IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 3, NO.l, MARCH 1993 349

SUBTRANSMISSION REDUCTION FOR INSTABILITY ANALYSIS James D. McCalley, Member John F. Dorsey, Sr Member Georgia Institute of Technology James F. Luini, Fellow R. Peter Mackin, Member Gerardo H. Molina Pacific Gas and Electric Company

ABS TRACT In this paper we specifically describe In this paper, we present a new method for creating power the subtransmission reduction problem and how it differs flow subtransmission equivalents to be used in voltage instabil- from other more traditional reduction problems, ity analysis. We present the motivating reasons for perform- the effects of the PDCI outage and why accurate subtrans- ing subtransmission reduction, and we show how the subtrans- mission representation in PG&E’s system is necessary, mission reduction problem differs from more traditional reduc- the new subtransmission reduction method and the asso- tion problems. Criteria for an acceptably reduced subtrans- ciated software developed, and mission system are stated, and Pacific Gas and Electric Com- pany’s (PG&E’s) reduction method is presented which utilizes the testing done to validate the new method. the program LODRED (from the EPRI Dynamic Equivalenc- ing Reduction Software) to perform load bus elimination and a 2 TERMINOLOGY new program called GALRED to perform generator bus aggre- For purposes of reduction, a network may be divided into the gation. Unlike most reduction methods, this method produces following three sub-systems: equivalents that are independent of base case and flows and consequently highly accurate for voltage instabilty analysis. The study system: the portion of the network to be re- Validation of the new method is performed using a 71% reduced tained. For the subtransmission reduction problem, the model (29% of its original size) of PG&E’s transmission system study system is the high voltage system (230 to 500 kV) to simulate bipole outage of the Pacific DC Intertie. The external system: the portion of the system to be re- duced. For the subtransmission reduction problem, the ex- KEYWORDS: Subtransmission, network reduction, voltage ternal system is the subtransmission system (60, 70, and inst ability. 115 kV). Although the subtransmission system is com- monly thought of as being “internal” to the transmission grid of a utility, we use the term “external” here to remain 1 INTRODUCTION in keeping with the network reduction literature. In this paper, we present a new method for creating power The boundary system: the portion of the external system flow subtransmission equivalents to be used in voltage insta- that is retained to connect the study system with the re- bility analysis. This new method was originally motivated by mainder of the external system. In most cases, the bound- the need to create more accurate power flow models to assess a ary system consists of the low voltage buses of transformers potential voltage instability problem in Pacific Gas & Electric connecting the high voltage system to the low voltage sys- Company’s (PG&E) system resulting from loss of the Pacific tem. However, other subtransmission buses may also be DC Intertie (PDCI). However, the resulting models may also retained and consequently serve as boundary buses. be used to more accurately assess any voltage instability or 3 THE SUBTRANSMISSION REDUCTION thermal overload problem on the high voltage system typically PROBLEM analyzed by power flow simulation. In the past, many utilities have used simple methods for rep- Bulk transmission analysis of many disturbances in the West- resenting subtransmission systems in bulk transmission anal- ern Systems Coordinating Council’s (WSCC) system, including ysis. Two related developments in the power industry have bipole outage of the PDCI, requires that most of the entire high made some of these methods obsolete. First, the increase in voltage grid be modeled. To reduce computation time, simplify non-utility generation, much of which is connected to the sub- data management, and use less computer memory, most mem- transmission system, gives subtransmission systems more - ber systems represent the effects of their subtransmission system age control and other dynamic characteristics. Second, the in- using reduced equivalent models. crease in wheeling and subsequent decrease in transmission ca- A reduction problem more traditional than subtransmission pacity margin, attributable to a more competitive environment, system reduction is that of reducing networks normally at the require that models exhibit higher accuracy for determining sys- same voltage level as the study system and interconnected with tem behavior so that transmission can be utilized safely and the study system via tie lines. We refer to this type of reduc- most economically [I]. tion problem as tie-line reduction. Subtransmission system re- duction differs from tie-line reduction in that the former allows only limited expansion of the boundary system to form a buffer zone. Because tie-line interconnections are normally few (from 1 to perhaps 9 or lo), tie-line reduction may always be pushed 92 WM 129-7 PWRS A paper recommended and approved further and further away from the study system by enlarging by the IEEE Power System Engineering Committee of the boundary system with little increase in model size. For the IEEE Power Engineering Society for presentation example, a 1 bus per connection enlargement of the boundary at the IEEE/PES 1992 Winter Meeting, New York, New system for a 2 tie-line system requires that model size increase York, January 26 - 30, 1992. Manuscript submitted August 28, 1991; made available for printing by only 2 buses. Boundary network enlargement can therefore December 31, 1991. allow relatively inaccurate tie-line reduction methods to be used without dramatically affecting study system accuracy. In con- trast, subtransmission systems are typically interconnected with

