UPFC Controller Design for Power System Stabilization with Improved Genetic Algorithm Stella Morris, P.K. Dash*, K.P. Basu, Multimedia University, Cybety:aya,Malaysia A.M. Sharaf, University of New Brunswick, Canada Abstract - This paper presents the genetic optimization of the parameters of a fuzzy PI controller for UPFC( Unified Power flow controller) using an improved genetic algorithm to enhance the transient stability performance of power systems. The function based Takagi-Sugeno-Kang (TSK) fuzzy controller uses minimum number of rules(hvo rules) and generates the proportional action which by one-to-two inference mapping gives a variable gain PI controller. This single input function based scheme dispenses the gain dependency of the proportional or integral gains and generates independent control actions. Computer simulation results on a 2-area 4-machine 12-bus power system with an UPFC installed between buses sand r confirm the efficacy of the new approach in damping the local-mode and inter-area mode of oscillations. Keywords: Unified power flow controller (UPFC), TakagiSugeno-Kang (TSK) fuzzy control, one-to-two inference mapping, crossover, mutation, fitness function, improved genetic algorithm.
A unified power flow controller is a multi-functional FACTS controller with the primary function of power flow control plus possible secondary duties of voltage support, transient stability improvement and oscillations damping, etc [1-2]. It basically consists of two voltage source inverters (VSIs) connected back-to-back with an interconnecting dc storage capacitor. One VSI is connected to the system bus by a shunt transformer and the other VSI is connected to the transmission line by a series transformer. The power balance between the series and shunt connected VSIs is a prerequisite to maintain a constant voltage across the dc capacitor connected between the two VSIs. The series inverter is used to inject a controlled voltage, in series with the line and thereby to force the power flow to a desired value. The shunt inverter is controlled in such a way as to provide precisely the right amount of real power at its dc terminals to meet the real power needs of the series inverter and to regulate the dc bus voltage. The basic circuit arrangement of UPFC is r,hown in Fig.1. Series Transformer
I. INTRODUCTION
W'
ith the development of power systems, especially the opening of electric energy markets, it becomes more and more important to control the power flow along the transmission line and thus to meet the needs of power transfer. Different types of FACTS devices like static compensator (STATCOM), unified power flow controller (UPFC), thyristor controlled static compensator (TCSC), etc. have been studied to show their effectiveness in improving voltage and angle stability and reducing electromechanical oscillations in interconnected power systems [ 1-41, Amongst the several FACTS devices studied for power system stabilization, STATCOM and UPFC provide the most significant performance in damping out low frequency multimodal oscillations in interconnected power systems. As utilities increase power exchanges over a fixed network, interarea oscillations are more likely to happen, even under nominal operating conditions. For many years power system stabilizers (PSSs) have been one of the most common controls used to damp out oscillations and to offset the negative damping of the automatic voltage regulators. The major role of PSS is to introduce a modulating signal acting through the excitation system to add rotor oscillation damping. Even though power system stabilizer is the main damping control, during some operating conditions, this device may not produce enough damping especially to inter-area mode and therefore, there is an increasing interest in using FACTS devices like STATCOM and UPFC to aid in damping of these oscillations.
* Corresponding author P.K.Dash was with M.M.U., Malaysia earlier and now with Silicon Institute of Technology, Bhubaneswar, India. , Email:
[email protected] 0-7 803-7906-3/03/$17.00 02003 IEEE.
Lpzi-tJ Fig. I Basic circuit arran;:ement of an UPFC
belween buses sand r.
Various control strategies for controlling the UPFC have been reported recently[!i-9]. The PI regulators used for controlling UPFC suffer from the inadequacies of providing suitable control and transient stability enhancement over a wide range of system operating conditions. The simple but effective fuzzy logic based controllers reported in the literature are mostly on either Mamdani or TSK fuzzy inferencing schemes. These controllers rely on the performance of the plant or the system and mostly centers around two input variables, the error e andJ'e) (f(e) is function of the error e), To circumvent the above mentioned difficulties and to have a robust nonlinear fuzzy PI controller for UPFC, this paper uses a single-input error ;dgnal(e). The control action of this proposed fuzzy controller is made variable by a rule base comprising a minimum number of rules(two rules), independent tuning parameters and a function based consequent part.
