DISTRIBUTION SYSTEMS EXPANSION PLANNING BY MULTI-STAGE BRANCH EXCHANGE TAKING FAULT INTO CONSIDERATION

Koichi Nara Ibaraki University

Hitachi, Ibaraki 316 JAPAN

Hijiro Kuwabara Kinki University

Kure, Hiroshima 737-01 JAPAN

Taiji Satoh Yamaguchi Univ. Ube, Yamaguchi

JAPAN

Kenichi Aoki Hiroshima Pref. Univ.

Minoru Kitagawa Chugoku Electric Power Co. Ohno, Saeki-gun, Hiroshima

JAPAN Shobara, Hiroshima

JAPAN

ABSTRACT- The expansion planning problem in distribution system is an extremely complex combinatorial optimization problem if it takes faults into consideration. So, it is very difficult to find a practically usable solu­tion of the problem. In this paper, the au­thors propose a new algorithm that can take fault cases into account. The algorithm is based on the multi-stage branch exchange method, and the solution is rather ~ccurate when compared with an existing method. The proposed method can take all the predetermined severe fault cases into consideration simulta­neously, and can help in avoiding inst al la­tion of excess facilities without affecting the ability for fault restoration. The validi­ty and the effectiveness of the proposed algo­rithm are demonstrated by several numerical examples.

Keywords : Distribution system operation and control, Transmission planning method, Fault restoration

1. INTRODUCTION

With the rapid increase of electrical energy consumption, the distribution system expansion planning problem has become an important problem in electric power indus­tries. Many approaches for the problem have been developed and proposed[l-14]. However, the expansion planning problem in distribution system becomes an extremely complex combinato­rial optimization problem if it takes faults into consideration. So, it is very difficult to find a practically usable solution of the problem. In the distribution planning area, there had been no publication which explicitly takes fault cases into consideration until the authors proposed the method based on the single-stage branch exchange method[l4]. Al though, this method is highly efficient in finding a sub-optimal planning, the calcula­tion results are approximated ones, and even feasible solutions occasionally cannot be found through the method for real scale prob­lems. In the meantime, the authors also have proposed what has been called the multi-stage branch exchange method[l3] so as to find more accurate solutions than the single stage branch exchange algorithm.

In this paper, the authors propose a new algorithm based on the multi ~stage branch exchange that can take fault cases into ac­count. The calculation results are rather accurate when compared with those of the existing method. The proposed method can take all the predetermined severe fault cases into consideration simultaneously, and can help avoiding installation of excess facilities without affecting the ability for fault resto­ration. In other words, an expansion _plan is developed under which all load sections except the fault one can be supplied "l:_h!'ough the

fault state reconfiguration in any predeter­mined case. In the algorithm, the whole prob­lem is decomposed into sub-problems according to predetermined fault cases, and is effi­ciently solved by using the multi-stage branch exchange algorithm and the decomposition­coordination technique.

In Chapter 2, the problem is precisely defined and mathematically formulated. The detailed solution algorithm through multi­stage branch exchange method is shown in Chapter 3. The validity and the effectiveness of the proposed algorithm are demonstrated by several numerical examples compared with those of the existing method in Chapter 4.

2. DESCRIPTION OF THE PROBLEM

2.1 Illustration of the Problem In the open loop radial distribution

system, although many loops are constructed by installed facilities, the system is operated as a radial system by opening several section­alizing switches. Therefore, even in the case of a fault, power can be supplied to the black-out area by closing appropriate open sectionalizing switches.

The problem here is to . find such a mini­mum cost expansion plan that can supply power to all the load sections except that in which the fault exists, even in the event of any fault.

To solve the problem, if one of the existing methods is simply applied to every predetermined fault case independently, and all the facilities which are found to be necessary to supply power to all the load sections are installed, then the degree of redundancy of the installed facilities becomes large, and extra installation cost _becomes necessary. So as to avoid such extra cost, all the constraints of each fault case must be formulated simultaneously, and the problem must be solved in one lump .

