OTTEN METHOD FOR PARTITIONING

POWER SYSTEM STATE ESTIMATION NETWORK

Ibrahim 0 . Habiballah Victor H. Quintana

Department of Electrical and Computer Engineering

University of Waterloo

Waterloo, Ontario

Canada N2L 301

Abstract - This paper presents the development of a heuristic algorithm to partition an observable power-system-state­estimation (PSSE) network into two or more observable subnet­works. The proposed algorithm partitions a spanning tree of an observable PSSE network, that guarantees the observability of all partitioned subnetworks. The partitioning technique is based on Otten's eigensolution approach that is used for placement of modules in single-stack-layout circuits. Otten's method provides a fu 11 screen of all possibilities a spanning tree can be partitioned into. The performance of this heuristic algorithm is evaluated by using several IEEE PSSE networks.

Keywords: partitioning; eigensolutions; state estimation.

1. INTRODUCTION

Power system state estimation (PSSE) can be defined as an algo­rithm to obtain the best estimate of the power system states, given a redundant set of measurements. These measurements involve voltage magnitudes, line power flows and/or node power injec­tions [ 1].

With the growth in size of existing power systems, many tech­niques and algorithms have been developed to reduce the com­plexity associated with the "integrated" high-order state estima­tion problem by decomposing it into lower-order subproblems. Many of the decomposition techniques and algorithms presented in the literatures are based on partitioning an observable PSSE network into k observable subnetworks [2-S]. However, none of these decomposition techniques and algorithms discuss how such partitioning can take place. It is the main aim of this paper to pro­vide a partitioning algorithm that is suitable for use in any of the existing PSSE decomposition techniques (e.g., decentralized two­level state estimation approaches).

The term "PSSE network", which will be used frequently throughout this paper, referred to the network graph of a power system and the corresponding redundant measurement meters dis­tributed around the network. The term "observable" is very important in PSSE problems. A necessary condition for observa­bility is the existence, in the graph of the PSSE network, of a full rank spanning tree [6]. If the system network is unobservable then the system is unsolvable [7]. Similarly, if one or more of the partitioned system subnetworks (whose overall system is observ­able) is unobservable, then it is impossible to consider these parti­tioned subnetworks. In this case, an alternative partitioning, whose subnetworks are all observable (i.e., every subnetwork has a full rank subspanning tree), should be found.

'697

This paper presents the development of a heuristic algorithm to partition an observable PSSE network into two or more observ­able subnetworks. The proposed heuristic algorithm partitions a spanning tree of an observable PSSE network. Partitioning a span­ning tree guarantees the observability of all the subnetworks (i.e., there is no need to check for the observability of the partitioned subnetworks). The partitioning algorithm is based on Otten's eigensolution approach as it is used for placement modules in single-stack-layout circuits. Otten's method provides a full screen of all possibilities a spanning tree can be partitioned into. The per­formance of this heuristic algorithm is evaluated by using four IEEE PSSE networks: 14-bus, 24-bus, 30-bus, and 118-bus power systems.

This paper is organized as follows. Next section describes Otten's eigensolution approach as it is used to place the modules in a single-stack-layout circuit. Section 3 discuss the development of the proposed partitioning algorithm supported with a simple IEEE example. Section 4 presents the results of partitioning three IEEE PSSE networks into observable subnetworks.

2. OTTEN'S METHOD FOR PLACEMENT

2.1. Single Stack Circuit Layout

Consider the problem of laying out groups of modules (e.g., cells, gates, op-amps, ... etc) connected by nets (e.g., wires) into stacks where the total wire length is to be minimized. Minimizing wire lengths tends to minimize the number of channels (i.e., track width) required for placement the net wires. This, in turn, reduces the total area of a chip.

Many approaches have been introduced in the literature to solve the problem of laying out groups of modules connected by nets into single stacks. One approach for solving this problem is to start with a randomly generated layout and then employ inter­change heuristic to minimize the total wire length of the layout. Another approach is the eigenvector-based method to generate good initial stack layouts where the total wire length is small. A module interchange algorithm is then used on the generated initial stack layout to further reduce the total wire length [8]. There are other eigensolution-based methods introduced for special layout problems; Otten's eigensolution method is such a method [9]. It works efficiently with perfect circuit layouts (i.e., perfect netlists).