1540
11. SYSTEM MODEL
A . UPFC Modeling
The schematic diagram of an UPFC installed between the buses s and r of a multi-machine power system is shown in Fig.1. It is composed of an excitation transformer (AT), a boosting transformer (ST), a DC link capacitor, and y o threephase GTO based voltage source converters. Under the assumption that a VSI can provide balanced voltages, the series and the shunt VSI can be modeled as simple voltage sources. Fig2 shows the simplified model of UPFC where p..lV,I and pShlV,(are the voltages induced across ST and AT respectively. The transformers have been modeled as simple reactance. The voltage sources with the transformer reactance can be converted as current injections at the appropriate buses. In Fig.3, the UPFC is represented as two current sources (one positive and the other negative) connected across the buses s and r.
Fig2 UPFC equivalent circuit with cantmlled voltage sources
Fig.3 UPFC equivalent circuit with cantmlled current SOU~CCS
where xse,xSh
= series and shunt reactances of UPFC converter transformers, respectively. B,,, BSh = series and shunt susceptances of UPFC converter transformers, respectively. pra, (S. = series voltage magnitude ratio (Vsr / IV,I ) and angle of VsTwith respect to V,. psn.aXh= shunt voltage magnitude ratio (VATI /V,I ) and angle of VATwith respect to V,. These current injections can be converted to appropriate real and reactive power injections at the respective buses. The real and reactive powers injected at the buses s and r are P, = P , I~V, I' B, sin ad + P,* I V, 1' E,, sin a,e Q, = prhI V, I' B,, cos a,, - 4, I V, 1' +P, I V, I' E, cos a,.
P,=-P,,B,~IV,I/V,I~~~(B,.+~,)
Q, = -p,J.. I vs II V I cos@,.
+ase)
However, if some real power loss is accounted for
1
Yo, = P se cos a ye Y,,
voy = P ,
e = tan
-1
s i n a s c Y , d + P.,.
(
~ , = ~ , ~ I ~ , I ' B , , ~ ~ n ~ , ~ + ~ ~ ~ I ~ , I ' ~ , , ~ i (2) n~,,+l~,I./~,I
1541
), Y ,
iXq
(I)
I, = real component of shunt converter cument. For constancy of dc link voltage, the following relation should be satisfied.
- p I Esin =
a r cv,q
COSa,,
Y,
'
, / = :
(9)
~
& and obtained as
of the shunt voltage control circuit can be
The dynamic equations of the UPFC are centered around the dc link capacitor. The dynamics of the D.C. link voltage neglecting losses can be revresented as
This controller is more sensitive to noisy data than a conventional PI controller, where independent tuning of P and I gains are done. To have a robust nonlinear fuzzy controller, a closed form fuzzy logic controller is realized using function based analytical approach. The TSK model is chosen for the consequent part of the fiuzzy rule base and only one input (error) is used for deriving the fuzzy controt magnitude. In order to achieve an independent tuning property of the gains, one-to-two inference mapping is used as shown in Fig.5. The conventional PI controller is shown in Fig.4.
(17) where I, is the in-phase component (with respect to V,) of the current drawn by the shunt converter. Further to control the DC link voltage, the current I, is obtained using a simple PI controller as (18) I . , = K , (v, - v, K , ,6 - Y d C
1.
p,
Fig.4 Conventional PI controller (Both P and Q controllen)
1.
The value ofthe parameters K,I, Kp2, Kit, Ki2;KP, Ki, KI, T I , Kpdo and Kidc are chosen as KPI=0.1, Kp2=0.1, Ki, =O.OI, Ki2 =O.Ol, K, =0.1, Ki =O.l, K, = I , TI =O.l, Kpdc=O.I , and Kidc=O. I
C. Synchronous Machine Model Each synchronous generator is modeled as a 3’d order model equipped with a simple automatic voltage regulator (AVR) for excitation control. A PSS is also used for controlling the local modal oscillations as mentioned in the introduction. The dynamics of each synchronous machine are: p 6 = W - U,, p= d / dt (differential operator), nf
po = -(Pm
n
- P,)
PE; =(E,do+AE,d -E; -(xd - x ; ) i d ) I z ~ , pAEfd =Ke(Vref - V t and -6.0 5 Efd < 6.0
+U)/%,
-AEfd/Z,
P, = E;iq + ( x q - xi))idiq
Fig.5 Fuzzy PI Controller with I to 2 mapping (Both P and Q controllers)
The gains of the PI controller are optimized by taking ITAE (1-: tle(t)bt) performance index Two independent Fuzzy proportional actions are generated and this results in two independent equivalent gains. The design of fm!y proportional controllers (Fuzzy PI or Fuzzy P2) can be realized using the TSK scheme for the consequent parts.in the fuzzy rule base. The two rules used are: Rule 1 : If error [el is PB then u p = f , ( e )
If error
is AZ then u p = f , ( e )
The control U in (19) is obtained from PSS control loop as U = K,(s.t,/l + s . t , ) . { ( l + s . t , ) / ( l + s . t , ) } A ~ (20)
Rule 2 :
where all the gains and time constants are Appendix.