0885-8950/93$03.00 0 1992 IEEE different parts of the system between Grizzly substation and Table Mountain substation. Even with these remedial mea sues, PG&E’s 500 kV system (PACI plus COTP) experiences a large instantaneous power surge which can remain up to 20% above the pre-disturbance level until automatic generation con- trol (AGC) operates to reduce the high flows. These high flows dramatically depress voltages throughout PG&E’s high voltage and subtransmission systems which can result in voltage insta bility problems, especially when pre-disturbance PDCI, PACI, t f’ and COTP flows are heavy. - Assuming the system survives the initial transient following a PDCI bipole outage, it is most susceptible to voltage insta- -7 -rYWN /F bility problems for l to 3 minutes after the disturbance. This ROUND UT is because load tap changers (LTCs) act to restore distribution E voltages and consequently loadings are brought back to their pre-disturbance levels during this time interval. PG&E and v-D,, Bonneville Power Administration (BPA) engineers have devel- oped a study procedure to analyze system response during the YOUNTAIN 1 to 3 minute time period following a PDCI outage [2]. This FOUR CORNERS procedure involves using a power flow program to simulate the disturbance. Although the LTCs are not modeled, constant MVA load characteristics for all loads allows simulation of ‘worst case’ behavior of the LTCs. Other appropriate inputs are made 6 WFORNIA to the program to account for the effects of remedial actions Figure 1: Simplied Representation of the WSCC Grid. and generator frequency control. The margin, i.e., the prox- imity to voltage instability, is determined by drawing a Q-V the study system at many different points. A 1 bus per con- curve for each post-disturbance power flow case. This curve is nection enlargement of the boundary system may require that drawn by modeling a fictitious synchronous condenser at the the model size increase by 30 or 40 buses. Therefore, increasing most voltage-sensitive bus to find reactive requirements for dif- the boundary system can cause a prohibitive amount of model ferent voltages at that bus. The ‘nose’ of the resulting Q-V size increase and cannot be used to effectively relieve inaccura- curve represents the point of voltage instability. The horizontal cies in the equivalent. This distinction requires a high degree of ‘distance’ of the nose to the zero-MVAR vertical axis, measured accuracy in the method used for subtransmission reduction. in MVARS, is therefore an indicator of the proximity to voltage instability. Figure 8 at the end of this paper shows examples of 4 THE PDCI OUTAGE these curves. PG&E’s subtransmission system can heavily influence the re- PG&E’s desire to increase the accuracy of its subtransmission sults of this outage for two reasons: equivalents was originally motivated by a need to determine maximum allowable simultaneous transfers on the Pacific DC a The subtransmission system is a large reactive resource Intertie (PDCI) and the Pacific AC Intertie (PACI). These flows due to the substantial amount of generation connected to are limited by a voltage instability problem caused by bipole it. outage of the PDCI. Simulation of this outage has become a a Portions of the subtransmission network are in parallel benchmark at PG&E for assessing power flow model accuracy with the PACI and therefore carry some of the additional because of its severity in terms of increased flows and decreased power injected into PG&E’s system from the PDCI outage. voltages throughout the system. Power flow models that are accurate for this outage are assumed accurate for other less 5 PG&E CRITERIA FOR SUBTRANSMISSION severe outages. In this section, we describe the PDCI outage REDUCTION and how it is analyzed. The PDCI, rated at 3100 MW, f500 kV, presently operates Criteria used at PG&E for an acceptably reduced subtrans- in parallel with the PACI, rated at 3200 MW, 500 kV. Also, mission system are: there is another AC intertie under construction, the California- Oregon Transmission Project (COTP), that would increase the 1. The effects of the reduced system on the study system total AC transfer capability to 4800 MW. For the purposes of should be represented accurately for base case conditions this paper, the eastern portion can be simplistically represented and disturbance conditions. The equivalent should retain as a single intertie connecting the Northwest with Southern Cal- high accuracy for voltage deviations of up to 10% caused by ifornia through Montana, Idaho, Utah, and Arizona/New Mex- increased flow. To retain this accuracy, the reduced system ico. In addition, there is an AC tie from Idaho to the Northwest must accurately model the voltage control characteristics and a DC tie from Utah to . Figure 1 illus- of the generation connected to the subtransmission system trates this system. PG&E’s system consists of the Northern and those portions of the subtransmission network that are California area from Round Mountain substation to Midway in parallel with the high voltage network. substation. 2. The reduction should decrease the number of buses by at A bipole outage of the PDCI activates remedial measures in- least 70% and, to avoid Y-bus matrix fill-up problems, the cluding generation dropping and brake insertion in the North- reduction should decrease the number of branches by at west and high speed series and shunt capacitor insertion in least 60%. This means that the 1696 bus, 1998 branch 351

