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2006 International Conference on Power System Technology
Power System Transient Stability Improvement Using Fuzzy Controlled STATCOM M. R. Zolghadri, A. Ghafori and M. Ehsan
Abstract-- In this paper a fuzzy logic based controller for STATCOM is used to improve power system transient stability. As opposed to the modern control theory, fuzzy logic design is not based on the mathematical model of the process. The controller designed using fuzzy logic implements human reasoning that can be programmed into fuzzy logic language (membership functions, rules and the rule interpretation). The nonlinear fuzzy controller is proposed to supply a supplementary control signal to STATCOM to increase the critical clearing time and overcome the uncertainties existing in the power systems. Proposed method is implemented in a single machine infinite bus system and the results are compared with conventional energy function based controllers. Index Terms— STATCOM, Fuzzy Logic Controller, Critical Clearing Time, Energy Function
T
I. INTRODUCTION
HE nonlinear behavior of power transfer in power system is the origin of using Lyapunov stability criteria for transient stability analysis. Introducing a lyapunov energy function including sufficient details of power system elements is usually a very difficult task. This is why engineers use energy-like functions to calculate critical clearing time and analyze the transient stability. Aylett introduced an energy-like lyapunov function for the first time in 1958 [1]. This function has been used to calculate transient stability analysis by many researchers and has been modified for several times. Athey et al. calculated this function using the concept of Center Of Angle (COA) formulation [2]. To calculate this function, system loads are considered as constant impedances and network is reduced to generator buses and the transfer conductance is neglected. It means that the network is assumed lossless. The major problem is that conductance play an important role in stability analysis. Considering this effect, we can not prove that energy-like function is a lyapunov function. Introducing extended invariance principle by Rodriguez et al. [3], it has been possible to prove the lossy power system transient stability by a sound theory. Nowadays, power electronic devices are widely used in industry. Application of these devices in power systems, This work was supported in part by the Sharif University of Technology. Ali Reza Ghafoori is preparing his MS in Electrical Engineering in the school of Electrical Engineering, Sharif University of Technology, Tehran Iran(
[email protected]). MohammadReza. Zolghadri is with the school of Electrical Engineering, Sharif University of Technology, Tehran Iran(
[email protected]). Mehdi Ehsan is with the school of Electrical Engineering, Sharif University of Technology, Tehran Iran(
[email protected]).
1-4244-0111-9/06/$20.00©2006 IEEE.
leads to better performance of system in many aspects. Voltage regulation, voltage stability and power system stability can be improved by using FACTS devices and their proper control. STATCOM is one of the parallel FACTS devices that is usually used for voltage regulation . It can also be used to improve power system stability by injection of reactive power to the network [4]. Its performance is analyzed by different methods. In the simplest form, transient stability is analyzed by equal area method but when system grows bigger, this method will not lead to good responses. Energy function or energy-like lyapunov function method can be used to analyze system stability and design controllers to improve system performance in existence of FACTS devices . Power systems are large-scale systems and there exists many uncertainties in calculating their parameters. Fuzzy logic provides powerful tools to consider these uncertainties and their modeling. All of previous methods are almost model based, it means that we should identify system dynamics [5]. As we know, this is a difficult task, especially when we want to model exactly the system. Usually fuzzy logic controllers are designed base on system behavior and there is no need to obtain exact model of the system. Fuzzy logic control is an appealing alternative to conventional control methods when system follow some general operating characteristics and a detailed process understanding is unavailable or traditional system model becomes overly complex. In this paper a fuzzy logic controller is proposed to apply a suitable control signal to the STATCOM to improve power system transient stability. In part II, STATCOM model for transient stability analysis is expressed and the effect of line resistance on power curve is described. In section III, energy function method is analyzed with considering line resistance in existence of STATCOM. Then a fuzzy logic controller is designed to control STATCOM. Finally, in section IV, both methods are applied to single machine infinite bus (SMIB) system and simulation results are expressed. II. MODELING STATCOM can be represented by a controlled shunt current source as shown in Fig.1. The STATCOM current is always in quadrature with its terminal voltage and can be written as: j (δ ±90) I STATCOM = I STATCOM e k (1) Positive and negative signs are for inductive and capacitive modes respectively. In capacitive mode the voltage magnitude and angle of bus k can be expressed by
2
[6]: Vk =
E ′X 2 cos(δ − δ k ) + VX1 cos δ k + X1X 2 I STATCOM X1 + X 2
δ k = tan −1(
E ' X 2 sin δ
VX1 + E ' X 2 cos δ
(2) (3)
)
The output power of the machine can be written as: E 'Vk sin(δ − δ k ) Pc = X1 E1∠δ
jX 1
V k ∠δ k
jX 2
(4)
E∠ 0
Fig. 3: Effect of line resistance on machine terminal power
I STATCOM
Fig. 1: Single machine infinite bus system with STATCOM Power
Pe post ( I STATCOM > 0) Pe post ( I STATCOM = 0) Pe post ( I STATCOM < 0)
Fig. 4: Effect of line resistance on the infinite bus terminal power curve Pe
Angle J
Fig. 2: The effect of STATCOM on electrical power curves
The maximum transffered power of generator is increased when STATCOM is in the capacitive mode and is decreased when it is in the inductive mode. Fig.2 shows the effect of STATCOM on the output power curve of generator for SMIB system graphically. Suppose that the transfer conductance is considered in SMIB system of Fig. 1 without STATCOM in the middle. Output active power of machine can be written as: Pe =
uE12 + EE1 (−u cos δ + sin δ ) (u 2 + 1) X T
(5)
Where XT=X1+X2 , u=RT/XT , RT=R1+R2 Effect of line resistance on electrical output power of machine can be shown in Fig. 3. As it is shown in Fig. 3 when u increases, first the electrical power increases and then it decreases. The power delivered at infinite bus terminal can be derived as: Pe =
− uE12 + EE1 (u cos δ + sin δ ) (u 2 + 1) X T
(6)
Effect of line resistance on electrical power delivered at infinite bus is shown in Fig. 4. The effect of line resistance on transient stability can be analyzed according to Fig. 5.
