PSPowerFlowAnalysis Dr. M.W.Renton Page 1 of 8 05/02/16 Power Systems (Analysis) Power-flow Analysis Power flow studies provide the essential information required to ensure efficient operation of a power system and are invaluable in planning and designing any changes to power system layout. Information obtained from a power flow study includes the magnitude (V) and phase angle (of the voltage at each bus and thereal power (P) and reactive power (Q) in each line. Statement of the power flow problem Although power flow studies can be based on either [Zbus] or [Ybus] here attention is confined to the admittance form of the power flow equations. The elements in [Ybus] are in general complex and of the form: Yij  Yij /ij ( Gij  jBij ) At a representative bus i, the bus voltage may be written in complex form as: Vi  Vi / i = Vi (cos i  j sin i ) From the general admittance equations of an N-bus system, the current injected into the network at bus i is: I Y V Y V Y V Y V i i i iN N in n  1 1  2 2   + =  n =1 N The net real and reactive powers injected into the network at bus i are defined as Pi and Qi, thus, in terms of the bus voltage and injected current at bus i : P i jQi Vi Ii Vi YinV n   *  *  n =1 N (see Appendix A on page 8) (Eqn 1) Expressing V and Y in polar form: P i jQi Vi i Yin V n     / in n / n =1 N /        Y VV in i n n in n i 1 N /            YinViVn j Y V V n in n i in i n n in n i 1 1 N N cos(   ) sin(   )PSPowerFlowAnalysis Dr. M.W.Renton Page 2 of 8 05/02/16 Power Systems (Analysis) Equating the real and reactive (imaginary) parts: P Y V V i in i n n  in  n  i N1 cos(   ) (Eqn 2) Qi YinViVn n   in  n  i N1 sin(   ) (Eqn 3) Equations 2 and 3 are the power flow equations representing respectively the real and reactive powers entering the network at bus i. The complexity of the power flow problem is illustrated by these equations, as they show that the power at a given bus i is a function of the voltage magnitude and phase angle at all the buses in the system. The real and reactive power entering the network at a bus can be related to the generated and demanded (load) powers at the bus. Figure 1 : Power notation at bus i Referring to Figure 1, if Pgi and Qgi denote the real and reactive power generated at bus i while Pdi and Qdi denote the real and reactive power demanded by the load then the power injected into the network is the difference between the generated and demanded powers. Thus: Pi = Pgi - Pdi and Qi = Qgi - Qdi Rewriting the power flow equations at bus i they become: cos( ) N 1 in n i n Pgi  Pdi  YinViVn     (Eqn 4) sin( ) N 1 in n i n Qgi  Qdi  YinViVn     (Eqn 5) The power flow problem is to solve equations 4 and 5 for the unknown bus voltages which will give the required real and reactive power at every bus. G i Pi , Qi Pgi , Qgi Pdi , QdiPSPowerFlowAnalysis Dr. M.W.Renton Page 3 of 8 05/02/16 Power Systems (Analysis) Bus Types Equations 4 and 5 show that at each bus there are 6 variables: Pg, Qg, Pd, Qd, V and In any power flow study the load demand at each bus is known, i.e. it is specified as input data, leaving potentially four unknowns at each bus: Pg, Qg, V and Since at each bus there are two equations only which cannot be solved for 4 unknowns; therefore at each bus there is a requirement to specify a further two variables leaving only two unknowns. As far as the mathematics is concerned, any two of the four variables may be specified; however considering the physical network and the variables over which there is physical control, the accepted practice in power flow studies is to identify three types of buses. For each bus type the two specified and two unknown variables are summarised in Table 1. Bus Type Specified Unknown Number slack V = 1.0 ,  Pg , Qg 1 generator Pg , V Qg ,  ~ 15% load Pg = 0 , Qg = 0 V ,  ~ 85% Table 1 : Accepted bus types Load bus At each non-generator bus, called a load bus, Pg and Qg are zero. The real and reactive power, Pd and Qd, drawn from the system by the load are known from historical records, load forecasts etc. For this reason a load bus is often referred to as a P-Q bus. The unknown quantities to be determined for a load bus are V and Typically 80- 85% of the buses in a network are load buses. Generator bus At any bus at which a generator is connected, there are two direct-control actions possible. The real power generated Pg and voltage magnitude V can be controlled by adjusting respectively the prime mover input and the generator excitation. Thus at a generator bus Pg and V are specified and Qg and  are the unknowns to be determined. Generator buses are alternatively known as voltage-controlled or P-V buses. Slack bus The branch currents in the network cannot be calculated until the bus voltages are determined. It therefore follows that the I2R transmission