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AbstractStatic compensators can be used for voltage

regulation in the presence of flicker or transients on parallel

connected loads in a distribution system. This paper presents a

general modeling strategy that allows design of a fast voltage

magnitude controller accounting for system parameters. The co-

ordinate transform used in modeling the system facilitates

extraction of linearized system dynamics with the help of circuit

simulators. It is also shown that the problem of voltage

regulation using instantaneous reactive current is non-minimum

phase for certain operating conditions, thereby limiting its

dynamic response irrespective of the control design method.

Simulation results with a controller designed using input output

bode plot are presented to prove the efficacy of the proposedmethod.

Index TermsFlicker mitigation, Modeling, STATCOM,

STATCOM control, Voltage regulation.

I. INTRODUCTION

ast load voltage regulation is required to compensate for

time varying loads such as electric arc furnaces,

fluctuating output power of wind generation systems, and

transients on parallel connected loads (e.g. line start of

induction motors) [1]-[3]. Reactive power compensation is

commonly used for flicker mitigation and load voltage

regulation. Due to their high control bandwidth, StaticCompensators (STATCOMs) based on three phase pulse

width modulated converters, have been proposed in [1], [4]-

[8] for this application. For effecting fast control, the

STATCOM is usually modeled using the dq axis theory for

balanced three phase systems, which allows definition of

instantaneous reactive current [9].

Most literature on STATCOM control concentrates on

control of STATCOM output current and dc bus voltage

regulation for a given reactive current reference. Regulation

of load voltage is achieved using a PID controller that

generates the reactive current reference (e.g. [10]). For this

paper it is assumed that the STATCOM can be modeled as acontrolled current source. The problem addressed here is to

design a voltage magnitude controller that will generate the

reactive current reference for the STATCOM. Other than

experimental procedures like Zeigler and Nichols [11], to the

A. K. Jain and N. Mohan are with the Department of Electrical and

Computer Engineering, University of Minnesota, Minneapolis, MN 55455.

(phone: 612-624-7309, fax: 612-625-4583 E-mail: [email protected],

[email protected]).

A. Behal is with the Department of Bioengineering, Clemson University,

Clemson, SC 29634-0915. (E-mail: [email protected]).

authors knowledge, there is no standard procedure for

designing a load voltage controller that ensures the required

bandwidth and robustness to system variations. In [4], a small

signal model of the system was derived by transforming the

equivalent impedance of the ac system to the dq frame.

However, the angular frequency of the dq frame was assumed

to be constant and equal to the steady state line frequency.

This paper describes a modeling strategy, similar to that used

for field oriented control of three phase ac machines, that

allows voltage controller design of STATCOMs accounting

for system parameters. Although a simple system is

considered here, it is indicated how the method described can

be extended to more complex systems. Using linear and

nonlinear systems approach it is shown that system is non-

minimum phase for certain operating conditions, and thus has

an inherent limitation on achievable dynamic response. A

voltage controller is designed using input to output bode plots

of the system linearized about various operating points.

Simulation results showing the controller performance are

presented.

Section II describes system representation. Section III

discusses the non-minimum phase nature of the system.

Controller design and simulation results are presented in

Section IV.

II. SYSTEM MODEL

A. System Description

The system considered here is a simplified model of a load

supplied on a distribution system. A STATCOM is connected

at the load bus to regulate the voltage magnitude. One phase

of the model is shown in Fig. 1(a). It consists of: the source

modeled as an infinite bus with inductive source impedance,

the load modeled by a resistance, the STATCOM modeled as

a controllable current source, and a coupling capacitor. The

coupling capacitor is included for two reasons: (1) if a

capacitor is not included then the line current and the

STATCOM output current are not independent [12], (2) a real

STATCOM will always have an L-C filter at its output (e.g.

[6]). It is assumed that the source, load, and STATCOM are

balanced three phase systems. The STATCOM current

dynamics have been neglected so it is represented as a

controlled current source.

