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International Journal of Advanced Research in Engineering and Technology (IJARET)
Volume 11, Issue 4, April 2020, pp. 219229, Article ID: IJARET_11_04_022
Available online athttp://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=11&IType=4
ISSN Print: 09766480 and ISSN Online: 09766499
© IAEME Publication Scopus Indexed
MULTIMACHINE POWER SYSTEM STABILITY
ENHANCEMENT WITH UPFC USING LINEAR
QUADRATIC REGULATOR TECHNIQUES
Brijesh Kumar Dubey
Department of Electrical and Electronics Engineering,
Pranveer Singh Institute of Technology, Kanpur, India
Dr. N.K. Singh
Department of Computer Science Engineering,
Director, ITM Gida, Gorakhpur, India
ABSTRACT
It is well known that for computer simulation and analysis of power systems both
planning and operation are necessary. Computer simulation requires an appropriate
mathematical model that many interrelated linear, nonlinear, differential and
algebraic equations of the system. Such mathematical model is needed for analysis and
improves power system dynamic stability performance and also design a suitable
controller. This paper provides comprehensive development procedure and final forms
of mathematical models of a power system installed with UPFC and controller UPFC
using linear quadratic regulator techniques for stability improvement. The impacts of
control strategy on power system multi machine installed with UPFC, without UPFC
and with controller UPFC at different loading and operating conditions are discussed.
The accuracy of the developed models is verified through comparing the study results
with those obtained from detailed MATLAB programming. In this paper settling time
analysis also have been done for justification of the stability improvement.
Keywords: Modelling; LQR; UPFC; Eigen value analysis; dynamic stability; Power
oscillation damping controller
Cite this Article: Brijesh Kumar Dubey and Dr. N.K. Singh, Multimachine Power
System Stability Enhancement with UPFC using Linear Quadratic Regulator
Techniques, International Journal of Advanced Research in Engineering and
Technology (IJARET), 11(4), 2020, pp. 219229.
http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=11&IType=4
1. INTRODUCTION
The expansion of electric power systems are not always favorable, because along with the
complexity of the network, the damping torque of the whole system is also reduce and the result
can make the power system unstable. On the other hand, the main drawbacks of the electrical
power systems are deteriorating voltage profiles and issues related to power system stability
Multimachine Power System Stability Enhancement with UPFC using Linear Quadratic Regulator
Techniques
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and security; the major cause of this problem is the overloading of the electrical power
transmission lines [1]. The main parameters which are responsible to determine the transmitted
electrical power over a power transmission line are power transmission line impedance, the
receiving and sending end voltages, and phase angle between the two voltages. Therefore,
controlling one are more of these power transmission parameters, it is possible to control the
active and reactive power flow over a power transmission lines. To improve power system
stability power system stabilizers have been widely used. The equilibrium of the dynamics of a
largescale power system is uncertain due to nonlinear and interconnected system, and physical
limitations of the controllers are also present. Controllers are designed as a part of excitation
system of the generator and feedback linearization scheme is developed by Deqiang Gan and
et.al [9]. However, due to some drawbacks of the conventional PSSs, the need for finding a
better substitution still remains. Therefore, in this paper, the application of the FACTS devices
such as unified power flow controller (UPFC) to improve dynamic stability of a multimachine
electric power system is presented and a supplementary stabilizer based on the FACTS device
is incorporated. Investigations involve the analysis of the linearized state space equations of the
power system dynamics. To damp out the low power oscillation frequency and increase system
oscillations stability, the installation of Power System Stabilizer (PSS) is both economical and
effective [2]. Recently appeared FACTS (Flexible AC Transmission System)based stabilizer
offer an alternative way in damping power system oscillation. The primary function of the
FACTS controllers is not only it Damping Duty, but also to increase the overall power system
oscillation damping characteristics [3]. The objective of this paper is to design a UPFC based
Power Oscillation Damping (POD) controller to damp the low frequency electromechanical
oscillations over wide range of operating conditions [46]. The objective and steps involved are
as follows:
• The model of the multi machine power system installed with UPFC is obtained by
linearizing the nonlinear equations around a nominal operating point.