PG&E full transmission model should be reduced to no more than 500 buses and 800 branches. 0 3. The equivalent elements should be in a form compatible with existing power flow and stability software, and they should represent realistic elements so as not to cause confu- ‘24 sion among analysts unfamiliar with reduction techniques. 4. Although the analysis and results described in this paper la= l.bjO.1 0.67+J.067 0.33+10.33 are motivated by the need to perform voltage instability analysis, it is desirable for the reduced system to be accu- Figure 2: Example of Load Distribution Performed by LODRED rate for transient and oscillatory analysis as well. We describe the new reduction method in the following three where subscript 1 refers to retained buses, subscript 2 refers to sections. PQ buses to-be eliminated, and f1 is the vector of retained bus injections. IS is the vector of yonstant MVA loads converted 6 AN OVERVIEW OF THE REDUCTION PROCESS to current sinks, according to Is, = (S,/E)*,where IS,,S,, and are the converted current sink, constant power load, The network reduction process has two steps. The first step and voltage, respectively, for external bus i. Application of the eliminates non-generation buses (PQ buses) using the Electric gaussian elimination formula to eliminate the bottom row yields Power Research Institute (EPRI) program called LODRED [lo]. This software uses the Ward Injection technique for PQ buses [Ill = [K’l] [Vl] + [I;] based on gaussian elimination. LODRED was originally part of The matrix [Y:,] is the Y-bus of the equivalent network: the EPRI Dynamic Equivalencing Software, a software package developed by Podmore and Germond in the mid 1970’s and used [YA] = [Yll] - [K2] [Yzzl-’ [Y21] (3) to produce equivalents for transient stability analysis. Because we have made minor modifications to LODRED to enable it to The vector [I;] is the vector of equivalent current sinks on the produce equivalents for voltage instabilty analysis, and to show retained buses: why it cannot be used for generator (PV) bus elimination, we give a brief description of LODRED in Section 7. [I;] = - [F12] [%2]-’ [is] (4) The second step aggregates two or more PV buses into one equivalent bus. Several different techniques have been proposed After elimination, [I$]is converted back to a vector of equivalent to perform PV bus aggregation, the most promising of which constant power loads according to S, = cpGj,where is the are the Radial Equivalent Independent (REI) method [12], the voltage at retained bus j, and S, is the constant power load to power conservation method of Podmore and Germond [lo], and be added to the existing load at retained bus j. various boundary matching techniques [3]. However, none of The effect of equation 4 is to distribute load from each elim- these techniques were deemed appropriate for voltage insta- inated bus to the adjacent retained buses in proportion to the bility analysis because they all compute the values of certain electrical proximity of the eliminated bus to each adjacent re- crucial equivalent elements, i.e., line impedances, shunt admit- tained bus. For example, in Figure 2, if buses 1, 2, and 3 are tances, and/or bus injections as a function of base case (pre- retained and bus 4 is eliminated, the load distribution would be disturbance) bus voltages. When bus voltages substantially de- viate from their base case conditions, these elements no longer maintain accuracy. To solve this problem, a new program called GALRED was written to perform PV bus aggregation. This program is described in Section 8. fs3 = &Is4 = -(I +jo.i) = .33 +j.033 (6) 64 0.3 - j3 7 PQ BUS ELIMINATION USING LODRED where K4 is the self admittance term for bus 4, given by

The description of LODRED given here is adapted from [lo], Y44 = %4+&4+Ysh4 = (0.2-j2)+(0.1-j1)+0 = 0.3-j3. (7) pp 4-24 to 4-25. We have modified the program so that all loads retain their constant MVA identity after reduction in- As explained in [3], the Ward Injection method is not appro- stead of being divided between constant MVA, constant current, priate for eliminating PV buses because it is unable to faithfully and constant impedance representation. This modification is in keeping with the ‘worst case’ assumption used in voltage insta- ‘If a constant impedance shunt were connected to bus 4, i.e., if Ysh4 # 0, then it would be distributed by equation 3 to the retained bility analysis during the l to 3 minute time frame that load buses in the same proportion as the constant MVA load. However, tap changers hold voltages to their pre-disturbance level so as the inclusion of the shunt in the self term of the eliminated bus, in to make loads appear as constant MVA [Z]. Additionally, code tics case, of Ysh4 in ~44(equation 7) introduces some inaccuracy in was added to enable LODRED and GALRED to aggregate load the load and shunt distribution for the following reason. In effect, the voltage and frequency sensitivities simultaneous with, but sep- shunt causes the ground node to become adjacent to bus 4, but load arate from the aggregation of the loads. A future paper will and shunt cannot be modeled at the ground node. Consequently, the report on this and other dynamics-related work. Assuming all total distributed load and shunt to buses 2 and 3 is not :qual_ to the original load and shunt. Inspection of the expressions fcr I,2 ,-Is3, and loads are constant MVA loads, the nodal admittance equation Y44 in equations 5, 6, and 7 will verify this fact since Is2 + Is3 # Is4 can be written as if Y.h4 # 0. In the case of the PG&E system reduction, described in Section 9, the effects of this phenomenon have not been substantial probably because most buses with large shunts were retained. See [4] for additional information. 352

