Electrical Engineering

, Volume 101, Issue 3, pp 943–967 | Cite as

Multi-band power oscillation damping controller for power system supported by static VAR compensator

  • Wesley PeresEmail author
Original Paper


Low-frequency electromechanical oscillations damping is powerfully crucial in power system operation. In order to fulfill this requirement, power oscillation damping (POD) controllers are installed in synchronous generators and flexible alternating current transmission system devices. These controllers can have either a conventional fixed structure composed by stages of gain and phase compensation or a multi-band modern structure (PSS4B) composed by three bands that correspond to a specific frequency range (low, intermediate and high frequency). In the PSS4B structure, each band consists of two branches based on differential filters (with a gain, lead–lag blocks and a hybrid block). This paper investigates the application of PSS4B as a POD controller for the static VAR compensator (PSS4B-SVC-POD) to damp oscillations in multi-machine power systems. The proposed PSS4B-SVC-POD and power system stabilizers (fitted on generators) are simultaneously designed through an optimization approach aiming at maximizing the closed-loop damping ratio. Modal analysis, frequency responses and time simulations for the well-known two-area four-generator power system show the good performance of the proposed controller fitted on static VAR compensator.


Power oscillation damping controller Static VAR compensator Multi-band PSS4B Power system stabilizers Coordinated design 

List of symbols



Automatic voltage regulator


Flexible alternating current transmission systems


Multi-band power system stabilizer (PSS4B)


Power oscillation damping


Power sensitivity model


Particle swarm optimization


Power system stabilizers


Modern multi-band power system stabilizer (MB-PSS)


Proposed POD controller based on the multi-band PSS4B


Proposed POD controller for SVC based on the multi-band PSS4B


Static synchronous series compensator


Static synchronous compensator


Static VAR compensator


Conventional POD fitted on SVC


Thyristor-controlled reactor


Thyristor-controlled series compensator

Number of devices


Number of busses


Number of generators


Number of conventional POD controllers


Number of multi-band PSS4B-POD controllers


Number of conventional power system stabilizers


Number of static VAR compensators

Power system variables

\( \varOmega_{k} \)

Set of nodes connected to bus k


Damping constant of generator k (pu)


Field voltage (exciter output) of generator k (pu)


Internal voltage of generator k, proportional to the rotor field flux in the direct axis (pu)


Inertia constant of generator k (s)


d-axis stator current of generator k (pu)

KAk, TAk

Gain (pu) and time constant (s) of AVR at generator k


Gain (pu) and time constant (s) of the dynamic model of SVC i

PGk, QGk

Real and reactive generated powers at node k (pu)

PLk, QLk

Real and reactive power loads at node k (pu)

Pkm, Qkm

Real and reactive power flows from node k to node m (pu)


Mechanical power of generator k (pu)


Injected reactive power at node k (pu)

Td0 k

d-axis open-circuit time constant of generator k (s)


Supplementary stabilizing signal (POD output) of SVC i (pu)


Supplementary stabilizing signal (PSS output) of generator k (pu)


Reference voltage of AVR of generator k (pu)


Reference voltage of SVC i (pu)


Voltage magnitude at node k (pu)


d-axis synchronous reactance of generator k (pu)


d-axis transient reactance of generator k (pu)


q-axis synchronous reactance of generator k (pu)


Susceptance of SVC (pu)


Internal angle of generator k (degrees)


Voltage phase at node k (degrees)


Synchronous speed (rad/sec)


Deviation operator (used in linearized system of equations)

\( \Delta \omega_{puk} \)

Rotor speed deviation of generator k (pu)

\( \varvec{f}\left( {\varvec{x},\varvec{z},\varvec{u}} \right) \)

Set of first-order nonlinear differential equations

\( \varvec{g}\left( {\varvec{x},\varvec{z},\varvec{u}} \right) \)

Set of nonlinear algebraic equations


Vector of input variables


Vector of state variables


Vector of algebraic variables

Modal analysis

\( \xi_{\hbox{min} } \)

Damping ratio associated with the dominant eigenvalue in closed-loop operation

\( \varvec{J}_{1} ,\varvec{ J}_{2} ,\varvec{J}_{3} ,\varvec{J}_{4} \)

Derivative matrices


State space matrix


Input matrix

\( \lambda = \sigma \pm j\omega_{d} \)

Complex eigenvalue with real (\( \sigma \)) and imaginary (\( j\omega_{d} \)) components

\( \xi \)

Damping ratio of any complex eigenvalue

Power system controllers

FLi, FIi, FHi

Low-, intermediate- and high-band central frequencies of PSS4B-POD i (Hz)

KH11 i, KH17 i

High-band first lead–lag blocks coefficients of PSS4B-POD i (pu)

KH1 i, KH2 i

High-band differential filter gains of PSS4B-POD i (pu)

KI11 i, KI17 i

Intermediate-band first lead–lag blocks coefficients of PSS4B-POD i (pu)

KI1 i, KI2 i

Intermediate-band differential filter gains of PSS4B-POD i (pu)

KL11 i, KL17 i

Low-band first lead–lag blocks coefficients of PSS4B-POD i (pu)

KL1 i, KL2 i

Low-band differential filter gains of PSS4B-POD i (pu)

KLi, KIi, KHi, KGi

Low-band, intermediate-band, high-band and series gains of PSS4B-POD i (pu)


Gain parameter of conventional POD i (pu)


Gain parameter of conventional PSS k (pu)


Time constants of conventional POD i (s)


Time constants of conventional PSS k (s)

TH1 i, TH2 i, TH7 i, TH8 i

High-band time constants of PSS4B-POD i (s)

TI1 i, TI2 i, TI7 i, TI8 i

Intermediate-band time constants of PSS4B-POD i (s)

TL1 i, TL2 i, TL7 i, TL8 i

Low-band time constants of PSS4B-POD i (s)


Washout time constant of conventional POD i (s)


Washout time constant of conventional PSS k (s)


Algebraic variables of conventional PSS k (pu)

V1 ckV2 ckVPSS k

State variables of conventional PSS k (pu)


Algebraic variables of PSS4B-POD i (pu)


Algebraic variables of conventional POD i (pu)

V1iV2iV3iV4iV5 iV6 i

State variables of PSS4B-POD i (pu)

V1 iV2 iVPODi

State variables of conventional POD i (pu)


Algebraic variables of PSS4B-POD i (pu)


Supplementary stabilizing signal (POD output) of SVC i (pu)


Output voltage of conventional PSS k (supplementary stabilizing signal) (pu)


Input signal of POD i (pu)

Metaheuristics and optimization

c1, c2

Positive acceleration constants (PSO)


Best location in history associated with the ith particle (PSO)


Maximum number of generations


Inertia weight bounds (PSO)


Inertia weight (PSO)

\( \varvec{x}_{\varvec{i}}^{\varvec{t}} \), \( \varvec{v}_{\varvec{i}}^{\varvec{t}} \)

Position and velocity of the ith individual in the tth generation


Best location among all particles in history (PSO)



This work was supported by the following Brazilian Agencies: FAPEMIG (APQ-02245-18), CNPq and Capes (Finance Code 001). Besides, the technical support from GOCES (Optimization, Control and Power System Stability Research GroupUFSJBrazil) is greatly acknowledged.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electrical Engineering (DEPEL)Federal University of São João del-Rei (UFSJ)São João del-ReiBrazil

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