2.2. Otten's Eigensolutions Method

Let P be defined as an m x /1 module-net matrix with m modules and n nets. Minimizing the track width (i.e., number of tracks) required for net placement corresponds to finding a permutation of the module-net matrix such that the bandwidth of the diagonal­ized matrix is ordered as small as possible [9]. The ordered band­diagonal matrix is called a perfect matrix P. Otten has presented an eigensolution approach to find this perfect matrix P (i .e., opti­maly permuting the modules and nets of the P matrix).

2.2.1. Finding a Perfect Matrix

Let P be defined as an m x n module-net matrix with m modules and /1 nets. To find a perfect matrix P, Otten [9] defined the fol­lowing two diagonal matrices Cmm and Dn,, with elements

l r12

C;; = i_pij i=l,2, ... . ,m (1) ·=I

[m r/2 djj= ~pij j=l,2, .... ,n (2) •=1

and the two matrices B,,"' and Amm as

B=CPD (3)

A =CB BT C-1 (4)

Otten [9] has shown that all eigenvalues of A are real and non­negative, and its largest eigenvalue is A.1 = l, and corresponding

eigenvector em .

For any eigenvalue of A, say A., we have

Ax= A.x (5)

where x is corresponding eigenvector. Premultiplying Eq. (5) by c-1, we have

. C-IAx = A_C-lx (6)

Substituting for the value of A from Eq. (4) into Eq. (6) yields

BBT (C-1x) = A.(C-1x) (7)

It is clear from Eqs. (5) and (7) that if (A., x) is an eigenvalue and eigenvector of A , then (A., c-1x) is an eigenvalue and eigenvector of BBT. The solution belonging to A.1 (i.e., the largest eigenvalue)

is a trivial solution because it assigns the same position to all rows and columns [9]. Otten has selected the second largest eigenvalue as a solution.

The following heuristic procedure produces a perfect matrix P (i .e., reduces the number of tracks in a netlist).

1) Find the eigenvector v = C-1x that corresponds to the second largest eigenvalue of BBT, as given by Eq. (7).

2) Find r = Cv = x , and order the components of r from largest to smallest. The coordinates of the ordered vector r_ gives the correct module sequence of the perfect matrix P.

3) Find k = D2PT r , and order the components of k from largest to smallest The coordinates of the ordered vector k gives the correct net sequence of the perfect matrix P.

Example

Let's consider the following 5-module, 7-net single-stack layout example shown in Figure 1 [8].

module

2

3

4

5

2

I 13

tracks (channels)

nets (wires)

16 I

Figure 1 single stack layout

In this example, all nets (wires) connect two modules except nets 3 and 7 that connect three modules. The total wire (net) length can be calculated by adding up the number of wires that fill each gap between the modules; total wire length of the 5-module 7-net example shown in Figure l is 17 units. The number of channels required to connect these modules are 7 (i.e. 7 tracks). The module-net matrix P is

698

1000100 0010010

P= 0 1 l 1 0 0 1 1100101 0011011

The total wire length of the above example can be minimized by optimaly permuting the modules and nets in Figure 1. According to Eqs. (1), (2) and (3), the C, D, and BBT matrices are as follows

D=

C=

1

l 10.) 1

1(2)

10.) -,h. 1 "'1(2) ~

"'1(3) 1 "Tc2)

2

1

2 1 2

1(2) -,h. 1 '1(2) ~

"\/(3)

0.5 0 0

0.35 0

0 0.42 0.12

0 0.29

0 0.12 0.42 0.21 0.29

0.35 0

0.21 0.46 0.08

0 0.29 0.29 0.08 0.42

The second largest eigenvalue of BBT is Ai= 0.765, and the

corresponding eigenvector v2 is

V2 = (0.626, --0.418, --0.205, 0.470, --0.412f

The vector r = Cv, therefore, is

,. = [0.443, --0.295, -0.103, 0.235, --0.206f

The optimal module ordering (largest to smallest) of the perfect matrix P is

1-4-3-5-2

The vector k = D2PT r is

k = [0.339, 0.066, --0.201, --0.154, 0.339, --0.251, --0.025f

The optimal net ordering (largest to smallest) of the perfect matrix P is

1-5-2-7-4-3-6

With these optimal ordering of modules and nets, the perfect matrix P will be

1 1 0 0 0 0 0 1 1 1 1 0 0 0

P= 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1

It can be observed that the perfect matrix P (i.e, the permuted module-net matrix) has been ordered such that the diagonal bandwidth is as small as possible. The optimal permutation of the 5-module, 7-net example is illustrated in Figure 2; total wire length, after permutation, is 9 units. The number of channels (tracks) required to connect these modules are 4 (note that the track width has been reduced to the minimum).