• To present systematic approach for designing UPFC based power oscillation damping
controller.
• Eigenvalue analysis technique has been used as this is a powerful tool for analyzing
oscillatory instability and yields information about the frequency and damping of each
oscillation mode.
• Design a POD controller using linear quadratic regulator technique which places the
eigenvalue corresponding to mode of oscillation at desired location such that
eigenvalues get placed within a vertical degree of stability.
• To demonstrate the effectiveness of the designed POD controller under different
controlling parameter.
2. INVESTIGATED SYSTEM
Figure 1 depicts a multi machine power system installed with unified power flow controller
between bus 2 and bus 3 on the transmission line. It consists of the components:
Excitation transformer (ET), (ii) Boosting transformer (BT), (iii) Two threephase GTO
based voltage source converters (VSCs), (iv) Dc link capacitor nased on pulse width modulation
converters (assumed) [1011]. This paper considers three identical machines with same rating
and operating conditions. The system parameters are given in AppendixA.
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Vt
It
VEt
XtE
IB Xb
VB
Xe
ET
Ii
VSCE
Cdc
VSCBBT
XBv
meδe mb δb
I0
IEt
qE
BUS 2
BUS 3
t
qE t
qE
BUS 1
G2
G1
G3
Figure 1 UPFC with MMIB power system
2.1. Unified Power Flow Control
The UPFC is the best controller, because over the line real and reactive power flows and the
bus voltage, it provides independent control [12], Between the converter dc link provides a path
to exchange active power. The input control parameter discussed here are, me, mb , δe, δb of the
UPFC. Here me is amplitude modulation ratio of shunt VSCE, δe is phase angle of shunt VSC
E, mb is amplitude modulation ratio of series VSCB and δb represents phase angle of series
VSCB [5]. The phase angle control method is more efficient than Amplitude modulation
control methods [13]. Power system which comprises a multiple synchronous generators
connected to an infinite bus through a transmission line and stepping up transformer. The
generators are assumed to have Automatic Voltage Regulator (AVR) controlling its terminal
voltage. The UPFC is used in this study to just analyze the purpose of power system stability
and its characteristics. The UPFC can fulfill the multiple control objectives.
Figure 2 Variation of Settling Time with Phase Angle
Figure 3 Variation of Settling Time with Modulation Index
G1, 0.3, 1.0058G1, 0.4, 1.4158
G1, 0.5, 1.8568G1, 0.6, 2.3344
G2, 0.3, 1.7544
G2, 0.4, 2.4183
G2, 0.5, 3.0233G2, 0.6, 3.2889
G3, 0.3, 1.2186
G3, 0.4, 1.1498
G3, 0.5, 1.1168G3, 0.6, 1.0873
Variation in Settling Time with variation in phase angle of the converters
for G1, G2, G3
G1 G2 G3
Phase angle
G1, 0.3, 2.3218
G1, 0.4, 1.1913G1, 0.5, 1.8997 G1, 0.6, 1.8997
G2, 0.3, 4.3069
G2, 0.4, 3.4188 G2, 0.5, 3.0233 G2, 0.6, 3.0882
G3, 0.3, 0.944
G3, 0.4, 1.0361 G3, 0.5, 1.1168 G3, 0.6, 1.1913
Variation in Settling Time with variation in modulation index of the
converters for G1, G2, G3
G1 G2 G3
Modulation Index
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2.2. PhillipsHeffron model for Multimachine System with UPFC
A singleline diagram of 3generator installed with a UPFC is shown in Figure 1. Figure 4
shows the PhillipsHeffron model of power system installed with UPFC, developed by Wang
[6, 14] with the modification of the basic PhillipsHeffron model including UPFC. Around a
nominal operating point, by linearizing the nonlinear model, this model has been developed.
