Net work One-line Di FIGURE 63.3 Conceptual one-line diagram of a four-bus power system A conceptual representation of a one-line diagram for a four-bus power system is shown in Fig. 63.3. For generality, a generator and a load are shown connected to each bus. The following notation applies 1= Complex complex power flow into bus I from the generator Complex complex power flow into the load from bus Comparable quantities for the complex power generations and loads are obvious for each of the three other The positive sequence network for the power system represented by the one-line diagram of Fig 63.3 is shown in Fig. 63. 4. The boxes symbolize the combination of generation and load Network texts refer to this network as a five-node network. (The balanced nature of the system allows analysis using only the positive quence network; reducing each three-phase bus to a single node. The reference or ground represents the fifth node. )However, in power systems literature it is usually referred to as a four-bus network or power system. For the network of Fig 63.4, we define the following additional notation: S,=SGi- Spi Net complex power injected at bus 1 I1 Net positive sequence phasor current injected at bus 1 VI= Positive sequence phasor voltage at bus The standard node voltage equations for the network can be written in terms of the quantities at bus 1 (defined above)and comparable quantities at the other buses. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC A conceptual representation of a one-line diagram for a four-bus power system is shown in Fig. 63.3. For generality, a generator and a load are shown connected to each bus. The following notation applies: – SG1 = Complex complex power flow into bus 1 from the generator – SD1 = Complex complex power flow into the load from bus 1 Comparable quantities for the complex power generations and loads are obvious for each of the three other buses. The positive sequence network for the power system represented by the one-line diagram of Fig. 63.3 is shown in Fig. 63.4. The boxes symbolize the combination of generation and load. Network texts refer to this network as a five-node network. (The balanced nature of the system allows analysis using only the positive sequence network; reducing each three-phase bus to a single node. The reference or ground represents the fifth node.) However, in power systems literature it is usually referred to as a four-bus network or power system. For the network of Fig. 63.4, we define the following additional notation: – S1 = – SG1 – – SD1 Net complex power injected at bus 1 – I1 = Net positive sequence phasor current injected at bus 1 – V1 = Positive sequence phasor voltage at bus 1 The standard node voltage equations for the network can be written in terms of the quantities at bus 1 (defined above) and comparable quantities at the other buses. FIGURE 63.3 Conceptual one-line diagram of a four-bus power system
Net work PassIve Component Neutral FIGURE 63.4 Positive sequence network for the system of Fig 63.3 II=YV+Yv2+Yv3+ Yv (63.2) 2=Y21V1+Y2y2+Y23V3+Y24V4 (63.3) YV+Yv+Y. L4=Y4V+Y2v2+Yv,+Y4V4 (63.5) The admittances in Eqs. (63. 2)through(63.5), Yi> are the ijth entries of the bus admittance matrix for the ower system. The unknown voltages could be found using linear algebra if the four currents I.I were known. However, these currents are not known. Rather, something is known about the complex power and voltage at each bus. The complex power injected into bus k of the power system is defined by the relationship between complex power, voltage, and current given by Eq (63.6 S, =VI Therefore (63.7) By substituting this result into the nodal equations and rearranging, the basic power flow equations for the four-bus system are given as Eqs. (63. 8)through(63. 11) SGI-SDI=VY,V,+Y12 V2+Y,,+Y, (63.8) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC (63.2) (63.3) (63.4) (63.5) The admittances in Eqs. (63.2) through (63.5), – Yij, are the ijth entries of the bus admittance matrix for the power system. The unknown voltages could be found using linear algebra if the four currents – I1… – I4 were known. However, these currents are not known. Rather, something is known about the complex power and voltage at each bus. The complex power injected into bus k of the power system is defined by the relationship between complex power, voltage, and current given by Eq. (63.6). (63.6) Therefore, (63.7) By substituting this result into the nodal equations and rearranging, the basic power flow equations for the four-bus system are given as Eqs. (63.8) through (63.11) (63.8) FIGURE 63.4 Positive sequence network for the system of Fig. 63.3. I YV YV YV YV 1 11 1 12 2 13 3 14 4 =+++ I YV YV YV YV 2 21 1 22 2 23 3 24 4 =+++ I YV YV YV YV 3 31 1 32 2 33 3 34 4 =+++ I YV YV YV YV 4 41 1 42 2 43 3 44 4 =+++ S VI k kk * = I S V S S V k k * k * Gk * Dk * k * = = − SG1 – S V YV YV YV YV * D1 * 1 * 11 1 12 2 13 3 14 4 = +++ [ ]
-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 simultaneous 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 calculation 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 = +++ [ ]