5

4 1 2 7

3 3

5 6

2

Figure 2 optimal stack layout

3. PARTITIONING HEURISTIC ALGORITHM

In this section the Otten's method for placement single-stack lay­out circuits , presented in the previous section, is utilized to parti­tion a spanning tree of an observable PSSE network into two or more subspanning trees. The optimal partitioning of a spanning tree of a topological observable PSSE network (i.e., every sub­spanning tree is of full rank) guarantees the topological observa­bility of the partitioned subnetworks.

Otten's method can provide the planner or the dispatcher with a full screen of all possibilities the spanning tree can be parti­tioned into. The following heuristic algorithm is proposed to par­tition a spanning tree of an observable PSSE network into k sub­spanning trees.

Algorithm

[l] Obtain a spanning tree of an observable PSSE network.

[2] Transfer the spanning tree to a single stack layout circuit by replacing every node and branch in the spanning tree by a module and net, respectively.

[3] Obtain the perfect matrix P of the spanning tree using Otten's approach, as explained in the previous section.

[4] Partition the perfect matrix Pinto k blocks with block sizes of m 1, m2, m3, . . ., mk such that the number of net cuts between

every two consecutive blocks is ONE.

[5] In case the number of net cuts between any two consecutive blocks is greater than ONE, change the size of every two of those consecutive blocks in such away that they do not exceed a prespecified limit and such that the number of net cuts between those particular blocks gets to ONE.

[6] In case the number of net cuts between any of the two modi­fied consecutive blocks remains greater than ONE, STOP; the existing spanning tree cannot be partitioned optimaly. Find another possible spanning tree and go back to Step 2. If there is no more possible spanning trees that can be partitioned opti­maly then STOP; the system can not be partitioned into k observable subnetworks.

[7] In case the perfect matrix P is partitioned optimaly (i.e., the number of net cuts between every two consecutive blocks is ONE), then each block represents a subspanning tree and the modules number for each block represents the corresponding nodes number of each of the partitioned subspanning trees.

699

Once the spanning tree is optirnaly partitioned into k subspanning trees (i .e., the total number of cuts is k - 1), then the actual inter­connected lines between the k subnetworks can be obtained directly from the original network.

Partitioning of IEEE 14-bus Network

Let us consider the partitioning of the IEEE 14-bus observable PSSE network, illustrated in Figure 3, into 3 subnetworks with block sizes of m1 = m2 = 5 and m3 = 4.

Let us allow changes in the size of any block by removing or adding nodes, such that the maximum difference of nodes between any two blocks is lWO. The connection of a possible spanning tree of the IEEE 14-bus network is given in Table 1.

12 14

6 9

4

2 3

~ injection measurement

• line-flow measurement

Figure 3 IEEE 14-bus observable PSSE network

Table 1 A Spanning Tree of the IEEE 14-bus

Branch From To Branch From To

1 1 2 8 6 10 2 1 5 9 6 12 3 2 3 10 6 13 4 3 4 11 7 8 5 4 7 12 9 11 6 4 9 13 9 14 7 5 6

According to Step 2 of the proposed Algorithm, every node and branch of the spanning tree given in Table l will be considered as a module and net, respectively. The module-net matrix P, accord­ingly, is given as shown in Figure 4.

N E T S l 2 3 4 5 6 7 8 9 10111213

Ml 1100000000000 2 1 0 l 0 0 0 0 0 0 0 0 0 0

0 3 0 0 1 l 0 0 0 0 0 0 0 0 0 4 0 0 0 l 1 1 0 0 0 0 0 0 0

D 5 0 1 0 0 0 0 1 0 0 0 0 0 0 6 0 0 0 0 0 0 1 1 1 1 0 0 0

P= U 7 0 0 0 0 1 0 0 0 0 0 1 0 0 8 0 0 0 0 0 0 0 0 0 0 1 0 0

L 9 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0000000100000

EllOOOOOOOOOOO 1 0 1 0000000010000

s 13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Figure 4 Module-net matrix P

After applying Otten's eigensolutions to this P matrix, the optimal module ordering (largest to smallest) of the perfect matrix f is

11 - 14 - 9 - 8 - 7 - 4 - 3 - 2 - 1 - 5 - 6 - 10 - 13 - 12

700

The optimal net ordering (largest to smallest) of the perfect matrix f is

12 - 13 - 11 - 6 - 5 - 4 - 3 - l - 2 - 7 - 8 - 10 - 9

With these optimal ordering of the modules and nets, the perfect matrix P is given in Figure 5.