2.2 Formulation of the Problem The distribution system model shown in

Fig.l is used to formulate the problem[9] . . In Fig.l, nodes show substations, transformers and load points. The branches represent elec­trical connections between nodes. Each branch has its impedance, current capacity and in­stallation cost . This model is introduced to each fault case to formulate the constraint equations. To formulate the problem, the following is also assumed:

( 1) Candidates for facilities to be in-stalled are known beforehand,

( 2) The load is a lumped constant current load of which the power factor is 1.0,

( 3) Fault cases which should be considered are pre-determined.

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Notations

• : source node - - - : potent.ial feeder

@ : transfonner node

~ : substation node

0 :load node

- :existing feeder

I

substation installation cost \

T31 I

~ feeder cost

~ -------T22 __ __.

---~-------0 /

T21

feeder :x------­installation1 cost -(fixed cost) -------

I I I

Fig.I Distribution system planning model

Then, the problem in which faults are taken into consideration can be formulated as the following linear mixed-integer programming problem[14].

[Objective function] m

Min. Zc =. 1:cjwj J=l

. [Constraints]

(a) Installed facility constraint

(b) Constraint for radial configuration m

( l)

( 2)

1:Yjt = n-1 (with connected graph) (3) j=l

(c) Power flow availability

+ I+ Myjt-Xjt -Sjt 0

- I-Myjt-xjt -sjt 0

(j=l,2, ... ,m

(d) Kirchhoff's current law

AtXt = D,

( t= 1, 2, ... , T

(e) Node-voltage constraints

+ ' V++ V+- O (vit-Vktl-Rjxjt +Sjt -Sjt =

- V-+ V-- 0 (vit-vktl-Rjxjt +sjt -sjt =

( t = 1,2, ... ,T, j = 1,2, ... ,m

(f) Current capacity constraints

- + IM+ IM+_o xrxjt -Sjt +a jt -

- - IM- IM- 0 xrxjt -sjt +a jt =

IM+_o IM- 0 a jt - , a jt ':' '

t = 1,2; ... ,T, j = 1,2, ... ,m

(g) Voltage drop constraints

v. t-v-s. tVM+a . tVM=O, l. - l. l.

VM · (ait =0)

t = 1,2, ... ,T, i = 1,2, ... ,n)

( 4)

( 5)

( 6)

( 7)

( 8)

(h} Nonnegative and integer constraint

+ - 0 xjt' xjt' vit' sjt• ajt~ '

YjtE{O, 1), wj€{0, 1}

( t = 1,2, ... ,T, j = 1,2, ... ,m) (9)

where, m Number of branches n Number of nodes including source node T Number of predetermined fault cases D Demand vector (element di represents demand at

node i) At (n-1) x m node-branch incidence matrix of fault

case t cj Installation cost of branch j;

cj=O for existing facility wj 0-1 Integer variable;

wj=l : if fac~lity j is installed W·=O : otherwise

Yjt: 0-1 in~eger decision variable; Yjt=l · : if facility j is used in fault

case t Y·t=O : otherwise

xjt:Power flow in branch j in fault case t; X·t = X·t+-X·t-

J tjt+:torward power flow x.t-:Inverse power flow

vit: Voltage at nod~ i in fault case t xj : Capacity of branch j R· Impedance of branch j yJ Allowable voltage drop M Large constant

I+ I- v++ v+- v-+ V-- IM+ IM­sj ,Sj ,Sj ,sj ,Sj ,sj ,Sj ,Sj •

sjVM : Slack variables

a ·=a .IM+ IM- VM • Violation variables J J , a j , Oj •

Total installation cost Small value to avoid the zero divide or multi­plication

3. SOLUTION ALGORITHM

3.1 Brief Explanation of the Algorithm The coefficient matrix of the equations of

the above formulation is schematically shown in Fig. 2, for the example of the 3 fault cases. Coefficients exist only in the shaded area. Since coefficients exist in limited places, we can apply the decomposition-coordi­nat1on strategy to solve the problem[l4]. That is, by hypothetically giving the installed facilities (or by determining the value of w . ), the whole problem can be decomposed into sdb-problems according to the fault case, and we optimize the each sub-problem by using the multi-stage branch exchange algoritlun. The

fault case 1·

fault case 2

fault case 3

facility installation cost

Min . Cil

Eq. ()I Eq . ( 9)

Eq. ( 2)

Fig.2 Schematic expression of the problem

132

whole solution algorithm consists of the following steps.