represent the reactive capabilities of PV buses under distur- 8.2 The Equivalent Shunt Element bance conditions. Even if a means were devised to distribute generator reactive capabilities from eliminated buses in a man- Constant impedance shunts in the unreduced external net- ner similar to the load of the example in Figure 2,the resulting work consist of capacitive and inductive bus shunts, capacitive equivalent would be undesirable for transient stability analysis. line charging, and inductive transformer magnetizing losses. For The reason is that the number of PV buses would increase, and each group of PV buses aggregated by GALRED, the values for each new PV bus would require that the original generator dy- these external network elements are summed and the aggregate namic data be decomposed to create two new sets of dynamic modeled on the equivalent bus, according to: data. Besides being a difficult task, this would dramatically in- crease the order of the model. Therefore a different algorithm N M is required for PV bus elimination, and this is the function of the program GALRED. I=,

8 PV BUS AGGREGATION USING GALRED where Bsheq is the equivalent bus shunt, Bdh, is the bus shunt susceptance on external bus i, and BchargcJ is the total line The distinguishing feature of GALRED is that it seeks to charging or transformer magnetizing susceptance for branch j. form an equivalent that is accurate under base conditions and There are N buses and M branches in each group. The M disturbance conditions by preserving the physical structure of branches include branches connecting external buses to bound- the network. In preserving the physical structure of the net- ary buses, but the N buses do not include the boundary buses. work, GALRED models constant impedance shunts and branch impedances by condition-independent, similar elements. By 8.3 Equivalent Bus to Boundary Bus Connections ‘condition-independent’, we mean that base conditions, i.e., base flows, voltages, and angles are not used in the calcula- Because each group of external buses is reduced to 1 equiva- tions. By ‘similar’, we mean that shunt elements in the ex- lent bus, equivalent branches must be formed between that bus ternal network are reduced to equivalent shunt elements only, and the boundary buses. In this subsection, we describe the and that branch elements in the external network are reduced algorithms used in determining the impedance values for these to equivalent branch elements only, i.e., there is no interchange equivalent branches. A pre-processing routine in GALRED re- between voltage-sensitive elements in the external network and duces all multi-branch connections between buses in the exter- current-sensitive elements in the equivdent or current-sensitive nal network to a single branch by paralleling them. elements in the external network and voltage-sensitive elements GALRED contains two separate branch impedance determin- in the equivalent. ing algorithms. Which one is used depends on how many bound- Inputs to GALRED are network data output from LODRED ary buses a group has. If there is only 1 boundary bus, the and a set of groups, each group consisting of two or more PV equivalent bus will be connected by a single branch radially from buses to be reduced to 1 equivalent PV bus. It should be noted this boundary bus, and we compute an equivalent impedance that GALRED does not automatically perform the groupings. that best preserves the impedance seen looking from the bound- At PG&E, initial groupings are made using a coherency recog- ary bus to one or more buses in the group. If there are two or nition routine [6] based on electrical proximity. Some adjust- more boundary buses, we compute equivalent impedances so as ment of these groupings by an engineer familiar with the system to best preserve the impedances seen looking from one bound- is usually necessary to satisfy GALRED’s topological require- ary bus to another. For the subtransmission reduction problem, ments, as explained in Section 8.3.2. the effects of the latter algorithm are particularly crucial in that the impedances of subtransmission paths parallel to the high 8.1 Bus Data Aggregation voltage system are accurately maintained.