121311 6 5 4 3 l 2 7 8 10 9

111000000000000 14 0 l 0 0 0 0 0 0 0 0 0 0 0 9 l 1 0 l 0 0 0 0 0 0 0 0 0 ~lock 80010000000000 #1 7 0 0 l 0 l 0 0 0 0 0 0 0 0

4 0 0 0 1 l l 0 0 0 0 0 0 0 P=30000011000000

2 0 0 0 0 0 0 l l 0 0 0 0 0 atock 1 0 0 0 0 0 0 0 l l 0 0 0 0 #2 5 0 0 0 0 0 0 0 0 1 l 0 0 0

6 0 0 0 0 0 0 0 0 0 1 1 1 l 1C 0 0 0 0 0 0 0 0 0 0 l 0 0 !Block 13 0 0 0 0 0 0 0 0 0 0 0 l 0 # 3 12 0 0 0 0 0 0 0 0 0 0 0 0 l

Figure 5 Perfect module-net matrix P

Let us partition the perfect matrix P into 3 blocks with block sizes of m1 = m2 = 5 and m3 = 4, as shown in Figure 5. Although, the

number of net cuts between the last two consecutive blocks is ONE (net 7), the number of net cuts between the first two con­secutive blocks is greater than ONE (nets 5 and 6 are cut). There­fore, the size of the first two blocks must be changed as presented in Step 5 of the proposed Algorithm.

It can be observed from the perfect matrix f (shown in Figure 5) that if the size of block 1 is increased by one (i.e., removing module 4 from block 2 and adding it to block l), then the number of net cuts between the first two consecutive blocks become ONE, as shown in Figure 6. Notice, from Figure 6, that the difference between any two blocks after the new changes in the sizes of the first two blocks is not greater than lWO (i.e., the difference doesn't violate the prespecified limit).

121311 6 5 4 3 l 2 7 8 109

11 l 0 0 0 0 0 0 0 0 0 0 0 0 14 0 l 0 0 0 0 0 0 0 0 0 0 0 9 1 1 O 1 0 0 0 0 0 0 0 0 0 !Block 80010000000000 #1 7 0 0 1 0 1 0 0 0 0 0 0 0 0 4 0 0 0 1 1 1 0 0 0 0 0 0 0

3 0 0 0 0 0 1 l 0 0 0 0 0 0 P = 2 0 0 0 0 0 0 1 l 0 0 0 0 0 Block

l 0 0 0 0 0 0 0 l 1 0 0 0 0 #2 5 0 0 0 0 0 0 0 0 1 1 0 0 0

6 0 0 0 0 0 0 0 0 0 1 1 1 1 1 O O O 0 0 0 0 0 0 0 0 1 0 0 Block 13 0 0 0 0 0 0 0 0 0 0 0 l 0 # 3 1~ 0 0 0 0 0 0 0 0 0 0 0 0 l

Figure 6 Optimal partition of P

Since the number of net cuts between any two consecutive blocks is now equal ONE, the modified block sizes m1 = 6 ,

m2 = 4 and m3 = 4 are the optimal solution. The partitioned nodes

of the spanning tree corresponding to the partitioned modules of the matrix fare given in Table 2.

Table 2 Optimal Partition of 14-bus Spanning Tree (k = 3)

Block Nodes

#I 4,7,8,9,11,14

#2 1,2,3 ,5

#3 6,10,12,13

The optimal partitioning obtained in Table 2 cuts 6 lines of the original IEEE 14-bus network, as shown in Figure 7, and yet the three partitioned subnetworks are all observable (i.e., every sub­network has a full rank subspanning tree as shown in Figure 6).