The parameters of the model depend on the system parameters and the operating condition. In
this model [Δu] is the column vector while [Wpu], [Wqu], [Wvu] and [Wcu] are the row vectors. Where,
[Δu] = [Δme Δδe Δmb Δδb]T, [Wpu] = [Wpe Wpde Wpb Wpdb]
[Wqu] =[Wqe Wqde Wqb Wqdb],[Wvu] = [Wve Wvde Wvb Wvdb]
[Wcu] = [Wce Wcde Wcb Wcdb]
The control parameters of the UPFC in which me and mb are the amplitude modulation ratio
of shunt and series converters respectively. By controlling me, the voltage at a bus where UPFC
is installed, is controlled through reactive power compensation. The magnitude of series
injected voltage can be controlled by controlling mb. δe and δb are the phase angle of shunt and
series converters respectively. Phase angle of the shunt converter, which regulates the dc
voltage at dc link, phase angle of the series converter when controlled results in the real power
exchange control [1516].
1
(sM+D)
ω0
s
Δω
W1
W4 W5

W2
W6
 Ex(s)1
(K3+sTd0)
WpuW8 Wqu Wqd Wvu Wvd Wpd
Wcu
1
(W9+s)W7
Δδ
ΔEfd
Δu
Δu
ΔE’q
ΔVdc
Figure 4 PhillipsHeffron model of power system installed with UPFC
2.3. Model Analysis
The constant parameters of the model computed for nominal operating condition and system
parameters are
Table 1 Model Analysis data
W1 = 0.1372 Wpb = 0.0715 Wpde = 0.2514
W2 = 0.4350 Wqb = 0.0223 Wqde =  0.2243
W3 = 0.4727 Wvb = 0.0145 Wvde = 0.0782
W4 = 0.0598 Wpe = 0.7860 Wcb = 0.1763
W5 = 0.0159 Wqe = 0.2451 Wce = 0.0018
W6 = 0.5092 Wve = 0.1597 Wcdb = 37.7306
W7 = 80.9318 Wpdb = 0.0229 Wcde = 77.2952
W8 = 21.0677 Wqdb =  0.0204 Wpd = 0.4287
W9 = 40.6634 Wvdb = 0.0061 Wqd = 0.1337
Wvd = 0.0871
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2.4. Design of POD Controller (Linear Quadratic Regulator Technique)
The linearized statespace model of multi machine power system is obtained by phillipheffron
model as expressed by,
�̇� = 𝐴𝑋 + 𝐵𝑈 (1)
Where A and B are the matrices of the system and input respectively. X is the system state
vector, and U is the input state vector. The matrices A and B are constant under the assumption
of system linearity. If we use state feedback, that is, if we set U= KX where K is the chosen
gain matrix, the equation (1) becomes [7],[8],
�̇� = (𝐴 − 𝐵𝐾)𝑋 (2)
And the problem is to allocate any set of eigenvalues to closed loop matrix by choosing the
gain matrix K [9].
Steps for controller design
• Required data – A, B, Q, R and N matrix
• Data for calculation K matrix, P matrix and eigen value
• Set N matrix is zero
• Q and R are the positive definite real symmetric matrix.
• Calculate the P matrix
• Find the value of K matrix
UPFCPower
System
POD
Controller
y(t)u(t)
x(t)
uin(t)
uoPOD(t)
Figure 5 Generalized Diagram of POD Controller
3. SIMULATION RESULTS UNDER DIFFERENT SYSTEMS AND AT
VARIOUS LOADING CONDITIONS
The proposed model of UPFC in multi machine power system figure 1 has been used in order
to study the damping performance. To study the performance of the proposed controller,
simulation results under different system conditions and at various loading conditions i.e. at
normal operating point corresponding to line loading of 1.0 pu and at 20 percent decrease and
increase in line loading are shown. It can be readily seen that the proposed controller performs
better in terms of reduction of overshoot and settling time than system without UPFC and with
system with UPFC only. This is consistent with the eigenvalues analysis results. Simulation
results with variation in system state, rotor angle (δ) of generator is only considered.