[Step 0)

[Step l]

[Step 2)

[Step 3]

[Step 4)

[Step 5)

Initial expanded branches (or used branches) are determined so as to construct an initial tree configu­ration, Multi-stage branch exchange for each fault case, Reduction of the sum of constraint violations of all the fault cases by single facility expansion, Reduction of the sum of constraint violations of all the fault cases by multi-facility expansion , Reduction of the sum of con­straint violations by using all candidate facilities , Reduction of installation cost by removing installed but unnecessary facilities.

The general flowchart of the solution algorithm is shown in Fig.3.

In every step of the above algorithm, the multi-stage branch exchange algorithm[l3] is employed. The multi-stage branch exchange is effective for escaping from a local minimum which single-stage branch exchange reaches. In this method, the second-stage branch exchange is employed as it is satisfactory from the tradeoff of the computational burden and the solution accuracy.

Remembering that the distribution system ordinarily is a radial system, the basic concept of the (single-stage) branch exchange can be stated as follows: from the initial radial configuration, a loop is constructed by adding one branch, and then the loop is brought back to a radial configuration by removing another branch from the loop. The second-stage branch exchange involves the following operations. After completing the single-stage branch exchange, when no further branch exchange can improve the objective function value, a branch exchange is forced to be executed and a new radial configuration is created. Of course, here, the objective func­tion value becomes worse. Then, again, another branch exchange is executed from the new radial configuration according to the branch exchange criteria. The resulting radial system

~ step o _c_o_n_s_t_r_u_c_t_i_n_i_t-ial treeconTi-g~1rati01~]

Step Multi-stage branch exchange for eac h I fault case __J

Step 2

~tep 3

infeasible feasible

Reduce sum of constraint violations by single faci I ity -~'i . .,,.P_a_ns_i_o_n ___ ~--~

infeasible

Reduce su11 of constraint violations by uulti-faci lily expansion

infeasibl e feasihle

Step 4 Reduce sum of constraint violations hy

Step 5

candidate faci 1 i ti es feasible

Reduce i nsta 1.1 al ion cnsL by removing i nsla 11 ed unnecessary far: i Ii li es

END

Fig.3 Flowchart of the solution algorithm

is accepted if the objective function value . is better than that of the initial radial config­uration. If this operation is executed suc­cessfully, then we can succeed in escaping from the local minimum. Of course, the higher stage may be executed in the same manner if necessary.

3.2 Detailed Explanation of the Algorithm Details of each step of the sol_ution

algorithm can be explained as follows.

Step O: Initial tree configuration Any initial tree configuratio~ _ma_Y be

selected by only using existing facilities or initially expanded ones . As this algorithm basically is an approximate solution metho~. a final solution depends on an initial solution . Therefore It must be noted that we have to apply thi's algorithm to several initial tree configurations, and have to select the best expansion pattern.

Step 1: Multi stage branch exchange for each fault case

The whole problem can be decomposed into sub-problems according to fault cases. The system configuration of each fault case is optimized independently by the multi-stage branch exchange method. Note that candid~tes for the multi-stage branch exchange are exist­ing facilities and facilities ".'hich : are as­sumed to be installed up to this point. The second-stage branch exchange is employed as the multi-stage branch exchange. The general flowchart of the employed second-stage branch exchange is shown in Fig·.4. In Fig.4, the

133

Uo

llo

No

START

Execute SINGLE-STAGE a Igor i thm for finding tree-I

For• a loop by adding the tie branch vh ich shows best

criteria to tree-I

Open the loop by re110v i ng the

branch vh ich shovs best

criteria and tree-2

is obtained

Construct a loop by aclding the branch which shows best

criteria to lree-2

Open the loop by re•o• ing the branch vh i eh sho\IS best er i ter i a fro•

the loop and tree-3 is obtained .

The objective of tree-3 is better than that of tree-I?

llo Predeter11 i ned number of

candidates for •ak ing a loop fro• tree-2 is checked?

Yes

Predeter•ined number of

candidates for ;oak i ng

tree-2 is checked?

Yes

A 11 the candidate branches

for constructing a loop

fro• tree-I are tried?