Block 1

14

6 9

Block 2

4

Table 3 IEEE 24-bus Spanning Tree

Branch From To Branch From To

1 1 6 13 8 22 2 2 3 14 9 15 3 2 24 15 9 II 4 3 22 16 JO 15 5 4 18 17 II 14 6 4 19 18 12 23 7 5 17 19 13 18 8 5 20 20 15 22 9 6 18 21 16 18

JO 6 20 22 17 23 11 7 24 23 19 21 12 7 13

Table 4 IEEE 30-bus Spanning Tree

Branch From To Branch From To

1 1 2 16 12 13 2 1 3 17 12 14 3 2 4 18 12 15 4 2 5 19 12 16 5 5 7 20 15 18 6 6 7 21 15 23 7 6 8 22 16 17 8 6 9 23 18 19 9 6 10 24 22 24

10 6 28 25 24 25 11 9 11 26 25 26 12 JO 17 27 25 27 13 10 20 28 27 29 14 10 21 29 27 30 15 10 22

The original and modified block sizes of the three networks are given in Table 6; this table shows that only the IEEE 24-bus network has maintained its block sizes as required, while the block sizes of the other two IEEE networks has to be changed to

Figure 7 Optimal partition of IEEE 14-bus obtain optimality (i.e., number of cuts is k - 1).

4. RESULTS AND DISCUSSION

In this section the performance of the proposed algorithm is evaluated by using three different IEEE standard networks: 24-bus, 30-bus, and 118-bus. The IEEE 24-bus network is to be partitioned into two subnetworks with block sizes of m1 = m2 = 12. The IEEE 30-bus network is to be partitioned into

three subnetworks with block sizes of m1 = m2 = m3 = JO. The

IEEE 118-bus network is to be partitioned into five subnetworks with block sizes of m1 = m2 = m3 = 24 and m4 = m5 = 23. We

allow changes in the size of any block, by removing or adding nodes, such that the maximum allowable difference of nodes between any two blocks of the IEEE 24-bus is ONE, IEEE 30-bus is TWO, and the IEEE 118-bus is FOUR.

The spanning tree connections of the three networks are given in Tables 3, 4 and 5, respectively. The partitioning technique pro­posed in this paper is applied, and the optimal partitioning of the three networks is obtained as discussed below.

701

As mentioned earlier, Otten's method gives a full screen of all possibilities a spanning tree can be partitioned into. It can be observed from Table 6 that while the IEEE 24-bus and IEEE 30-bus networks have only one possibility for partitioning, the IEEE 118-bus network can be partitioned into four different ways. The new block sizes of the IEEE 30-bus and 118-bus networks are within the prespecified limit.

The partitioned nodes of the IEEE 24-bus, 30-bus, 118-bus 1,

118-bus 2, 118-bus 3, and 118-bus 4 networks are given in Tables 7, 8, 9, 10, 11 and 12, respectively. One can test that the parti­tioned subnetworks are all observable.

Table 13 shows the number of branches and actual lines that have been cut after the optimal partition was obtained. The number of branch-cuts proves that the spanning tree of each of the IEEE networks has been partitioned optimally (i.e., number of branch cuts is equal to k - 1, where k is the number of partitioned blocks in each network). The number of line-cuts represents the actual number of lines that cut the original network. It can, also, be observed from Table 13 that although there are four different ways for partitioning the IEEE 118-bus network, the number of

Table 5 IEEE 118-bus Spanning Tree

Branch From To Branch From To 1 1 2 60 51 58 2 1 3 61 52 53 3 2 12 62 53 54 4 3 5 63 54 55 5 4 11 64 54 56 6 5 6 65 55 59 7 5 8 66 59 60 8 5 11 67 59 63 9 6 7 68 60 62

10 8 9 69 61 62 II 8 30 70 61 64 12 9 10 71 62 67 13 11 13 72 65 66 14 12 14 73 68 69 15 12 16 74 68 116 16 12 117 75 69 75 17 13 15 76 69 77 18 15 17 77 70 74 19 15 33 78 70 75 20 17 18 79 70 71 21 17 31 80 71 73 22 17 113 81 75 118 23 19 20 82 76 118 24 19 34 83 77 78 25 20 21 84 77 82 26 21 22 85 78 79 27 22 23 86 79 80 28 23 24 87 80 81 29 23 25 88 80 97 30 23 32 89 82 83 31 24 72 90 82 96 32 25 26 91 83 84 33 26 30 92 84 85 34 27 28 93 85 86 35 27 32 94 85 88 36 27 115 95 86 87 37 28 29 96 88 89 38 30 38 97 89 90 39 32 114 98 91 92 40 33 37 99 92 93 41 34 43 100 92 100 42 35 36 101 92 102 43 35 37 102 93 94 44 37 39 103 94 95 45 38 65 104 95 96 46 39 40 105 98 100 47 40 42 106 99 100 48 41 42 107 100 101 49 42 49 108 100 106 50 43 44 109 100 103 51 45 49 I 10 103 104 52 46 48 111 103 110 53 47 49 112 105 108 54 47 69 113 106 107 55 48 49 114 108 109 56 49 50 115 109 110 57 49 51 116 110 Ill 58 50 57 117 110 112 59 51 52

