Identification of Coherent Generators Using Fuzzy C-Means ...

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highly effective in determining the coherent groups of generators and in constructing thedynamic equivalent of power system with high accuracy.3. Mathematical ModelThe classical model is the simplest model used in studies of power system dynamics andrequires a minimum amount of data. This model is based on the following assumptions.1. Mechanical power input is constant.2. Damping or asynchronous power is negligible.3. Constant-voltage-behind-transient-reactance model for the synchronous machinesis valid.4. The mechanical rotor angle of a machine coincides with the angle of the voltagebehind the transient reactance.5. Loads are presented by passive impedances.The swing equation for machine number i can be described in a linearized form as follows:d ΔωiMi=ΔPmi −ΔPei − Diωi, i = 1,2,..., n (1)dtd Δδidt=Δ ω , i = 1,2,..., n(2)iThe power into the network at node i, which is the electrical power output of machine i , isgiven as follows:P = Re( E I )(3)i i in2ei=i ii+ ∑ i j ijcos( θij − δi + δj), = 1,2,...,j = 1j≠in2i ii i j ijδi δj ijδi δjj = 1j≠iP E G E E Y i n∑ (4)= E G + E E [ B sin( − ) + G cos( − )], i = 1,2,..., nThe set of equations (1) and (2) are sets of n-coupled nonlinear second-order differentialequations. It can be rearranged in state space model for n generators as follows:4
⎡ 1⎤⎢⎥⎡Δδ ⎤1⎡ 1⎤⎡Δδ1 ⎤ ⎢⎥⎢ ⎥ ⎢ ⎥⎢δ⎥Δ ⎢ ⎥⎢ δ2⎥ ⎢ ⎥⎢Δ2⎥ ⎢⎥1⎢ ⎥ ⎢ 0 ⎥⎢ ⎥ ⎢⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥Δ ⎢ δ1 δn ⎥ ⎢ ⎥⎢Δn ⎥ ⎢⎥⎢ ⎥ ⎢ ⎥⎢ ⎥1 = + ⎢ ⎥[ Δ Pm 1 Δ Pm2 Δ Pn⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢]M⎥1⎢Δ ω K1⎥ ⎢ 11 K1 n D1 ⎥⎢Δω1 ⎥ ⎢⎥⎢Δ ω⎥ ⎢K 2 21 K2 n D ⎥⎢2Δω⎥ ⎢ 1 ⎥2⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ M ⎥2⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢⎥⎢ ⎥Δω⎢Kn n1Knn D⎥⎢nΔω⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ n⎦⎢1⎥⎢⎥⎢⎣Mn⎥⎦(5.a) (5.b)Or X = AX + BUTWhere X = [ Δδ1, Δδ2,..., Δδn, Δω1, Δω2,..., Δ ωn] , U = [ ΔPm1, ΔPm2,..., Δ Pmn]and the diagonal and off-diagonal elements K ii , K ij are as follows:1 ∂ΔPeiKij= − i ≠ jM ∂ΔδKiiij1=− [ EEYi j ij sin( δi −δ j− θij)] (6)Mi1 ∂ΔPei=−M ∂Δδii∑ i j ik i k ik(7)k = 1k ≠in1=− [ [ EEY sin( δ −δ −θ)]]MiA static reduction is performed on the original power system in order to obtain thegenerator nodes only. This reduction is obtained by the following expression.red1Y ( YmmYmrY −rrYrm)= − (8)Where subscript m denotes the generating bus and r denotes the load bus. The busredadmittance matrix used in eqn. (6) and eqn. (7) is Y as model is used after staticreduction of power system. It is important view to represent the synchronous machine inhigh order state space model taking in consideration the modeling of excitation system andthe modeling of speed-governor system. The excitation system that is used is IEEE Type -1system which is shown in Fig. (2).Fig. (2) IEEE Type one excitation system5
Page 1 and 2: Mahdi M. M. El-ariniAhmed FathyJ. E
Page 3: Fig. (1) Partition of the studied P
Page 7 and 8: 4. Proposed TechniqueIn this sectio
Page 9 and 10: Input the coherencymeasure C ij tol
Page 11 and 12: Rankingof S 1Table (1) the ascendin
Page 13 and 14: parameters of the equivalent machin
Page 15 and 16: Table (8) The effect of Changing th
Page 17 and 18: The second technique is changing th

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