losses in the system cannot be pre-specified; they can only be calculated once the power flow equations have been solved for the bus voltages. If the real power at all generator buses is pre-specified such that the real power generated matches exactly the real power demand of the loads, there will be no slack in the real power generation to supply the losses.PSPowerFlowAnalysis Dr. M.W.Renton Page 4 of 8 05/02/16 Power Systems (Analysis) A similar argument applies to the reactive power balance. The amount of reactive power to be generated to satisfy the load demand can be pre-specified, but, the reactive power associated with shunt capacitors and reactors cannot be evaluated. Also, the reactive I2X "losses" in the series reactances of the lines cannot be determined until the currents are known. Again therefore some slack in the reactive power generation is required. To allow the slack necessary to meet the losses, one generator type bus is chosen as the slack bus, at which the real and reactive powers are not designated. Since Pg and Qg are the unknown variables at the slack bus, it follows that V and  must be specified (since a maximum of two unknowns only are allowed at any bus). These are set normally at 1.0 per unit and 0o. Thus the slack bus, alternatively known as the swing or reference bus, is essentially a generator bus with no power constraints so that a wide range of Pg and Qg values are possible. State Variables The bus-voltage magnitudes |V| and angles which are not specified as part of the input data are termed the state variables. Once the state variables have been calculated the state of the system is known and all other quantities, which depend on the state variables, can be calculated. P and Q at the slack bus, Q at each generator bus, branch currents and power losses are all examples of dependent quantities. A summary of the state variables associated with an N-bus system with one slack bus and Ng generator buses is presented in Table 2. Bus type Specified quantities V ,  state variables/bus No. of buses Total state variables slack V ,  0 1 0 generator V , P i.e. Ng Ng load P, Q 2 (i.e. V ,  N-Ng-1 2(N-Ng-1) totals N 2N-Ng-2 Table 2 : Summary of state variables In terms of state variables, the power flow problem can be thought as one of determining all state variables by solving an equal number of power flow equations, i.e. a need to solve 2N-Ng-2 equations for 2N-Ng-2 state variables.PSPowerFlowAnalysis Dr. M.W.Renton Page 5 of 8 05/02/16 Power Systems (Analysis) Solution of the Power Flow Equations Because of the different types of data specified for the various bus types, the power flow equations defined by equations 1 to 5 cannot be solved by any formal algebraic method. It is necessary to employ numerical, iterative techniques. Such methods compute progressively more accurate estimates of the unknowns (the state variables) until results are within the required degree of accuracy. When this is achieved in a finite number of iterations, the solution is said to converge. Two methods used commonly are the Gauss-Seidel method and the Newton-Raphson method. Almost all commercial power flow programs (e.g. ERACS) are based on one (or both) of these methods. This investigation is limited to the Gauss-Seidel method. [The Gauss-Seidel method is computationally less efficient and may take longer to converge than the Newton-Raphson method (particularly when solving large systems) but it is simpler to apply and understand. Illustrating the Gauss-Seidel method is justified on the grounds that it provides an insight into the application of an iterative technique for dealing with the power flow problem and, once one method is understood, transition to the other method is straightforward.] The Gauss-Seidel Method To illustrate this method, consider a simple 4-bus system with bus 1 designated the slack bus. From Eqn 1: P i jQi Vi YinV n   *  n =1 N Since the number of buses N = 4 the equation can be expanded: P i  jQi  Vi* [Yi1V1  Yi2V2  Yi3V3  Yi4V4 ] At the slack bus (bus 1 in this example) V1 and  1 are known, therefore, computation of the state variables begins at bus 2 (i.e. with i = 2). Hence from above: P 2  jQ2  V2* [Y21V1  Y22V2  Y23V3  Y24V4 ] Solving for V2 gives: 1 { * [ 21 1 23 3 24 4 ]} 2 2 2 22 2 Y V Y V Y V V P jQ Y V      Eqn (a) Corresponding equations may be written for each bus. Thus: at bus 3: 1 { * [ 31 1 32 2 34 4 ]} 3 3 3 33 3 Y V Y V Y V V P jQ Y V      Eqn (b)PSPowerFlowAnalysis Dr. M.W.Renton Page 6 of 8 05/02/16 Power Systems (Analysis) at bus 4: 1 { * [ 41 1 42 2 43 3]} 4 4 4 44 4 Y V Y V Y V V P jQ Y V      Eqn (c) Rather than separating the real and reactive parts of each of these equations, corresponding to equations 2 and 3, the Gauss-Seidel method solves for the complex voltages