Table 6 Subnetwork Block Sizes

IEEE No. of No. of Nodes in No. of Nodes in

Systems Blocks Original Blocks New Blocks

24-Bus 2 12,12 12,12

30-Bus 3 10,10,10 11,10,9

118-Bus 1 5 24,24,24,23,23 21,25,25,22,25

118-Bus 2 5 24,24,24,23,23 21,25,24,23,25

118-Bus 3 5 24,24,24,23,23 21,25,23,24,25

118-Bus 4 5 24,24,24,23,23 21 ,25,25,25,22

702

line-cuts between the partitioned subnetworks happen to be the same. Table 14 shows the total CPU time required for partition­ing the three IEEE networks; computations are perfonned on a multiuser DEC Statation 5000/200.

Table 7 IEEE 24-bus Subnetwork Nodes

Block No. of Nodes Nodes

# l 12 2,3,7,8,9,10,11,13, 14,15,22,24

#2 12 1,4,5,6,12, 16, 17,18, 19,20,21,23

Table 8 IEEE 30-bus Subnetwork Nodes

Block No. of Nodes Nodes

#1 11 1,2,3,4,5,6,7' 8,9,11 ,28

#2 10 10,20,21,22,24, 25,26,27,29,30

#3 9 12,13,14,15,16, 17,18,19,23

Table 9 IEEE 118-bus 1 Subnetwork Nodes

Block No. of Nodes Nodes

#1 21 91,92,93,94,95,96,98,99, 100, 101,102,103,104, 105, 106, 107,

108,109,110,111,112

#2 25 68,69,70,71,73,74,75,76,77, 78,79,80,81,82,83,84,85,86,

87 ,88,89,90,97,116, I I 8

#3 25 39,40,41,42,45,46,47 ,48,49' 50,51,52,53,54,55,56,57 ,58,

59,60,61,62,63,64,67

#4 22 1,2,3,4,5,6,7, 11, 12, 13, 14,15, 16,17' 18,31, 33,35,36,37,113, 117

#5 25 8,9, 10, 19,20,21,22,23,24,25' 26,27 ,28,29,30,32,34,38,43,

44,65,66,72, 114, 115

Table 10 IEEE 118-bus 2 Subnetwork Nodes

Block No. of Nodes Nodes

#1 21 91,92,93,94,95,96,98,99, 100, 101, l 02, 103, 104, 105,106, 107'

I 08, 109,110, 111, 112

#2 25 68,69,70,71,73,7 4,75,76,77' 78,79,80,81,82,83,84,85,86,

87,88,89,90,97,116,118

#3 24 40,41,42,45,46,47,48,49, 50,51,52,53,54,55,56,57 ,58,

59,60,61,62,63,64,67

#4 . 23 1,2,3,4,5,6,7,11,12, 13, 14,15, 16,17' 18,31,

33,35,36,37,39,113, 117

#5 25 8,9, I 0, 19,20,21,22,23,24,25, 26,27 ,28,29,30,32,34,38,43,

44,65,66,72,114,115

Table 11 IEEE 118-bus 3 Subnetwork Nodes

Block No. of Nodes Nodes

#I 21 91,92,93,94,95,96,98,99, 100, 101,102,103,104,105,106, 107,

108,109,110,111,112

#2 25 68,69,70,71 ,73,74,75,76,77' 78,79,80,81,82,83,84,85,86,

87,88,89,90,97'116, 118

#3 23 41,42,45,46,47,48,49,50, 51,52,53,54,55,56,57 ,58,

59,60,61,62,63,64,67

#4 24 1,2,3,4,5,6,7' 11, 12, 13, 14, 15,16, 17 ,18,31 ,33, 35,36,37,39,40,113,117

