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926 Samuel N. Ndubisi and Marcel .U. Agu STEP 1: Determination of the degree of firing (DOF) of the rules The (DOF) of the ith rule consequent is a scalar value which equals the minimum of the two antecedent membership degrees. Table 1: Decision table of 121 rules Integral of voltage error VI Voltage error Ve NV NL NB NM NS ZR PS PM PB PL PV NV NV NV NV NV NV NV NL NB NM NS ZR NL NV NV NV NV NV NL NB NM NS ZR PS NB NV NV NV NV NL NB NM NS ZR PS PM NM NV NV NV NL NB NM NS ZR PS PM PB NS NV NV NL NB NM NS ZR PS PM PB PL ZR NV NL NB NM NS ZR PS PM PB PL PV PS NL NB NM NS ZR PS PM PB PL PV PV PM NB NM NS ZR PS PM PB PL PV PV PV PB NM NS ZR PS PM PB PL PV PV PV PV PL NS ZR PS PM PB PL PV PV PV PV PV PV ZR PS PM PB PL PV PV PV PV PV PV For instance, if Ve is PM with a membership degree of 0.4 and VI is PS with a membership degree of 0.2 then the degree of firing of this rule is 0.2. STEP 2: Inference mechanism The inference mechanism consists of two processes called fuzzy implication and rule aggregation. The degree of firing of rule interacts with its consequent to provide the output of the rule, which is a fuzzy subset. The formulation used to determine how the DOF and the consequent fuzzy set interact to form the rule output is called a fuzzy implication. In fuzzy logic control the most commonly used method for inferring the rule output is mamdani method [9, 10, 11]. STEP 3: Defuzzification To obtain a crisp output value from the fuzzy set obtained in the previous step, a mechanism called defuzzification is used. In this work output u is defuzzified according to the membership functions shown in fig 2. Centre of area method [7] is used for defuzzification. Figure 2: Membership functions for output u Tuning: To achieve a fast response from the controller during faults, the FLC can be tunned with the parameters g0, g1 and g2 as shown in fig 4. 3. A Power System Dynamic Model In this paper, simplified dynamic model of a single-machine infinite bus power system is considered [12]. This model consists of a single synchronous machine connected through a parallel transmission
A Rule-Based Fuzzy Automatic Voltage Regulator for Power System Stability 927 line to a very large network approximated by an infinite bus. The classical third order single–axis dynamic model of the simple machine infinite bus power system shown in fig 3 has the equations as follows [12,13] 3.1. Mechanical equations () t = ω() -ω 0 � δ t (1) ω� () t = D ω0 ( ω() t -ω0 ) − ( Pe ( t) − Pm ) H 2H (2) The mechanical input power Pm is assumed to be constant in the excitation controller design. This implies that the governor action is slow enough not to have any significant impact on the machine dynamics. 3.2. Electrical generator dynamics ′ 1 T ′ ( ) () t = E () t − E () t Eq f q d 0 (3) 3.3. Electrical equations (assumed X ′ d = X q ) q () t = E′ q () t + ( X d − X ′ d ) I () t ′ f () t = K cU () t E′ d () () t Vs sinδ () t t = E d E f P I I e d q Q () t () t () t E′ q = X ′ ds −V s cosδ t X ′ () t () Vs sinδ = X ′ E′ q = ds ds () t () t V cosδ () t s X ′ ds () t = X I () t ′ E ad f () t = E′ () t − X ′ I () t 2 s V − X ′ [ ( ) ( () ) ] 2 1 2 2 + X I t Vt q d d d q ds (11) More details of the modeling can be seen in [14]. The definitions of the parameters are as indicated under symbols and abbreviations. X = X + X + X (12) ds ds d d T T L L X ′ = X ′ + X + X (13) X = X + X s T L (4) (5) (6) (7) (8) (9) (10)
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