. Yes EllO

~ "'

..., " n ., =r ~ "' I x "' n rt =r "' .. .. " "' .. "'

Yes

Fig.4 Second-stage branch exchange

t;erms "tree-1" fol lowing: · '

"tree-2" and "tree- 3" mean the the fault cases and no candidate branch found in Step 2, then Step 3 is executed.

is

tree-1: radial distribution system configura­tion before branch exchange,

tree-2: radial distribution system configura­tion after first-stage branch ex­change,

tree-3: radial distribution system configura­tion after second-stage branch ex­change.

The details of the multi-stage branch exchange can be shown in reference[l3].

If no constraint violation exists after the multi-stage branch exchange, the distribu­tion system can be operated successfully in any fault case by using the existing facili­ties. In this case, after the sum of voltage margins is maximized, [Step l] terminates, and Step 5 is executed. If constraint violations still exist, Step 2 is proceeded with.

Step 2: Reduction of constraint violations by single facility expansion

It must be noted that, in this step, con­straint violations still exist at least in one fault case. To decrease the sum of con­straint violations in all the fault cases, a new facility is added to the existing system one by one. The criteria of the branch expan­sion are expressed as eq.(10)-eq.(12).

Let superscripts "tree• and tree configuration and loop respectively in the following.

"loop " denote configuration,

[Criteria for selecting one facility expansion]

(I) In the case of z1 loop=O

min.

where, T

Z tree_ ,.. z tree 1 -,<.u

t=l T

Z loop_ ,... z loop 1 -,<.u

t= l T

Ll Zo ]· = L Ll Zto ]. . t= l .

T

sum of constraint the existing tree

(12)

violations of configurations

sum of constraint violations after constructing a loop

sum of deviations of voltage margin after constructing a loop

LlZ1 j= ELlZu j: sum of ' t=l ' violations

deviations of constraint after constructing a loop

One branch is selected for installing according to criteria (I) (III). After assuming the selected branch to be installed, Step 1 is carried out in each fault case. After executing Step 1, if we succeed in eliminating or decreasing the sum of constraint viola­tions, installation of the selected faci 1 i ty is decided upon ; otherwise, it is rejected. If we succeed in eliminating the constraint violations in all the fault cases, Step 5 is performed to decrease installation cost. Otherwise, the next candidate branch is selected according to criteria (I)-(III), and the same steps are repeated. If we fail to eliminate the constraint violations in some of

Step 3: Reduction of constraint violations by multiple facility expans i on

In this step, constraint violations still exist because they cannot be eliminated by the single facility expansion procedure in [Step 2]. Then, the following is tried.

Two branches, which are selected in turn according to criteria (I)- (I I I), are assumed to be simultaneously installed into the dis­tribution system. By utilizing the two new facilities, multi-stage branch exchange (Step 1) is carried out for each fault case so as to eliminate the constraint violations. After this procedure, if the sum of constraint violations is eliminated or decreased , instal­lation of the two facilities selected is decided upon; otherwise they are rejected. In case no constraint violations exist in any fault case, and the sum of voltage margins cannot be further improved upon, then, [Step 5] is carried out . Otherwise, the two

next best branches are selected, and this procedure is repeated. If the sum of con­straint violations cannot be eliminated in this step, then [Step 4] is performed.

Step 4: Reduction of constraint violations by using all candidate facilities

Even here, at least one fault case exists in which the constraint violations still remain. In this step, it is assumed that all the facilities are installed into the distri­bution system so as to eliminate the remaining constraint violations. Under this assumption, the multi-stage branch exchange is independ­ently carried out for the fault cases in which constraint violations still exist.

Since all the facilities are assumed to be installed, constraint violations may be elimi­nated in all the fault cases. Then, the facil­ity expansion pattern for each fault case may be obtained. The final facility expansion pattern is composed of the logical sum of the facilities which are used in at least one fault case. If a feasible expansion pattern is found , [Step 5] is executed.

If constraint violations any case even at this stage, failed to find a feasible algorithm terminates here.

still remain in the algorithm has solution and the

Step 5: Reduction of the installation cost Up to here, a feasible expansion planning

in which al 1 fault cases are taken into ac­count is obtained by installing several facil­ities in the preceding steps. However, we can reduce the installation cost by removing unnecessary facilities or by replacing the expensive facilities with cheaper ones, pro­vided these operations do not cause any con­straint violations.