The system dynamics are described by:

,

, , ,

s abc

s s s abc L abc s abc

diL R i v v

dt= + (1)

System Modeling and Control Design for Fast

Voltage Regulation Using STATCOMsAmit K. Jain, Student Member, IEEE, Aman Behal, and Ned Mohan, Fellow, IEEE

F


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,

, , ,

L abc

c s abc SC abc L L abc

dvC i i g v

dt= (2)

Here, ,s abci , ,SC abci , ,s abcv , and ,L abcv are vectors consisting of

the individual phase quantities denoted in Fig. 1(a), 1/L Lg R=

is the load conductance, sL is the source inductance, sR is the

source resistance, and cC is the coupling capacitor. Under the

assumption that zero sequence components are not present,

(1)-(2) can be transformed to an equivalent two phase system

by applying the following three to two phase transformation.0 2 /3 4 /3

,

j j j

s sa sb scv v e v e v e

= + + (3)

where, the complex number ,s s sv v j v = + . This is

followed by the rotational transformation:

, ,

j

s dq sd sq sv v j v e v

= + =

(4)

Applying the two transformations, (1) and (2) can be written

as

,

, , ,( )s dq

s s s s dq L dq s dq

diL R j L i v v

dt= + + (5)

( ), , , ,L dq

c s dq SC dq L c L dqdvC i i g j C v

dt= + (6)

where, given by /d dt = , is yet to be defined and may

be a function of time. The equivalent circuits corresponding

to the real (d-axis) and imaginary (q-axis) components of this

equation are shown in Fig. 1(b) and 1(c) respectively.

,s abci

sR

,s a bcvL

R ,SC abci

+

,L abcv

+

sL

cC

(a)

(b)

(c)

s sqL i

SCdi

sdi

sL

sdv

LR

+

+

Ldv

+

sR

c LqC v

cC

SCqi

sqi

sqv

s sdL i

Lqv

sL

LR

+

+

+

sR

c LdC v

cC

Fig. 1. (a) One phase of the system model, (b) d-axis equivalent circuit, (c) q-

axis equivalent circuit.

B. Choice of Reference Frame

The choice of reference frame is similar to that used for

field oriented control of three phase ac machines. The angle

used in (4) is chosen as ( )1tan / s sv v

= . Thus,

0 / 0Lq Lqv dv dt = (7)

Defining st = , where s is the frequency of the

infinite bus phase voltages, we get ,j

s dq sv V e

= , where sV is

the constant magnitude of the infinite bus voltage. The

relative orientations of the vectors ,L dqv , ,s dqv , and thereference frame chosen are shown in Fig. 2. Ignoring losses, a

STATCOM only supplies reactive power so that 0SCdi .

Using the conditions mentioned above the transformed system

equations can be rewritten as:

Ldc L Ld s d

dvC g v i

dt= + (8)

cossds Ld s sd s sq sdi

L v R i L i Vdt

= + + (9)

sinsq

s s sd s sq s

diL L i R i V

dt = (10)

s

d

dt

= (11)

( )sq SCq

c Ld

i i

C v

= (12)

where (12) has been derived using (7). Corresponding

equivalent circuits are shown in Fig. 3.

Since 0Lqv = , Ldv represents the instantaneous magnitude

of the phase voltages ,L abcv , which has to be controlled. The

reactive current absorbed by the STATCOM, SCqi , is the

control input. Since it has been assumed that negative

sequence components are not present, in steady state all the

state variables in (8)-(11) are constant. Effectively, thebalanced three phase ac system is transformed to an equivalent

dc system.

Lv

axis

sv

st

d axis

q axisaxis

Fig.2. Orientation of reference frames.

C. Advantages of the System Representation

The control problem is simplified to control of dcquantities as opposed to sinusoidally varying quantities, and

the input (STATCOM reactive current, SCqi ) and output (bus

voltage magnitude, Ldv ) are clearly defined on an

instantaneous basis. Equations (8)-(12) define the system

completely and can be used to design controllers using linear

or nonlinear techniques.