Case1: [pf=0.85, me =.5, δe =0.5, mb =0.5, δb =0.5, Load= 0.8 PU, D=4]
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Table 2 Settling Time and Overshoot for Multimachine system
Settling Time and Overshoot analysis
System Settling Time Overshoot
System Without UPFC (G1) 7.1299 3.3065
and without (G2) 9.8276 3.8700
controller UPFC (G3) 19.2541 3.4422
System With UPFC
(G1) 1.1128 3.1854
(G2) 3.0233 5.1764
(G3) 1.1168 3.3337
System With controller (G1) 0.4311 2.5789
UPFC (G2) 0.8307 2.6178
(G3) 0.772 2.5874
Table 3 Eigen values for Multimachine system.
Eigen Value Analysis
System Without UPFC and controller UPFC
Generator(G1) Generator(G2) Generator(G3)
28.3671 + 0.0000i 45.9363 + 0.0000i
28.3680 +
0.0000i
0.4291 +10.3486i 0.3185 + 9.1082i
0.1565
+10.3624i
0.4291 10.3486i 0.3185  9.1082i 0.1565 10.3624i
5.5149 + 0.0000i 4.3082 + 0.0000i 5.5083 + 0.0000i
0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i
System With UPFC only
88.4236 + 0.0000i 73.2065 + 0.0000i
88.4178 +
0.0000i
6.5628 +29.2773i
26.4028 +
0.0000i
6.5492
+29.2205i
6.5628 29.2773i
11.8046 +
0.0000i 6.5492 29.2205i
2.9793 + 4.8762i 1.1876 + 5.7775i 2.7204 + 5.0381i
2.9793  4.8762i 1.1876  5.7775i 2.7204  5.0381i
System With controller UPFC
7.5417 + 0.0000i 7.8567 + 0.0000i 7.5417 + 0.0000i
0.6782 + 0.0000i 0.4661 + 0.0000i 0.6782 + 0.0000i
0.0038 + 0.0074i 0.0044 + 0.0057i 0.0038 + 0.0074i
0.0038  0.0074i 0.0044  0.0057i 0.0038  0.0074i
0.0005 + 0.0000i 0.0005 + 0.0000i 0.0005 + 0.0000i
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Figure 6 Systems(G1, G2, G3) with and without (UPFC and UPFC controller)
Figure 7 Systems[G1 (Blue), G2 (Red), G3(Orange)] with UPFC controller
Figure 8 Systems[G1 (Blue), G2 (Red), G3(Orange)] with UPFC only
Case2: [pf=.85, me =.5, δe =.5, mb =.5, δb =.5, Load= .8 PU, D=8]
Table 4 Settling Time and Overshoot for Multimachine system
Settling Time and Overshoot analysis
System Settling Time Overshoot
System Without UPFC (G1) 3.7132 3.3197
and without (G2) 4.6897 3.6575
controller UPFC (G3) 7.1299 3.3065
System With UPFC
(G1) 1.4938 3.4864
(G2) 2.0513 4.789
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(G3) 1.1128 3.1854
System With controller (G1) 0.8120 2.5422
UPFC (G2) 0.8436 2.607
(G3) 0.4311 2.5789
Table 5 Eigen values for Multimachine system
Eigen Value Analysis
System Without UPFC and controller UPFC
Generator(G1) Generator(G2) Generator(G3)
0.8406 + 9.7780i 45.9362 + 0.0000i 28.3671 + 0.0000i
0.8406  9.7780i 0.6625 + 9.0878i 0.4291 +10.3486i
10.3168 + 6.1200i 0.6625  9.0878i 0.4291 10.3486i
10.3168  6.1200i 4.3099 + 0.0000i 5.5149 + 0.0000i
0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i
System With UPFC
78.2843 + 0.0000i 73.2089 + 0.0000i 88.4236 + 0.0000i
6.6486 +15.0308i 26.4507 + 0.0000i 6.5628 +29.2773i