Bus 1 is called a voltage controlled bus because it is apparent that the reactive power generation at bus 1 is being used to control the voltage magnitude. This type of bus is also referred to as a p-v bus because of the specified quantities. Typically, all generator buses are treated as voltage controlled buses Bus Classifications There are four quantities of interest associated with each bus: al power, 2. reactive power, Q 3. itude. v At every bus of the system two of these four quantities will be specified and the remaining two will be unknowns. Each of the system buses may be classified in accordance with the two quantities specified. The following classifications are typical Slack bus--The slack bus for the system is a single bus for which the voltage magnitude and angle are specified. The real and reactive power are unknowns. The bus selected as the slack bus must have a source of both real and reactive power, because the injected power at this bus must"swing"to take up the "slack "in the solution the best choice for the slack bus(since, in most power systems, many buses have eal and reactive power sources)requires experience with the particular system under study. The behavior of the solution is often influenced by the bus chosen. ( In the earlier discussion, the last bus was selected as the slack bus for convenience. Load bus(P-Q bus)-A load bus is defined as any bus of the system for which the real and reactive power specified Load buses may contain generators with specified real and reactive power outputs; however, wer as a Voltage controlled bus(P-V bus)Any bus for which the voltage magnitude and the injected real power are specified is classified as a voltage controlled (or P-v) bus. The injected reactive power is a variable(with pecified upper and lower bounds) in the power flow analysis. (A P-v bus must have a variable source of reactive power such as a generator or a capacitor bank Generalized Power Flow Development The more general(n bus)case is developed by extending the results of the simple four-bus example. Consider the case of an n-bus system and the corresponding n+l node positive sequence network. Assume that the buses are numbered such that the slack bus is numbered last. Direct extension of the earlier equations(writing the node voltage equations and making the same substitutions as in the four-bus case) yields the basic power flow The Basic Power Flow Equations(PFE s=B-jQk=VΣYV fork=1,2,3,,n-1 P-jQ=V2Y V (63.13) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Bus 1 is called a voltage controlled bus because it is apparent that the reactive power generation at bus 1 is being used to control the voltage magnitude. This type of bus is also referred to as a P-V bus because of the specified quantities. Typically, all generator buses are treated as voltage controlled buses. Bus Classifications There are four quantities of interest associated with each bus: 1. real power, P 2. reactive power, Q 3. voltage magnitude, V 4. voltage angle, δ At every bus of the system two of these four quantities will be specified and the remaining two will be unknowns. Each of the system buses may be classified in accordance with the two quantities specified. The following classifications are typical: • Slack bus—The slack bus for the system is a single bus for which the voltage magnitude and angle are specified. The real and reactive power are unknowns. The bus selected as the slack bus must have a source of both real and reactive power, because the injected power at this bus must “swing” to take up the “slack” in the solution. The best choice for the slack bus (since, in most power systems, many buses have real and reactive power sources) requires experience with the particular system under study. The behavior of the solution is often influenced by the bus chosen. (In the earlier discussion, the last bus was selected as the slack bus for convenience.) • Load bus (P-Q bus)—A load bus is defined as any bus of the system for which the real and reactive power are specified. Load buses may contain generators with specified real and reactive power outputs; however, it is often convenient to designate any bus with specified injected complex power as a load bus. • Voltage controlled bus (P-V bus)—Any bus for which the voltage magnitude and the injected real power are specified is classified as a voltage controlled (or P-V) bus. The injected reactive power is a variable (with specified upper and lower bounds) in the power flow analysis. (A P-V bus must have a variable source of reactive power such as a generator or a capacitor bank.) Generalized Power Flow Development The more general (n bus) case is developed by extending the results of the simple four-bus example. Consider the case of an n-bus system and the corresponding n+1 node positive sequence network. Assume that the buses are numbered such that the slack bus is numbered last. Direct extension of the earlier equations (writing the node voltage equations and making the same substitutions as in the four-bus case) yields the basic power flow equations in the general form. The Basic Power Flow Equations (PFE) (63.12) and (63.13) S P jQ V Y V for k = 1, 2, 3, , n – 1 k * k kk * ki i n i =1 =− = ∑ … P jQ V Y V n nn * ni i n i =1 − = ∑