2012 Fourth International Conference on Computational Intelligence and Communication Networks
Design of Decentralized PSSs for Multimachine Power System via Reduced Order Model Mahendra Kumar Dr. Rajeev Gupta M.Tech. Scholar Professor
[email protected] [email protected] Department of Electronics UCE, Rajasthan Technical University Kota – 324001, India with speed variations. The application of a PSS is to generate a supplementary stabilizing signal, which is applied to the excitation system or control loop of the generating unit to produce a positive damping. The most widely used conventional PSS is the lead-lag PSS, where the gain settings are fixed at certain value which are determined under particular operating conditions to result in optimal performance for that specific condition. However, they give poor performance under different synchronous generator loading conditions [3]. The static output feedback problem is one of the most investigated problems in control theory. The complete pole assignment and guaranteed closed loop stability is still not obtained by using static output feedback. Another approach to pole placement problem is to consider the potential of time varying fast output sampling feedback. It was shown by Chammas and Leondes [2] that a controllable and observable plant was discrete time pole assignable by periodically time-varying piecewise constant output feedback. Since the feedback gains are piecewise constant, their method could be easily implemented and indicated a new possibility. Such a control law can stabilize a much larger class of systems than the static output feedback [6]-[12]. In decentralized power system stabilizer, the control input for each machine should be function of the output of that machine only [2]. This can be achieved by designing a decentralized PSS using fast output sampling feedback technique in which the gain matrix should have all oơdiagonal terms zero or very small compare to diagonal terms. In decentralized PSS, to activate the proposed controller at the same instant, proper synchronization signal is required to be sent to all machines. Thus, the decentralized stabilizer design problem can be translated into a problem of diagonal gain matrix design for multi machine power system [13]. For a large power system, the order of the state matrix may be quite large. It would be diƥcult to work with these complex systems in their original form [2]. One of the ways to overcome this diƥculty is to develop a reduced order model for a large power system. Then a state feedback gain can be computed from the reduced model of the power system. This paper proposes the design of a power system stabilizer for multi-machine system using fast output sampling feedback via reduced order model.
Abstract-- The Power System Stabilizer (PSS) is added to excitation system to enhance the damping during low frequency oscillations. In this paper, the design of decentralized PSSs for 10 machines with 39 buses using fast output sampling method via reduced order model is proposed. In multi-machine power system the order of the states matrix is very large. The main objectives of order reduction is to design a controller of lower order which can effectively control the original high order system so that the overall system is of lower order and easy to understand. The state space matrices of the reduced order system are chosen such that the dominant eigenvalues of the full order system are unchanged. The other system parameters are chosen using the particle swarm optimization with objective function to minimize the mean squared errors between the outputs of the full order system and the outputs of the reduced order model when the inputs are unit step. Design of fast output sampling controllers via reduced order model using Particle Swarm Optimization (PSO) method is proposed for good damping enhancement for various operating points of multi-machine power systems. In fast output sampling technique, the nonlinear model of 10 machine and 39 bus system is linearized at different operating point and a linear model is obtained. A robust fast output sampling feedback gain which realizes output injection gain is obtained using LMI approach. This robust fast output sampling control is applied to non-linear model of a Multi-machine system at different operating points. This method gives very good results in the design of Power System Stabilizers and takes less computation time in operation of power system. Keywords-- Decentralized control, fast output sampling feedback, multi-machine system, nonlinear simulation, LMI, power system stabilizer, reduced order model , PSO.
I. INTRODUCTION Power system stabilizers were developed to aid in damping these oscillations via modulation of the generator excitation. This development has brought an improvement in the use of various tuning techniques and input signals and in the ability to deal with turbine-generator-shaft torsional modes of vibrations [1]. To provide damping, the stabilizers must produce a component of electrical torque on the rotor which is in phase 978-0-7695-4850-0/12 $26.00 © 2012 IEEE DOI 10.1109/CICN.2012.91
617
II. POWER SYSTEM STABILIZER
3.2 STATE SPACE MODEL OF 10 MACHINE AND 39 BUS POWER SYSTEMS The state space model of a 10-machine 39 bus system as shown in Fig. 3.1 can be obtained using machine data, line data and load flow data as given in [10] as ⋅
x = [ A]x + [ B](ΔV ref + ΔVs ),
Figure 2.1: Block-diagram of Convential Power System Stabilizer
y = [C ]x,
Implementation of a power system stabilizer implies adjustment of its frequency characteristic and gain to produce the desired damping of the system oscillations in the frequency range of 0.2 to 3.0 Hz. Convential Power System Stabilizer as shown in figure 2.1, where Kpss represents stabilizer gain and the stabilizer frequency characteristic is adjusted by varying the time constant Tw, T1, T2, T3 and T4. A power system stabilizer can be made more eơective if it is designed and applied with the knowledge of associated power characteristics. Power system stabilizer must provide adequate damping for the range of frequencies of the power system oscillation modes. Designed stabilizer must ensure for the robust performance and satisfactory operation with an external system reactance ranging from 20% to 80% on the unit rating [2, 15].
(4) (5)
Where x = [ x1 , x2 ,..., x10 ]T , & y = [ y1 , y2 ,...., y10 ]T .