#5 25 8,9, 10, 19,20,21,22,23,24,25, 26,27 ,28,29,30,32,34,38,43,

44,65,66,72, 114, 115

Table 12 IEEE 118-bus 4 Subnetwork Nodes

Block No. of Nodes Nodes

#I 21 91,92,93,94,95,96,98,99, 100, 101,102, 103,104, 105, 106, 107,

108, 109,110, 111,112

#2 25 68,69,70,71,73,74,75,76,77, 78,79,80,81,82,83,84,85,86,

87,88,89,90,97' 116, 118

#3 25 39,40,41,42,45,46,47 ,48,49, 50,51,52,53,54,55,56,57 ,58,

59,60,61,62,63,64,67

#4 25 1,2,3,4,5,6,7,8,9,10,11, 12,13, 14, 15, 16,17' 18,31,

33,35,36,37, 113, 117

#5 22 19 ,20,21,22,23,24,25,26 27 ,28,29,30,32,34,38,43,

44,65,66,72,114,115

Table 13 Number of Branch and Line Cuts

IEEE No. of No. of Systems Branch Cuts Line Cuts

24-bus 1 5

30-bus 2 7

I 18-bus1 4 29

l 18-bus1 4 29

118-bus' 4 29

l 18-bus4 4 29

Table 14 Total CPU Time for Partitioning

IEEE Systems CPU (sec)

24-bus 0.1

30-bus 0.2

118-bus 10.9

5. CONCLUSIONS

Partitioning an observable power-system-state-estimation (PSSE) network into two or more observable subnetworks is the first step required with any of the decentralized two-level state estimation approaches. This paper has developed a heuristic algorithm to obtain this type of partitioning. The proposed algorithm that parti­tions the spanning tree of an observable PSSE network guarantees the observability of all the subnetworks (i.e., there is no need to check for the observability of the partitioned subnetworks). The partitioning algorithm is based upon the Otten' s eigensolution approach. Otten ' s method provides a full screen of all possibilities a network can be partitioned into. The performance of this heuristic algorithm was evaluated by three IEEE PSSE network examples: 14-bus, 24-bus, 30-bus, and 118-bus.

REFERENCES

[l] F.C. Schweppe and E.J. Handschin, "Static State Estimation in Electrical Power Systems", IEEE Proceedings, Vol. 62, pp. 972-982, July 1974.

[2] T. Van Cutsem, M. Ribbens-Pavella, "Critical Survey of Hierarchical Methods for State Estimation of Electrical Power Systems", IEEE Trans. Power App. Syst., PAS-102, pp. 3415-3424, Oct. 1983.

[3] M.S . Kurzyn, "Aggregation Methods for Hierarchical static State Estimation in Large Scale Power Systems: A Survey", Proc. 9th PSCC, Cascais, Portugal, pp. 540-545, 1987.

[4] L. Holten, A. Gjelsvik, S. Aam, F.F. Wu, and W.H.E. Liu, "Comparison of Differents Methods for State Estimation", IEEE Trans. Power Syst., Vol. 3, No. 4, pp. 1798-1806, Nov. 1988.

[5] K.L. Lo, M.M. Salem, R.D. McColl and A.M. Moffatt, "Two­Level State Estimation for Large Power System", Part I "Algo­rithms", Part 2 "Computational Experience", IEE Proceedings, Vol. 135, Pt. C, No. 4, pp. 299-318, Nov. 1988.

[6] V.H. Quintana, A. Simoes-Costa, and A. Mandel, "Power Sys­tem Topological Observability Using a Direct Graph-Theoretic Approach", IEEE Trans. Power App. Syst., PAS-101, pp. 617-626, March 1982.

[7] K.A. Clements and B.F. Wollenberg, "An Algorithm for Obser­vability Determination in Power System State Estimation", IEEE PES Summer Meeting, San Francisco, CA, A 75 447-3, July 1975.

[8] G.S. Rowan and A. Vannelli, "An Eigenvector-based Approach for Multi-stack VLSI Layout", 29th Midwest Symposium on Circuits and Systems, Lincoln, Nebraska, Aug. 1986.

[9] R.H.J.M. Otten, "Eigensolutions in Top-Down Layout Design", International Symp. on Circuits and Systems, Rome, Italy, pp. 1017-1020, May 1982.

703