The transformed system can be represented in the form of

equivalent circuits using circuit simulators with analog

behavioral modeling capabilities. The dand q axis equivalent


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circuits shown in Fig. 1(b) and 1(c) can be easily derived from

the single phase circuit in Fig. 1(a). Inductors and capacitors

(a)

s sqL isdi

sL

sdvLR

+

+

Ldv

+

sR

cC

(b)

SCqi

LR

c LdC v

s sdL i+

sqi

sqv

+

Fig. 3. Simplified equivalent circuits: (a) daxis, (b) q axis.

are augmented by appropriate controlled voltage and current

sources connected in series and parallel respectively. Three

phase sinusoidal ac sources with constant amplitude arereplaced by controlled dc sources (e.g. cossd sv V = ,

sinsq sv V = ). Two additional equations corresponding to

(11)-(12) are needed to define and . Using these

principles, a more complicated system can be modeled

without writing all the state equations explicitly. For this

paper, Matlab/Simulink along with the Power System

Blockset has been used for modeling the equivalent circuits.

Fig. 4 shows the Simulink block diagram of the system.

Results obtained from the equivalent circuits based model

were verified with those obtained from direct modeling of (8)-

(12).

Fig. 4. Simulink block diagram of transformed system.

Since all the states are constant in steady state, operating

point calculation is possible by equating the state derivatives

to zero. This can be done by a dc bias point calculation in a

circuit simulator. The system can then be linearized about

calculated operating points to obtain either state space data or

bode plots of the required transfer functions of the linearized

system. Since simulators like Simulink can calculate

operating points and extract linearized model of the system

about the operating point, it is not necessary to write all thestate equations explicitly. This is important for systems

having a large number of states.

III. NON-MINIMUM PHASE NATURE OF SYSTEM

System data used in the rest of the paper is taken from [7].

However, a purely resistive load and a reasonable value for

the coupling capacitor have been assumed here. The system

parameters are:

sV (L-L rms) 415 V sL 9 mH

LR 32 sR 0.3

cC 1 F

This system was linearized around operating points

corresponding to a range of values of the reactive current

absorbed by the STATCOM, SCqI . Fig. 5 shows bode plots of

the small signal transfer function from ,sc qi to Ldv for the

linearized system. It is evident that for certain values of SCqI

the transfer function has one Right Half Plane (RHP) zero.

This non-minimum phase behavior was partially explained in

[4]. There, a linearized model was used and the STATCOM

current was assumed to be zero. As seen in Fig. 5, the system

is non-minimum phase only for certain values of SCqI .

Fig. 5. Bode plots of system for different values ofICq

.

A. Explanation of the Non-Minimum Phase Nature

Non-minimum phase nature of a system implies that initial

transient behavior of the system is counter intuitive. If the

input is increased as a step the immediate effect is a reduction

in the output. After an initial undershoot the output starts

increasing and reaches its steady state value. The peculiar


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behavior of our system can be understood from an inspection

of equivalent circuits shown in Fig. 3. Assume that at t = 0

SCqi is increased as a step. Since sqi cannot change

instantaneously, the current c LdC v reduces instantaneously.

Since Ldv cannot change instantaneously the immediate effect

is a reduction in . If the operating condition of the system

prior to the step change was such that (0) 0sqi > , then sdi

starts reducing due to reduction in value of the controlledvoltage source s sqL i . This leads to a decrease in Ldv . Once

sqi starts increasing, due to reduction of the controlled voltage

source s sdL i , sdi and Ldv start increasing. If the initial

system condition is such that (0) 0sqi < , then a reduction in

does not cause sdi to decrease. Thus, the system exhibits non-

minimum phase behavior only for (0) 0sqi > .

Fig. 6(a) and 6(b) show the simulated step response of the

uncontrolled open loop system for cases where (0) 0sqi > and

(0) 0sqi < respectively. It was numerically verified that the

small signal transfer functions obtained for operatingconditions with 0sqI > , had an RHP zero and those for

0sqI < did not.