The bus data aggregation performed by GALRED is straight- 8.3.1 Algorithm 1: For 1 Boundary Bus forward in that the real generation, maximum real generation, and reactive limits of the group are each summed, and the ag- For each group g having only 1 boundary bus, gregate of each respective quantity is modeled on the equivalent bus. Aggregation of constant impedance shunts is described in 1. Determine the number Ndc of decoupled subsystems com- the next section. The equivalent generator voltage set point is prising group g. A decoupled subsystem consists of a sub- computed by taking a weighted average of the generator voltage set of buses in group g that remain interconnected when set points of the group according to the following formula: all branches between the group g buses and the boundary bus are removed. The simplest example of a group com- prised of several decoupled subsystems is when all buses in a group are connected via their own step-up transformer to the same high side boundary bus. Figure 3 illustrates a The weight, Qmaz serves to quantify the effect, under stressed more complex example of this situation. conditions, of each generator on the voltage profile of the study 2. Determine the dominating bus jk for each decoupled sub- system. Q,, is used instead of Qgen because Qmaz repre- system k = 1, ..., Ndc of group g. The dominating bus is sents the generator’s ability to affect voltages under stressed the bus with the largest MVA injection (ISgen - Sloadl). conditions when voltages are low, typical of voltage instabil- This bus is responsible for the greatest flow in the subsys ity studies. For the less commonly studied light load condi- tem and thus the largest amount of losses. Compute ZiJL, tions when voltages are high, Qmin may be a more appro- the Thevenin impedance seen looking from the boundary priate weight, but equation 8 still works well since usually bus i to this bus jk. (See Section 8.3.3 below on how this Qmai,/Qmat, z Qmin,/Qminj for two generators i and j. impedance is computed.)

- --* 353

Boundary Bus where Nb is the number of boundary buses and therefore also the number of unknowns. There are three possibilities regarding solution of these equations: I ,+e---$-- ---+group, \ Nb = 2. There is 1 equation and two unknowns.

I t The equation is underconstrained. Here we let Zi, = I zj, = Zij/2. Nb = 3. There are 3 equations and 3 unknowns. The system of equations [A][Z,] = [Z] are solved exactly ystem using gaussian elimination and back substitution. Nb 2 4. There are more equations than unknowns. Figure 3: A group comprised of two decoupled subsystems. The system of equations [A][Z,] = [Z] is overcon- strained, and an exact answer is, in general, not possible. A solution [ZL] is therefore found that minimizes the least square error [I11 according to [AIT[A][Z:] = [AIT[Z]. These equations are solved using gaussian elimination and back substitution.

I 8.3.3 Computing Thevenin Impedances /o The Thevenin impedances in both of the above algorithms are computed assuming boundary bus i is a current source, Figure 4: Illustration of a group having boundary buses 1 and 3 not bus j is grounded, and the boundary bus to external system interconnected via buses of the group. An incorrect branches for all other boundary buses (if there are more than grouping of buses a and b with bus c will generate a one) are opened. It is assumed that each generator bus in the transmission path between boundary buses 1 and 3 external system is an independent current source and can be through the equivalent. GALRED will flag such a situation. opened. (Under this assumption, the Thevenin impedances are 3. Compute the impedance of the equivalent connection be- exact for stressed conditions typical of voltage instability when tween the boundary bus i and the equivalent bus e as the most generators are operating at their reactive limits; however, parallel combination of the Nde impedances computed in the Thevenin impedances are approximate when generators are the last step. regulating and acting as a voltage source. The fact that the testing reported in Section 9 indicates good performance of the 8.3.2 Algorithm 2: For 2 or More Boundary Buses equivalent under base conditions implies that the approxima- This algorithm is applied only to groups of external buses tion is a good one.) A Y-bus for the associated network is set that have 2 or more boundary buses. Also, all boundary buses up with dimension N + 1 (N external buses plus the current must be interconnected via the buses of the group. Otherwise, source boundary bus 2). The last row and column of this ma- the algorithm attempts to create a transmission path which does trix must correspond to boundary bus i. Gaussian elimination not exist in the unreduced network. The program GALRED will of this matrix produces an upper triangular matrix with the flag any group of external buses violating this requirement so admittance y1, as element (N + 1, N + 1). This admittance is that the user can regroup the buses. An example of a group inverted to yield the desired impedance from boundary bus z to violating this requirement is illustrated in Figure 4. bus j, Z,,. For each group g having Nb 2 2 boundary buses, 8.3.4 Equivalent Bus to Equivalent Bus Connections 1. Compute Z,,, the Thevenin impedances seen looking from boundary bus 2 through the external system to bound- An additional calculation is required if two groups g1 and g2 ary bus 3, for each pair of boundary buses, with all other are connected by a single branch (il, 22). Application of Algo- boundary bus to external bus connections open. (See Sec- rithm l or Algorithm 2 to each group results in an impedance tion 8.3.3 below on how these impedances are computed.) Z,,,,, from equivalent bus el to boundary bus i2 and an impedance Z,,,,, from equivalent bus e2 to boundary bus 2. Replace all external buses in the group g with a single il. equivalent bus e connected to each boundary bus k through GALRED calculates the desired impedance from el to e2 as a branch with a presently unknown impedance Zke. Ze,,e, = Ze,,,, + Ze,,,, - Z:,,:, 3. For each pair of boundary buses i, j,equate the sum of the two boundary bus to equivalent bus impedances Z,, + Z,, where Z,,,,,is the impedance of the original branch connected to the corresponding Thevenin impedances computed in between bus il and bus 22. Figure 5 illustrates this situation. two groups g1 and g2 are connected by 2 or more branches, Step 1, Z,, so that If a pre-processing routine in GALRED combines the inter-group Zse + Zje = Ztj branches in parallel. The resulting single branch is arbitrar- ily reconnected between one of the previously connected inter- These equations may be written in matrix form as group pairs of buses. In reducing PG&E’s subtransmission, as [AI [Zel= [ZI. described in Section 9, this situation occured only once. This 4. The number of equations formed in step 3 is equal to the situation occurs infrequently even in the relatively densely con- number of boundary bus pairs, which is Nb boundary buses nected subtransmission network because normally, if two sets of taken 2 at a time, or generator buses are interconnected by 2 or more branches, the two groups are stiffly connected and therefore combined into a single group. 354