directly from the equations as they are shown. Solution of the above equations for the voltages at buses 2, 3 and 4 is possible only if all other quantities are known. Thus it is assumed that in addition to the admittance terms, the real and reactive powers P and Q at buses 2, 3 and 4 and the voltage at the slack bus V1 / 1 are known. Solution proceeds as follows: (1) make initial estimates V2(0) , V3(0) and V4(0) for the unknown bus voltages. It is usual practice to set these initial estimates at 1 /0 per unit. Such initialisation is known as a flat start, a term descriptive of the uniform voltage profile assumed. (2) from equation (a) use the initial voltage estimates to make a corrected estimate V2 (1) of the voltage at bus 2, i.e. find 1 { (0)* [ 21 1 23 3(0) 24 4(0) ]} 2 2 2 22 (1) 2 Y V Y V Y V V P jQ Y V      Note: all quantities on the right-hand side are either fixed or are initial estimates. (3) use equations (b) and (c) together with the latest estimates of the bus voltages to revise the voltage estimates at buses 3 and 4, i.e find: 1 { (0)* [ 31 1 32 2(1) 34 4(0) ]} 3 3 3 33 (1) 3 Y V Y V Y V V P jQ Y V      and 1 { (0)* [ 41 1 42 2(1) 43 3(1) ]} 4 4 4 44 (1) 4 Y V Y V Y V V P jQ Y V      Note particularly that, as the corrected voltage is found at each bus, rather than waiting until the next iteration, it is used immediately to find the corrected voltage at the next bus. Thus, use the corrected voltage V2(1) rather than V2(0) when estimating V3(1) . Similarly when calculating V4(1) the most recent estimates of V2 and V3 (i.e. V2(1) and V3(1) ) should be used. This completes the first iteration. The entire process is repeated again and again until the change in the voltage at every bus is less than some pre-defined amount (small %).PSPowerFlowAnalysis Dr. M.W.Renton Page 7 of 8 05/02/16 Power Systems (Analysis) Generalising the above procedure, the Gauss-Seidel algorithm can be expressed as: { ( 1)} 1 ( ) 1 1 ( 1)* ( ) 1            j k N i j ij k j i j k ij i i i ii k i Y V Y V V P jQ Y V (Eqn 6) The subscript (k) signifies the number of the iteration and (k-1) the previous iteration. Voltage-controlled (P-V or Generator) buses Although not stated explicitly above, it can be deduced that Eqn 6 applies to load (P-Q) buses only since a solution is possible only if both P and Q. are known. At any voltagecontrolled buses, where the voltage magnitude rather than the reactive power is specified, an additional step is needed. From Eqn 1: P i jQi Vi YinV n   *  n =1 N hence the reactive power: Qi Vi YinV n   Im {  n =1 N * } (Eqn 7) where "Im" denotes "the imaginary part" of the expression. The value of reactive power Qi provided by Eqn 7 is substituted in Eqn 6 to give a new value of Vi at the voltagecontrolled bus. By definition, the voltage magnitude Vi at a voltage-controlled bus is fixed, although the angle  i may vary. Violation of the voltage magnitude must be avoided; thus a correction is applied to the magnitude of Vi obtained from Eqn 6 such that the voltage magnitude is restored to the original, specified value. Defining: Vi = the specified voltage magnitude at bus i; and V i (k) = the modified value of Vi returned by equation 6 during iteration k; then the corrected value is: V V V V i k i i k i , k ( ) ( ) corr  ( ) (Eqn 8) The value V i k ( , ) corr defined by Eqn 8 is the value carried forward to the next stage of the iterative process. The above procedure is applied to every voltage-controlled bus and is repeated for each iteration.PSPowerFlowAnalysis Dr. M.W.Renton Page 8 of 8 05/02/16 Power Systems (Analysis) It is important that the effect of Eqn 8 is appreciated. The modified value returned by Eqn 6 is a complex quantity which can be written: V V i k i k i ( )  ( ) /  (k) Substituting in Eqn 8: V V V V V V V V i k i i k i k i i k i k i k i i k ( , ) ( ) ( ) ( ) ( ) ( ) ( ) / / corr      Thus Eqn 8 restores the magnitude of the bus voltage to the original value Vi while allowing the voltage to adopt the new angle  ( i k) also defined by Eqn 8. Corresponding to Eqn 7, the algorithmic expression for the calculation of the reactive power Qi during iteration k is: Qi k Vi k YijV Y V i j j k ij j i N j ( )   Im (  *[ ( )  (k ]}    { 1)    1 1 1) (Eqn 9) Appendix A - Proof Note that Eqn 1 does not say that S = P - jQ. Apparent power S is not even mentioned. This equation says simply that if P is the real power and Q is the reactive power then: P - jQ = V*I This is entirely consistent with the definition of complex apparent power: S = P + jQ = VI* Proof Letting V = a + jb and I = c + jd then : S = P + jQ = VI* = (a + jb)(c - jd) = (ac + bd) +j(bc-ad) Thus P = ac + bd and Q = bc - ad V*I = (a - jb)(c + jd) = (ac + bd) - j(bc - ad) = P - jQ Thus it is confirmed that equation 1 is a statement that V*I yields P - jQ.