If an installed facility that is not used in at least one fault case exists, then the multi-stage branch exchange is carried out to eliminate this facility. To execute the step, the multi-stage branch exchange is carried out for every fault case by temporarily setting the cost of the selected branch somewhat higher, and that of other newly installed facilities somewhat lower. Since a feasible solution has already been found, constraint violations must not be created in the branch exchange procedures.

After the branch exchange, if the result-

134

ing cost is lower than the previous one, then the new installation pattern is employed. This procedure is repeated until no candidat~ facility can be found.

The resulting expansion planning is the sub-optimal one which we intended to find.

4. NUMERICAL RESULTS

The proposed algorithm is tested on FACOM­S3500 computer(4MIPS) by applying it to the 6.6 KV, 59 node, 69 branch distribution system shown in Fig. 5 (a). In this figure, solid lines show the initial tree configuration of the example system, and symbol "o" on the branch denotes installation candidate facili­ties. Other notations in this figure are the same as those in Fig . l. In Fig. 5 (a), solid lines with "a" are the facilities which are initially installed to construct the initial tree configuration. Initial installation cost of these 25 facilities is 1268.2[~:hypotheti­cal money unit]. The topological confi_guration of the initial system is the same as that of reference[l4]. However, so as to demonstrate the effectiveness of the proposed method, section loads in selected ten nodes are some­what increased to make the problem difficult to solve. The total load in the system is 1505.0[A] in this example, whereas 1375.0[A] in reference [14].

The calculation processes started from this initial tree configuration and the calcu­lation results are shown in Table 1 for three test cases. In Case 1, four fault cases are pre-determined for this system . They are one non-fault case and three transformer fault cases: #5, #6 and #58. In Case 2, five fault cases are incorporated; one non-fault case and four transformer fault cases: #3, #4, #5 and #58. In Case 3, the non-fault case and every transformer fault, i.e. #3, #4, #5, #6 and #58, are assumed. The last line of Table 1 shows the calculation time for each case. In order to demonstrate the effectiveness of this method, the calculation results given by the single-stage branch exchange method[l4] are shown in Table 2.

By comparison of Table 1 and Table 2, we can easily find that fat Case 1 and 3, a feasible solution cannot be derived through the single-stage method(Table 2), whereas a feasible solution can be found using this method( Table 1) al though calculation burdens are several ten times larger than those of the other method.

The authors have independently applied the multi-stage branch exchange algorithm[l3] to all the single transformer fault cases (non­fault case is included and the condition is the same as Case 3) of the system shown in Fig . I. From the calculation results, by making the logical sum of tne installed faci 1 i ties for each fault case, the authors found that the total installation cost is 1972.6(~]. This result can be compared with the calculation result of Case 3. Then, we can al so eas i 1 y find that the resulting cost of this method (1724.5[~]) is 248.1[~] cheaper than that of the above installation cost (1972.6(~]). This advantage comes from the fact that the all fault cases are processed simultaneously in one lump in this method.

For each of the tree cases in Table l, the calculation processes can be explained ·step by step as follows. For every case, in Step O or for the initial tree configuration, the number of newly installed facilities is 25 with the initial cost of 1268.2[~], and the sum of constraint violation is 2088.0.

[Case l] In Step 1, by applying multi-stage branch

exchange to the initial system, constraint violation is reduced to 1618. 0. To eliminate this violation, 4 branches are added in Step 2. Although the installation cost has in­creased to 1497.1, the violation is reduced to 13. 9. No further reduction of the violation can be achieved by multi-stage branch ex­change, and Step 3 is carried out. However, no improvement can be achieved in Step 3. In Step 4, 4 more branches are added, and the feasible solution can be found. The · installation cost here, 1895.5[il is reduced by removing 4 branches in Step 5, and the final feasible solution with the cost 1164.7 [~] is given.