G-

I ' we:e1 and e2 are A-- 'e1 ,i2 fiiious equivalent buses.

Figure 5: Illustration of addiional calculation for equivalent bus to equivalent bus connection.

I- The impedances of these two branches will not be included in the equivalent branch calculation between bus i and bus j. Therefore the branch bsses will be added in as equivalent bad.

Figure 6: Illustration of branches in external system having impedances not to be included in equivalent branch calculations. Figure 6: Reductkn of System 1 to System 2. 8.3.5 Branches Not Included in Equivalent Branches

The effects of impedances for some branches in the ex- to be reduced to System 2. The boundary buss, 1, 2, 3, and ternal system may not be included in the equivalent branch 4 are the low voltage buses of transformers connecting the high impedances. Branches of this sort exist when one or more ex- voltage system with the subtransmission system. We desire to ternal buses are connected radially from a bus in the path be- select the values of Zl,, Zz,, Z3,, and Z4, that best preserve tween bus i and bus j in the Thevenin impedance calculation. the impedances seen looking from each boundary bus through An example of this situation is illustrated in Figure 6. To com- the equivalent to another boundary bus. pensate for this situation, GALRED computes the losses in the 1. Compute the boundary bus to boundary bus impedances unreduced case for these branches and adds them in as con- with all other boundary bus to external system branches stant MVA load to the equivalent bus.Although this action vio- open. For Z12, we open branches 3-7 and 4-8. The corre- lates our principle of using only condition-independent, similar sponding Y-bus matrix is elements in the equivalent, the associated inaccuracy for dis- j87.5 -j50 -j25 0 -j12.5 turbance conditions is deemed small because (1) conditions for -j50 j161.1 0 -jlOO a radial external bus do not change dramatically, and (2) the -325 0 j58.3 -j33.3 -j12.50 -jlOO0 -j33.30 j133.30 j12.5O"0 = aggregate additional load associated with this action is small. [ ] [I] [ ;] 8.3.6 Transformers with Off-Nominal Tap Ratios Gaussian elimination yields Transformers in the external network are treated as lines un- less they have an off-nominal tap ratio and they form a bound- 187.5 -j50 -j25 0 -jl2.5 ary bus to external bus connection. In this case, the connecting [ i jlY.5 -j14.2 -jlOO -j7.1 0 j49.6 -j44.1 -j4.3 = I\ branch between that boundary bus and the equivalent bus has 0 j18.7 -j9.2 ] [ [ ] its impedance computed as above, but the branch is represented 0 0 j5.38 I] as a transformer with tap ratio equal to that of the correspond- ing boundary bus to external bus connection in the unreduced Therefore case. -j5.38v1 = 11 =+ Y12 = -j5.38 8.4 Example + 212 = & = j0.186

Because Algorithm 1, bus data aggregation, and constant The other impedances are computed similarly and are as impedance shunt aggregation are relatively straightforward, we follows: 213 = j0.214, Zlr = j0.201, 223 = j0.224, Zzi = only give an example for Algorithm 2. In Figure 7, System 1 is j0.280, and 234 = j0.231. 355

2. The bottom diagram of Figure 7 shows the external system Figure 8: Testing For HS96 Model; Q-V Curves replaced by the equivalent bus e with the four equivalent branches. 3. We now equate the sum of the two boundary bus to equiva- lent bus impedances to the corresponding impedances com- puted in Step 1.