(a) (b)

Fig. 6. Response of the uncontrolled system to a step increase inCq

i :

(a) (0) 0isq > (b) (0) 0isq < .

B. Nonlinear Systems Approach

According to [13] a nonlinear system is non-minimum phase

if the zero dynamics of the system are not asymptoticallystable. Assuming 0sqi , zero dynamics of the system are

derived as follows. First the output Ldv is set to a constant

value of LV in (8) which gives sd L Li g V= . These values of

Ldv and sdi are used in (9) to obtain SCqi in terms of sqi and

. Substituting the obtained value of SCqi in (10) and (11)

and using (12) for the definition of gives the followingzero dynamics.

( )1

(1 ) cos sinsq

s s sq L L L L s s s

sq

diL R i g V V g R V V

dt i = +

(13)

( )1

(1 ) cosL L s ss sq

dV g R V

dt L i

= + + (14)

Stability of the system described by (13) and (14) was studied

by calculating the equilibrium points and linearizing (13) and

(14) about them. Eigen values of the resulting state matrixwere calculated numerically with LV as a parameter. It was

found that for 0sqi > one eigen value is positive, while for

0sqi < both the eigen values are negative. Since the

linearized system corresponding to (13)(14) is unstable for

the case 0sqi > , the nonlinear system given by (13)(14) is

also unstable [13]. For 0sqi < , since the linearized system is

stable, (13)(14) is locally asymptotically stable. Thus, it may

be concluded that the system described by (8)(12) is non-

minimum phase for (0) 0sqi > .

IV. CONTROLLER DESIGN AND SIMULATION RESULTS

A. Control Design

Bode plots of the linearized system shown in Fig. 5 were

used along with the Single Input Single Output Design Tool in

Matlab to design the controller. The worst case bode plot,

corresponding to the minimum value of the RHP zero (1400

rad/s), was found at the maximum value of SCqI (10 A)

considered. A purely integral compensator with the

parameters indicated below was designed for effecting voltage

regulation. Open loop bode plots of the system with the

compensator are shown in Fig. 7.

IK 100 Gain Margin (min.) 9 dB

Gain crossover

frequency

55 Hz Phase Margin (min.) o78

Fig. 7. Bode plots of the open loop system with integral compensator for

different values ofICq

.


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B. Simulation Results

A three phase system was modeled in Simulink with the

parameters mentioned above. A block diagram of the closed

loop system implementation is shown in Fig. 8. The angle

used for abc to dq and dq to abc transforms is derived from

sensed load voltages by using ( )1tan / s sv v

= . SCdi is set

to zero while SCqi is connected to the output of the integral

compensator described above.

,s abcv

,s abci

sR

LR

,SC abci+

,L abcv

+

sL

cC

,L abcv

abc

todq abc

to

dq

0SCdi =

SCqi

IK

1

s

+

,Ld refv

Ldv

abc

to

1tan ( / )s sv v

Fig. 8. Block diagram of the simulated system

Fig. 9 Simulated response for a step change in voltage reference.

Fig. 9 shows the system response for step changes of 0.05

p.u. in the desired voltage magnitude. As expected from the

designed gain crossover frequency the voltage settles in less

than 15 ms. Fig. 9 shows the system response for a step

increase in load from 100% to 120% at t = 40 ms. As seen,

the load voltage magnitude is regulated to its reference value

in less than 15 ms.

A three phase balanced current source of magnitude 5A

and frequency of 10Hz was connected in parallel with the

load. This simulates the effect of pulsating load power. Fig.

11 shows the system response with and without the controlled

STATCOM. The variation of the phase voltage magnitudes is

Fig. 10. Simulated response for a step increase in load current.

Fig. 11. Response to parallel current injection at 10 Hz.