(6)
IV. FAST OUTPUT SAMPLING METHOD In this technique an output feedback law is used to realize a discrete state feedback gain by multirate observations of the output signal. The control signal is held constant during each sampling interval τ .
III. MULTIMACHINE POWER SYSTEM ANALYSIS Analysis of practical power system involves the simultaneous solution of equations consisting of synchronous machines and the associated excitation system and prime movers, interconnecting transmission network, static and dynamic load (motor loads), and other devices such as HVDC converters, static var compensators. The dynamics of the machine rotor circuits, excitation systems, prime mover and other devices are represented by differential equations. The result is that the complete system model consists of large number of ordinary differential and algebraic equations. Model 1.0 is assumed for synchronous machines by neglecting the damper windings. In addition, the following assumptions are made for simplicity [10]. 1. The loads are represented by constant impedances. 2. Transients saliency is ignored by considering xq=x’d. 3. Mechanical power is assumed to be constant. 4. Efd is single time constant AVR.
Figure 3.1: Single line diagram of 10 machines and 39 bus System [10]
y(t) L0
0
L1
Δ
Δ= τ Ν
L2
2Δ
t
3.1 GENERATOR EQUATIONS
The machine equations (for kth machine) are 1 ª − E 'qk + ( xdk − x 'dk )idk + E fdk º¼ , pE 'qk = T 'd 0 k ¬ pδ k = wB ( S mk − S mk 0 ), pS mk
u(t)
κτ−τ
(1)
κτ
κτ+τ
t
(2) Figure 4.1: Fast Output Sampling Method [2, 6]
1 = [ − Dk ( Smk − Smk 0 ) + Pmk − Pek ] (3) 2H
Let ( Φ,Γ, C ) be the system [6] sampled at rate
1
Δ
where Δ = τ N . Let υ denote the observability index of ( Φ, C ). N is chosen to be greater than or equal to υ . Output
618
measurements
are
taken
at
time
V. MODEL ORDER REDUCTION USING PSO TECHNIQUE
instants
t = lΔ, l = 0,1,......., N − 1 . The control signal u (t), which is
applied during the interval, kτ ≤ t E (k + 1)τ is then constructed as a linear combination of the last N output observations [6-8]. Consider a discrete -time system having at time t = kτ , the fast output samples
yk = [ y (kτ − τ )
PSO is an evolutionary computation technique developed by Eberhart and Kennedy [14] in 1995, which was inspired by the Social behavior of Bird flocking and fish schooling. Particle Swarm Optimization has more advantages over Genetic Algorithm as follows: PSO is easier to implement and has fewer parameters to adjust. Every particle in PSO remembers its own previous best value as well as the neighborhood best. PSO utilizes a population of particles that fly through the problem space with given velocities. Each particle has a memory and it is capable of remembering the best position in the search space ever visited by it. The Positions corresponding to the Best fitness is called Pbest (also called local best) and the global best out of all the particles in the population is called gbest. At each iteration, the velocities of the individual particles are updated according to the best position for the particle itself and the neighborhood best position [12].
y (kτ − τ + Δ) .. .. y (kτ − Δ)] (7)
Then a representation for this system is x(k + 1) = Φτ xk + τ uk ,
y (k + 1) = C0 xk + D0uk ,
(8.a) (8.b)
Where C0 and D0 are defined as
ªC º « » «C Φ » », C0 = «. « » «. » «C Φ N −1 » ¬ ¼
ª0 º «C Γ » « » D0 = «. » « » N −2 j «¬.C ¦ j =0 Φ Γ »¼
When in an iteration the value of gbest and pbest are calculated. Particle’s position and velocity are updated as follow for the next iteration: k k vidk +1 = vidk + c1× r1× ( pidk − xidk ) + c 2 × r 2 × ( p gd − xgd )
(15)
Let F be an initial state feedback gains such that the closed loop system matrix (Φτ + Γτ F ) has no eigenvalues at the origin. Then one can define a fictitious measurement matrix,
k +1 id
x
(10) (11)
To balance the global and local search the parameter ‘w’, called inertia weight, is introduced.
(9)
, which satisfies the fictitious measurement equation ~
y (k ) = C x(k ) The control law is of form
uk = Lyk
For the output feedback gain L to realize the effect of F it must satisfy
This parameter can be constant in the range [0.9, 1.2] or be as follows: w = w f + (( w f − wi )(max it − it ) / it ) (17)
xk +1 = (Φτ + Γτ F ) xk = (Φτ + Γτ LC ) xk that is LC = F
(12) To reduce this effect we relax the condition that L exactly satisfy the above linear equation and include a constraint on the L
where ‘it’ is the number of iteration, ‘max it’ is the defined number of maximum iterations. In that case, equation (15) is modified as below: k k vidk +1 = w.vidk + c1.r1.( pidk − xidk ) + c 2.r 2.( pgd − xgd )
L < ρ1 LD0 − F Γτ < ρ 2
(18) In this work, the fitness function is defined by the meansquared error
(13)
LC − F < ρ3
MSE =
LMI Formulation of above equations is ª− ρ1 2 I L º « T »