Zie + Zze = Zi2 = j0.186 Zie + Z3e = 213 = j0.214 z1e + z4e = Z14 = j0.201 Z2, + Z3e = 223 = j0.224 z2, + z4, = zz4 = 30.280 Z3e + Z4e = Z34 = 30.231 Writing in the matrix form [A][Ze]= [Z],we have '8300 -150 -100 -50 0 50 100 1100 - j0.186 Tesla 500 kV MVRRS (Negative for Load) 1010 ZIe j0.214 j0.201 1001 north and south ends, respectively, of PG&E's 500 kV system.' 0110 j0.224 The magnitudes of these power injections were determined 0101 j0.280 [ = from previous studies that modeled the entire WSCC sys- 0011 - j0.231 tem. Figures 8 and 9 display the results of this testing. In each figure, four curves are drawn so that comparison can be 4. Note that there are 4 unknowns and 6 equations above. made between the unreduced case, the 627 LODRED-reduced The system is overconstrained. The least squares min- case, the 484 LODRED,GENRED-reduced case, and the 484 imization problem is therefore written as [A]T[A][Z,]= LODRED,GALRED-reduced case. [AlT[Z1or Figure 8 shows a Q-V post-disturbance curves of the most voltage-sensitive bus in the system. Curve 8-B indicates that 3111 j0.601 the 627 LODRED-reduced model exhibits a very small amount 1311 of inaccuracy in that the point of voltage instability is only 1131 j0.669 22 MVARS from that of the 1654 unreduced case, curve 8-A. 1113 j0.712 [ This amount of inaccuracy, probably due to the shunt-related Gaussian elimination and back substitution of the last problem discussed in footnote 1, is deemed negligible for a 62% equation yields the final result: reduction. However, Curve 8-C indicates that the 484 bus LODRED,GENRED-reduced model exhibits a relatively large [ = [ j0.122]j0.078 amount of inaccuracy in that the point of voltage instability is 111 MVARS from that of the unreduced case. This amount of inaccuracy is deemed unacceptable, especially since the reduc- Z; e j0.112 tion is only an additional 9%. Curve 8-D shows that if GALRED Z; e j0.133 is used to perform the PV bus aggregation instead of GENRED, 9 TESTING the level of accuracy achieved by LODRED is retained. Here it is apparent that GALRED introduces essentially no additional inaccuracy over that of the LODRED model. Figure 9 illus- Testing of the subtransmission reduction procedure was per- trates a similar comparison as Figure 8 except that P-V curves formed on a 1696 bus model of the PG&E transmission system are drawn instead of Q-V curves. In these curves, the voltage of representing 1996 summer peak conditions (HS96). PQ bus elimination using the LODRED program reduced the model the most voltage-sensitive bus was recorded as a function of the real power injected into PG&E's 500 kV system. Again, Curve size by 63% to 627 buses. From the 627 bus reduced case, PV 9-B shows the inaccuracy at the point of voltage instability for bus aggregation was performed using the GALRED program the 627 bus LODRED-reduced case is very small (25 MW) and for another 9% reduction to 484 buses. Also, to provide a basis of comparison, the Podmore-Germond power conservation pro- in fact negligible, but Curve 9-C indicates that of the 484 bus LODRED-GENRED-reduced case is relatively large (95 MW) gram GENRED was used on the 627 bus case to form a second However, Curve 9-D shows that 484 bus case. The same inputs were used to both GALRED and therefore unacceptable. the 484 bus LODRED, GALRED-reduced case exhibits almost and GENRED consisting of 229 external buses in 85 different groups. Although most of the groups consisted of PV buses the same amount of inaccuracy as the 627 LODRED-reduced connected to the subtransmission system, a few of the groups case, indicating again that the GALRED program introduces consisted of PV buses connected radially from a bus in the high essentially no additional inaccuracy. Testing on a 1995 spring off-peak case (not illustrated) indicates similar results for the voltage system through a step-down transformer. Under base more lightly loaded conditions. conditions, the 627 bus case and both of the 484 bus cases were extremely accurate relative to the unreduced case. 'This simple method of simulating the PDCI outage does not ac- The test was Of the reduced to count for any remedial actions or governor under-frequency response. the unreduced under PDC1 Outage conditions. In all The results therefore do not reflect actual system capabilities; how- cases, the PDCI outage was simulated simplistically by mod- ever, they do well serve the task at hand of determining the accuracy ehg large positive and negative real power injections at the of the reduction methods. 356

