-5=V2Y2V1+Y2V2+Y2V3+Y2V Sp3=V Y,,+Y32V2 SG4-Sp4=VYV,+YV2+Y,V,+Y,v. (63.11) Examination of Eqs.(63.8)through(63 11)reveals that, except for the trivial case where the generation equals the load at every bus, the complex power outputs of the generators cannot be arbitrarily selected. In fact, the complex power output of at least one of the generators must be calculated last because it must take up the unknown"slack"due to the, as yet, uncalculated network losses. Further, losses cannot be calculated until the voltages are known. These observations are a result of the principle of conservation of complex power (i.e, the sum of the injected complex powers at the four system buses is equal to the system complex power losses Further examination of Eqs. (63.8)through(63. 11)indicates that it is not possible to solve these equations or the absolute phase angles of the phasor voltages. This simply means that the problem can only be solved to some arbitrary phase angle reference. In order to alleviate the dilemma outlined above, suppose Sga is arbitrarily allowed to float or swing(in order to take up the necessary slack caused by the losses)and that SGl Sgz, and Sa are specified (other cases will be considered shortly). Now, with the loads known, Eqs.(63.7)through(63. 10)are seen as four simulta neous nonlinear equations with complex coefficients in five unknowns Vi, V2, V3, va, and The problem of too many unknowns(which would result in an infinite number of solutions) is solved by ecifying another variable Designating bus 4 as the slack bus and specifying the voltage V, reduces the proble to four equations in four unknowns. The slack bus is chosen as the phase reference for all phasor calculations, its magnitude is constrained, and the complex power generation at this bus is free to take up the slack necessary in order to account for the system real and reactive power losses. The specification of the voltage V, decouples Eq (63. 11)from Eqs.(63.8)through(63. 10), allowing calcu lation of the slack bus complex power after solving the remaining equations. (This property carries over larger systems with any number of buses. )The example problem is reduced to solving only three equations lultaneously for the unknowns VI, V2, and V3. Similarly, for the case of n buses, it is necessary to solve n-1 simultaneous, complex coefficient, nonlinear equations. Systems of nonlinear equations, such as Eqs. (63. 8)through(63. 10), cannot(except in rare cases)be solved by closed-form techniques. Direct simulation was used extensively for many years; however, essentially all power flow analyses today are performed using iterative techniques on digital computers P-V Buses In all realistic cases, the voltage magnitude is specified at generator buses to take advantage of the generator's reactive power capability. Specifying the voltage magnitude at a generator bus requires a variable specified in the simple analysis discussed earlier to become an unknown (in order to bring the number of unknowns back into correspondence with the number of equations). Normally, the reactive power injected by the generator becomes a variable, leaving the real power and voltage magnitude as the specified quantities at the generator bus It was noted earlier that Eq (63. 11) is decoupled and only Eqs. (63.8)through(63.10)need be solved simultaneously. Although not immediately apparent, specifying the voltage magnitude at a bus and treating the bus reactive power injection as a variable results in retention of, effectively, the same number of complex unknowns. For example, if the voltage magnitude of bus I of the earlier four bus system is specified and the reactive power injection at bus 1 becomes a variable, Eqs.(63. 8)through(63. 10)again effectively have three complex unknowns. (The phasor voltages V2 and V3 at buses 2 and 3 are two complex unknowns and the angle 8, of the voltage at bus 1 plus the reactive power generation QGi at bus 1 result in the equivalent of a hird complex unknown) e 2000 by CRC Press LLC© 2000 by CRC Press LLC (63.9) (63.10) (63.11) Examination of Eqs. (63.8) through (63.11) reveals that, except for the trivial case where the generation equals the load at every bus, the complex power outputs of the generators cannot be arbitrarily selected. In fact, the complex power output of at least one of the generators must be calculated last because it must take up the unknown “slack” due to the, as yet, uncalculated network losses. Further, losses cannot be calculated until the voltages are known. These observations are a result of the principle of conservation of complex power (i.e., the sum of the injected complex powers at the four system buses is equal to the system complex power losses). Further examination of Eqs. (63.8) through (63.11) indicates that it is not possible to solve these equations for the absolute phase angles of the phasor voltages. This simply means that the problem can only be solved to some arbitrary phase angle reference. In order to alleviate the dilemma outlined above, suppose – SG4 is arbitrarily allowed to float or swing (in order to take up the necessary slack caused by the losses) and that – SG1, – SG2, and – SG3 are specified (other cases will be considered shortly). Now, with the loads known, Eqs. (63.7) through (63.10) are seen as four simulta￾neous nonlinear equations with complex coefficients in five unknowns – V1, – V2, – V3, – V4, and – SG4 . The problem of too many unknowns (which would result in an infinite number of solutions) is solved by specifying another variable. Designating bus 4 as the slack bus and specifying the voltage – V4 reduces the problem to four equations in four unknowns. The slack bus is chosen as the phase reference for all phasor calculations, its magnitude is constrained, and the complex power generation at this bus is free to take up the slack necessary in order to account for the system real and reactive power losses. The specification of the voltage – V4 decouples Eq. (63.11) from Eqs. (63.8) through (63.10), allowing calcu￾lation of the slack bus complex power after solving the remaining equations. (This property carries over to larger systems with any number of buses.) The example problem is reduced to solving only three equations simultaneously for the unknowns – V1, – V2 , and – V3 . Similarly, for the case of n buses, it is necessary to solve n-1 simultaneous, complex coefficient, nonlinear equations. Systems of nonlinear equations, such as Eqs. (63.8) through (63.10), cannot (except in rare cases) be solved by closed-form techniques. Direct simulation was used extensively for many years; however, essentially all power flow analyses today are performed using iterative techniques on digital computers. P-V Buses In all realistic cases, the voltage magnitude is specified at generator buses to take advantage of the generator’s reactive power capability. Specifying the voltage magnitude at a generator bus requires a variable specified in the simple analysis discussed earlier to become an unknown (in order to bring the number of unknowns back into correspondence with the number of equations). Normally, the reactive power injected by the generator becomes a variable, leaving the real power and voltage magnitude as the specified quantities at the generator bus. It was noted earlier that Eq. (63.11) is decoupled and only Eqs. (63.8) through (63.10) need be solved simultaneously. Although not immediately apparent, specifying the voltage magnitude at a bus and treating the bus reactive power injection as a variable results in retention of, effectively, the same number of complex unknowns. For example, if the voltage magnitude of bus 1 of the earlier four bus system is specified and the reactive power injection at bus 1 becomes a variable, Eqs. (63.8) through (63.10) again effectively have three complex unknowns. (The phasor voltages – V2 and – V3 at buses 2 and 3 are two complex unknowns and the angle δ1 of the voltage at bus 1 plus the reactive power generation QG1 at bus 1 result in the equivalent of a third complex unknown.) SG2 – S V YV YV YV YV * D2 * 2 * 21 1 22 2 23 3 24 4 = +++ [ ] SG3 – S V YV YV YV YV * D3 * 3 * 31 1 32 2 33 3 34 4 = +++ [ ] SG4 – S V YV YV YV YV * D4 * 4 * 41 1 42 2 43 3 44 4 = +++ [ ]