and thus large unbalanced conditions and frequency deviations are encountered. R EFERENCES [1] L.Gyugyi, “Power electronics in electric utilities: Static var compensators,” in Proc. IEEE, vol. 76, no. 4, 1988, pp. 483–494.

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