F
Pm2 Pm1
A
K
E I
B
H
C
D
G
c
L
M
N
a
b
O
delta
delta1 delta2
a: Electrical Power when u=0 b:Machine Terminal Power u>0 c:Infinite Bus Terminal Power u>0
Fig. 5: Effect of line resistance on transient stability
As we know, generator wants to deliver a constant active power to the infinite bus in the both lossless and lossy networks. According to Fig. 5, with constant power Pm1, the machine angle should increase from delta1 to delta2 so the input mechanical power should increase to Pm2. This results in steady state increase in machine angle and input mechanical power. When a fault occurs on the machine terminal bus, the output power decreases to zero until fault removal. In the lossy network, fault duration area increases from ACNL to FEOM. For transient stability criteria analysis (critical clearing time or stability margin), both fault duration and post fault duration areas means ACNL and CIG for lossless network and BDOM and EJK for lossy network should be considered. Increasing area ACNL to
3
FEOM decreases stability margin but overall stability margin depends on difference between area HGD and JKE too. If area JKE is less than or equal to HGD, stability margin will decrease otherwise it depends on how much area JKE is bigger than HGD. It can be seen that usually increasing u, decreases stability margin. The problem will be more serious when u is near one or significantly large. III. CONTROL STRATEGY In this section, first the Energy Function Based Controller (EFBC) is analyzed in the lossy network and then Mamdani fuzzy logic controller (FLC) and its design is expressed. These controllers are implemented to control a STATCOM in the power system. Controller design and adjustment is expressed for FLC. A. Energy Function in the lossy network In this part the energy function of system is obtained in existence of transfer conductance. In fig. 1 an STATCOM is located in the middle of the line in a single machine infinite bus system. As mentioned above, it can be proved that in a lossless network a proper control signal for STATCOM can make the derivative of transient energy function of system negative, so it improves the stability of system [7]. It is proved in [8] that the same transient energy function can be used to study the stability of a lossy system by extended invariance principle. Extended invariance principle dose not require the derivative of Lyapunov function, V, to be negative definite or semi negative definite. In this case V is called extended lyapunov function. (for more details about Extended Invariance Principle see [8 ,9]) It can be proved that the transient energy function used for the system containing STATCOM could be an extended lyapunov function too. The system in Fig. 1 can be expressed in the form of Fig. 6. Vk ∠δ k
( E1∠δ − θ1 ) y1
( E∞ ∠ − θ 2 ) y 2
I STATCOM
y1 + y2
After some mathematical calculations, Vk and δk can be obtained as: Vk =
E1 y1 sin(δ + θ1 − δ k ) + Ey 2 sin(θ 2 − δ k ) I + y1 + y 2 y1 + y 2
δ k = tan −1 − tan
−1
E1 y1 sin(δ + θ1 ) + I cos(δ k ) + E∞ y 2 cos θ 2 − E1 y1 cos(δ + θ1 ) − I sin(δ k ) + E∞ y 2 sin θ 2
Dynamic equations of the above system are: ⎧δ& = ω ⎨ ⎩Mω& = Pl − Tω Pl = P − C1k sin(δ − δ k ) − D1k cos(δ − δ k ) P = Pm − g1 E 2 Where the parameters are: δ: machine rotor angle M: inertia constant Pm: mechanical input power T: damping coefficient C1k=EVkb1 D1k=EVkg1 y1=g1+jb1=y1