Table 1 Numer i cal re sults of this algorithm

Case 1 Case 2 Case 3 Step E. F. c. v. Cost E. F. c. v. Cost E. F. c. v. Cost

0 25 2088. 0 1268 . 2 25 1760. 0 1268. 2 25 30 I 0. 0 1268. 2 I 25 1618 . 0 1268.2 25 812 . 0 1268 . 2 25 2062 . 0 1268. 2 2 29* 13. 9 14 79 . I 30* 0. 0 1556 . I 32* o. 0 1724.5 3 29 13 . 9 1479. 1 - - - - - -

4 3 3* o. a 1895. 5 - - - - - -5 29 o. 0 1164 . 7 29 o. 0 1536. 6 32 o. 0 1724. 5

CPU 25m - 55s 19m - 20s 40m - 18s Note: E.F . Total number of newly installed fac1l 1t1es ,

C. V. Sum of cons t raint viol a tions, CPU Co mputation t ime . "*" in E. F. column means that multi - stage branch exchange has effectively ~· orked in the step. •

Table 2 Numerical results of the me t hod in reference [ 14]

Case l Case 2 Case 3 Step E. F. C. V. Cost E. y. C. Y. Cost E. F. c. v. Cost

0 25 2088. 0 1268 . 2 25 1760 . 0 1268 . 2 25 30 I O. 0 1268 . 2 1 25 1618~ 0 1268 . 2 25 812 . 0 1268 . 2 25 2062. 0 1268 . 2 2 30 23 . 0 165 1. 8 31 o. 0 1764. 0 32 3. 0 1724. 6 3 30 23 . 0 1651. 8 - - - 32 3. 0 1724 . 6 4 31 23 . 0 1760 . 8 - - - 32 3. 0 1724. 6 5 31 ?/23/:D. 1760. 8 29 0. 0 1617. 0 32 :;:;rn:;: :o 1724 . 6

CPU OOm - 40s 03m - 23s OOm - 54s Note : Notations E. F., C. V. and CPU mean the same as those of Table 1.

135

(a) Initial configuration

Note Branches marked by "•" are the installed ones

(b) Calculation result by this algorithm

Note Branches marked by "•" are the installed ones

(c) Calculation result by reference[l4]

Fig.S System configurations of example systems

136

[Case 2] In Step 1, the · constraint violation is

reduced to 812.0, and it is eliminated by adding 5 branches in Step 2. Since the feasi­ble solution is found in Step 2, Step 5 is proceeded to without processing Steps 3 and 4. In Step 5, one branch is deleted from the system, and the final cost is reduced to 1536.6[~]. This cost can be compared with the cost 1617. 0 [~] of Case 2 in Table 2, since the feasible solution is found in this case. The final topological system configurations of these two cases are shown in Fig. 5 ( b) and (c). From these figures, it can be seen that the cost difference of these two cases mainly comes from the fact that unnecessary trans­former #4 is inevitably installed through the method proposed in reference [14].

[Case 3] All the constraint violations are elimi­

nated in Step 2 by installing 7 branches. Installation cost is 1724.5[~] which cannot be reduced even in Step 5 in this case.

In Case 3, computation time is twice as large as that of Case 2. This is because much computation burden is necessary in the multi­stage branch exchange procedures for eliminat­ing the constraint violations in the fault case of transformer #6. In Table 1, t~e typi­cal numerical results in which the multi-stage branch exchange is effective for finding the feasible solution are shown. It must be noted that we cannot necessarily find a better solution by the multi-stage branch exchange than that by single-stage branch exchange in most of the cases. In such a case, of course, the calculation result is the same as that through single-stage branch exchange although more calculation time is necessary for the multi-stage branch exchange processes.

5. CONCLUSIONS

In this paper, an expansion planning algo­rithm for distribution systems in which multi­ple fault cases are taken into account is pro­posed. The solution algorithm is developed by applying the decomposition-coordination method and the multi-stage branch exchange method. Through the multi-stage branch exchange meth­od, the accuracy of the calculation results is fairly improved. From the numerical example,

it can be found that the result of the pro­posed method is far better than that of the existing methods. Our task for the future will be to expand this algorithm to include multi­year expansion planning.

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[4]

[5]

T.Gonen and B.L.Foote : "Distribution-System Plan­ning Using Mixed-Integer Programming", IEE Proc., Vol.128, Part C, No.2, March (1981) D.I.San, D.R.Farris, P.J.Cote, R.R.Shoults and M.S.Chen : "Optimal Distribution Substation and

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137