6.6486 15.0308i 11.5876 + 0.0000i 6.5628 29.2773i
2.2194 + 7.5799i 1.6158 + 5.7241i 2.9793 + 4.8762i
2.2194  7.5799i 1.6158  5.7241i 2.9793  4.8762i
System With controller UPFC
5.4454 + 0.0000i 7.8567 + 0.0000i 7.5417 + 0.0000i
0.1653 + 0.0000i 0.4661 + 0.0000i 0.6782 + 0.0000i
0.0043 + 0.0066i 0.0045 + 0.0057i 0.0038 + 0.0074i
0.0043  0.0066i 0.0045  0.0057i 0.0038  0.0074i
0.0015 + 0.0000i 0.0005 + 0.0000i 0.0005 + 0.0000i
Figure 9 Systems(G1, G2, G3) with and without (UPFC and UPFC controller)
Figure 10 Systems[G1 (Blue), G2 (Red), G3(Orange)] with UPFC controller
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Figure 11 Systems[G1 (Blue), G2 (Red), G3(Orange)] with UPFC only
Case3: [pf=.85, me =.5, δe =.5, mb =.5, δb =.5, Load= 1.0 PU, D=4]
Table 6 Settling Time and Overshoot for Multimachine system
System Settling Time Overshoot
System Without UPFC (G1)8.1609 3.2915
and without (G2) 10.1817 3.5377
controller UPFC (G3)19.8206 3.2929
System With UPFC
(G1) 1.6356 3.4154
(G2)2.7273 4.6552
(G3) 1.0089 2.9598
System With controller (G1)0.7377 2.4151
UPFC (G2) 0.7759 2.4456
(G3) 0.7433 2.4956
Figure 12 Systems(G1, G2, G3) with and without (UPFC and UPFC controller)
Figure 13 Systems[G1 (Blue), G2 (Red), G3(Orange)] with UPFC controller
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Figure 14 Systems[G1 (Blue), G2 (Red), G3(Orange)] with UPFC only
4. CONCLUSION
In this paper, the power system low frequency electromechanical oscillation was damped via
linear quadratic regulator technique (using MATLAB tool) based POD controller when applied
independently with UPFC and investigated for multimachine power system. For the proposed
controller design problem, an eigenvaluebased objective function to maximize the system
damping ratio among all complex eigenvalues concept was developed.
Method have been described in this paper. As compared The effectiveness of the proposed
controller in damping the low frequency EM mode of oscillations and hence improving power
system stability have been verified through eigenvalue analysis and simulation results with
different system condition and under different line loading and without any line loading.
APPENDIX A
System DataGenerator data:
• [M] = [8, 8, 8] M J/ MVA;
• [Xd] = [1, 1, 1];
• [X’d] = [0.3, 0.3, 0.3];
• [Td0] = [5.0, 5.0, 5.0] sec;
• [Xq ] = [0.6, 0.6, 0.6];
• [δ ]= [0.6981, 0.6981, 0.6981] radian;
• [E’q ]=[1.0, 1.0, 1.0]
Excitation System data:
• [Ka] = [10, 10, 10];
• [Ta] = [0.01, 0.01, 0.01] sec.
Transformers data:
• [Xb] = [0.03, 0.03, 0.03];
• [Xe ]= [0.03, 0.03, 0.03]
Transmission line data:
• [XBV] = [0.3, 0.3, 0.3];
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• [XtE ] = [0.3, 0.3, 0.3]
Operating conditions:
• Vb= 1.0;
• pf = 0.85;
• Frequency = 50 Hz.
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