where the set PB stands for positive big and the set AZ stands for approximately zero, (e) and f,( e ) are given functions of the error e. The membership functions are shown in Fig.6 and are applied in the positive error domain. If the functions f,(e) and (e) are chosen as
given in
111. ANALYTICAL FUZZY PI CONTROLLER
Most of the existing fuzzy logic based PI regulators use two inputs such as erroe (e) and its derivative (e) or change
(Ae) and the control action is %
Ki
1;
f,
where the apparent proportional and integral gains
(21)
EI
and
f,
f,( e ) = x , e , and (I?) = x2 e (22) the closed form of solui:ion for the fuzzy inference is obtained as
%
Au = K ie + K , Ae
le1
u p
=e
1x2
+ (XI
- xz )
1.1
A sensitivity index is derived at
are obtained as functions of both inputs 1542
(23)
e = 0 , as
~
This control scheme is termed as TSK-I[IO]. It can be observed from (24) that the sensitivity function is greater than 0 and this is an important condition for system stability.
C.Seleciion Two chromosomes in the population will be selected to undergo genetic operations for reproduction by the method of spinning the roulette wheel[l2]. The selections can be done by assigning a probability qi to the chromosomep; (30) , i = 1,2 ,.... pap _ s i z e 4 I. =
f(pj)
cf b k )
pop
Si?#
~
,=I
The cumulative probability 0
e-
?
i j for
the chromosome p i is
defined as
Fig.6 Scaled error Membership function
IV. IMPROVED GENETIC OPTIMIZATION Genetic algorithm (GA) is a directed random search technique that is widely applied in optimization problems. Different selection schemes and genetic operators have been proposed to improve the performance of GA. In this paper, an improved CA[ 1 I ] has been used for the optimization of TSK-I fuzzy parameters(x,, x2, x3, and x4). First a population of chromosomes is created randomly. Then the chromosomes are evaluated by a defined fitness function. Next, some of the chromosomes are selected for performing genetic operations. The produced offspring replace their parents in the initial population. This process repeats until a user-defincd criterion is reached.
The selection process starts by randomly generating%a nonzero floating-point number, d E [o I ] . Then, the chromosome p i is chosen if ji-,< d 5 j . In the selection process, only two chromosomes will be selected to undergo the genetic operations.
D.Genetic Operations I ) Crossover: The crossover operation is mainly for exchanging information from the two parents. The two parents will produce one offspring. Initially, 4 chromosomes will be generated as follows: chl = P + * P
(32)
ch 2 = p (I - w ) + max (p., , p i )w rh3= p,,(l-w)+min(p,,q,)w c h 4 = (P,," + P,i" )(I - w)+ (PI t 2 X L xiax x t . 1 P max =
(33)
L
_I
A . Initial population The first set of population is generated randomly.
t,, P l ? " ' , P,,_,,;:, 1 P i = LJ PI, ...P;;-P,,,,..,,, p =
/,
(25)
I
(26)
i=1,2,..., pop-size; j=1.2 ,..., no-vars pura A,n 5 p i 2 para iwhere, pop-size = population size = 50. no-vars = number of variables to be tuned = 4 pi, =parameters to be tuned = minimum
value of the parameter
(27)
p.
'j
para
value of the parameter p
= maximum
1,
B. Evaluation Each chromosomc in the population is evaluated by a defined fitness function. The fitness function to evaluate a chromosome in the population can be obtained as follows:
"d
y ,d ( I ) = desired output at any given instant y, (t ) = actual output at any given instant number of input-output data pairs. I fitness function =
nd
=
~
I+e
(29)
[.a,
(34) P2)W
(35) (36)
(37) P mi" = [x i'" X,"i x 1:. x:)" 1 w = the weight to be determined by the users = 0.8 Among chi-ch4, the one with the largest fitness value is used as the offspring of the crossover operation and is (38) os = [ x , x 2 x, x 4 ] 2) Mufation: The best offspring of the crossover operation will then undergo the mutation operation. The mutation operation is to change the genes of the chromosomes. Three new offspring will be generated by the mutation operation mui , = [x, x 2 x, x , ] t [ b , A n , b , A n , b , A n , b , A n , ] 2
j = 1,2,3 (39) where b,, b2, b j , and br, can only take value of 0 or I . Anl, An2, An3, and Ana are randomly generated numbers such that para bin 5 (xi + ~ n , )