1.85% without the STATCOM and 1.44% with the

STATCOM. Although the STATCOM does compensate to

some extent, it is limited by the controller bandwidth. The 10

Hz parallel load appears at beat frequencies in the dq axis

frame. Since the open loop gain crossover frequency is only

55 Hz the STATCOM cannot completely compensate for the

disturbance. The controller bandwidth cannot be increased

any further due to the non-minimum phase nature of the

system. Thus, this limitation on the dynamic response is

imposed by system parameters.


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V. CONCLUSION

This paper has described a method for modeling

distribution systems in which a STATCOM is used for load

voltage regulation. The modeling strategy, similar to that used

for field oriented control of AC machines, suited to design of

voltage magnitude controller accounting for system

parameters. Although a simple system was considered in this

paper, the method presented can be applied to complex

systems. The particular co-ordinate transform used facilitates

extraction of linearized system dynamics with the help of

circuit simulators having analog behavioral modeling

capabilities. Towards this end Matlab/Simulink with the

Power System Blockset were used.

The system is non-minimum phase for certain operating

conditions. These operating conditions have been identified

and explanation of the non-minimum phase nature based on

both linear and nonlinear systems approach has been

provided. This characteristic of the system imposes restriction

on the dynamic response achievable with reactive current

compensation and therefore on the ability of STATCOMs to

regulate voltage in the presence of disturbances.A simple integral controller was designed utilizing bode

plots of the input to output transfer function. Simulation

results illustrating dynamic response of the controlled system

have been presented.

VI. REFERENCES

[1] G. D. Preville, Flicker mitigation: Application to a STATCOM, inProc. 2001 European Conference on Power Electronics and

Applications, CD-ROM.

[2] M. Gutierrez, G. Venkataramanan, A. Sundaram, Solid state flickercontroller using a pulse width modulated AC-AC converter, in Proc.

thirty-fifth IEEE Industry Applications Society Annual Meeting, 2000,

vol. 5, pp. 3158-3165.

[3] C. V. Moreno, H. A. Duarte, and J. U. Garcia, Propagation of flicker inelectric power networks due to wind energy conversions systems,IEEETransactions on Energy Conversion, vol.17, pp.267-72, June 2002.

[4] K. R. Padiyar, and A. M. Kulkarni, Design of reactive current andvoltage controller of static condenser, Electric Power and Energy

Systems, vol. 19, no. 6, pp. 397-410, 1997.

[5] C. Hochgraf, and R. H. Lasseter, STATCOM controls for operationwith unbalanced voltages, IEEE Transactions on Power Delivery,

vol.13, pp.538-44, April 1998.

[6] S. Chen, and G. Joos, Series and shunt active power conditioners forcompensating distribution system faults, in Proc. 2000 Canadian

Conference on Electrical and Computer Engineering, vol.2, pp.1182-

1186.

[7] P.S. Sensharma, K. R. Padiyar, and V. Ramanarayanan Analysis andperformance of distribution STATCOM for compensating Voltage

fluctuation, IEEE Transactions on Power Delivery, vol. 16, pp. 259-

264, April 2001[8] J. Dolezal, and J. Tlusty, Background of active power filter control forflicker suppression, in Proc. 2001 European Conference on Power

Electronics and Applications, CD-ROM.

[9] C. Schauder, and H. Mehta, Vector analysis and control of advancedstatic VAR compensators, IEE Proceedings-C Generation

Transmission & Distribution, vol.140, pp.299-306, July 1993.

[10] P. W. Lehn, and M. R. Iravani, Experimental evaluation of STATCOMclosed loop dynamics, IEEE Transactions on Power Delivery, vol.13,

pp.1378-84, Oct. 1998.

[11] K. Ogata, Modern Control Engineering, 3rd ed., New Jersey, Prentice-Hall, 1997.

[12] A. K. Jain, and N. Mohan, Fast voltage regulation using STATCOMs,(to appear) in Proc.Caribbean Colloquium on Power Quality, Dorado,

Puerto Rico, 2003.

[13] H. K. Khalil, Nonlinear Systems, 3rd, ed., Upper Saddle River, NewJersey, Prentice Hall, 2002, pp. 139-142 and pp.512-517.

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