[6] Dorsey, J., Schlueter, R., “Global and Local Dynamic Figure 9: Testing For HS96 Model; P-V Curves Equivalents Based on Structual Archetypes for Co- herency”, Paper 83 WM 047-8 Presented at the IEEE/PES Winter Meeting in New York, 1983. [7] Dy Liacco, T.E., Savulescu, S.C., Ramarao, K.A.,‘An On- Line Topological Equivalent of a Power System’, IEEE Transactions on Power Apparatus and Systems, Vol. PAS- 97, No 5, Sept/Oct 1978. [8] Alvarado, F., Elkonyaly, E., ‘Reduction in Power Systems’, Paper A 77 507-7 for the IEEE PES Summer Meeting in Mexico City, 1977. [9] Tinney, W., Powell, W., ‘The REI Approach to Power Network Equivalents’, Proc. PICA Conference, Toronto, I pp.314-320, May, 1977. 5800 5850 5900 5950 [lo] Podmore, R. Germond, A. ‘Development of Dynamic PACItCOTP Import (MW) Equivalents for Transient Stability Studies’, Final Report, EPRI EL-456, 1977. 10 CONCLUSIONS [I11 Strang, G. Linear Algebra and its Applications, Third Decreasing transmission capacity margins in today’s electric Edition, 1976 by Harcourt Brace Jovanovich, Inc. power systems have created the need for greater accuracy in the [12] Dimo, P., Groza, L., Ionescu, S., Ungureanu, B., Petcu, I., analysis procedures. In particular, disturbance analysis of the ‘The REI Equivalent, A General Model for the Analysis of PDCI bipole outage in the WSCC system has created the need Power System Behavior’, CIGRE Report 318, 1964. to more accurately model the subtransmission system in the northern California area. PG&E’s new method of producing BIOGRAPHICAL SKETCHES subtransmission equivalents allows accurate power flow analy- sis while conserving computation time and computer memory Dr. John F. Dorsey received the B.S. degree from Purdue and easing data management. The major improvement of the University in 1964, and the M.S. and Ph.D. degrees from Michi- new method over previous ones is that subtransmission voltage gan State University in 1969 and 1980, respectively. He is cur- control characteristics and paths parallel to the high voltage rently an Associate Professor of Electrical Engineering at the system may be accurately modeled for base conditions as well Georgia Institute of Technology, where he has been employed as disturbance conditions. In doing so, the new method creates since 1980. Dr. Dorsey is a senior member of the IEEE. equivalents having elements that are easily identifiable by users James F. Luini received the B.S. degree and the M.S. degrees and compatible with existing analysis software. Reduction of from the University of California at Berkeley in 1957 and 1964, a 1654 bus model of the PG&E transmission system to a 484 respectively. He is currently a supervising engineer at Pacific bus model shows that the method is highly accurate for voltage Gas and Electric Company, where he has been employed since instability analysis. 1958. Mr. Luini is a registered professional engineer (Control Systems) in the state of California, and he is a Fellow of the IEEE. References R. Peter Mackin received the B.S. degree in Civil Engineer- ing and the M.S. degree in Electrical Engineering from Montana [l] McCalley, J., Dorsey, J, &U, Z., Luini, J., Filippi, J., ‘A State University in 1981 and 1982, respectively. He is currently New Methodology for Determining Transmission Capacity an engineer at Pacific Gas and Electric Company, where he has Margin in Electric Power Systems’, A paper presented at been employed since 1983. Mr. Mackin is a registered profes- the February, 1991 IEEE Power Engineering Society Win- sional engineer (Electrical Engineering) in the state of Califor- ter Meeting in New York. nia, and he is a member of the IEEE. James D. McCalley received the B.S. and M.S. degrees from [2] Mittelstadt, W., Taylor, C., Klinger, M., Luini, J., Mc- Georgia Institute of Technology in 1982 and 1986, respectively. Calley, J., Mechenbier, J., ‘Voltage Instability Modeling He began working for Pacific Gas and Electric Company in 1986, and Solutions as Applied to the Pacific Intertie’, A paper from which he is currently on leave of absence to perform doc- presented at the 1990 CIGRE Meeting in Paris. toral studies at Georgia Institute of Technology. Mr. McCalley [3] Wu, F., Monticelli, A., ‘Critical Review of External Net- is a registered professional engineer (EE) in the states of Cali- work Modelling for Online Security Analysis’, Electical fornia and Georgia, and he is a member of the IEEE. Power & Energy Systems, Vol 5, No 4, 1983. Gerard0 H. Molina received the B.S. degree from Northwest- [4] Deckman, S., Pizzolante, A., Monticelli, A., Stott, B., Al- ern University in 1982, and the M.S. degree from M.I.T. in sac, O., ‘Numerical Testing of Power System Load Flow 1986. He is currently an engineer at Pacific Gas and Electric Equivalents’, IEEE Transactions on Power Appartus and Company, where he has been employed since 1989. Systems, Vol PAS-99, No 6, Nov/Dec 1980. [5] Monticelli, A., Deckman, S., Garcia, A., Stott, B., ‘Real Time External Equivalents for Static Security Analysis’, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-98, No 2, March/April 1979.