dynamic, branch exchange method for optimal system planning

G.J.Pebonis M.P. Palpadopoulos

Indexing ierms: Distribution sy.stem plunning, Brunch exclzung6: Dynumic progumming

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Absdract: The optimisation of distribution system planning constitutes a combinatorial

The application of accurate methods

is faced in two steps: first, a set of eco omical and technically accepted system con igurations is constructed: second, the optimal seq ence of system configurations, for a mu1 iyear study period, is determined based on

i Y. the \ set of economical configurations. The

osed method is fully dynamic, models

seasions, are used to model load variations, while the I installation or construction of capacitors, volt ge regulators, lines and substations is opt iLised.

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1 Inproduction

The prioblem of optimal planning of radial distribution system1 can be briefly described as the selection of a sequence of system configurations for successive years, minim sing the total cost while satisfying the opera- tional lconstraints. Since 1980, the problem has been formullated as a large scale nonlinear mixed integer progra ming problem. However, because of its size and CO plexity, only systems of small size can be faced using F hese formulations [ 1-41. Heuristic methods have been presented [4-61 which are intended to find an eco- nomicil sequence of configurations that satisfy the operat onal constraints for the study period. Although these 1 usually branch exchange) methods are more effectiye when applied on systems of realistic size, they

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are pseudodynamic and incorporate other simplifying assumptioins that reduce their efficiency.

A heuristic switch exchange method was presented in [7, 81 for the optimal reconfiguration of distribution networks, for energy losses reduction and load balanc- ing. In the present paper, a dynamic, branch exchange method for the optimisation of multiyear distribution system planning. is presented. The problem is solved in two steps: (i) the ‘static subproblem’ is addressed. Based on the method presented in [7, 81, a set of network configura- tions, which present economical and technically accepted operation for different one- or multiyear sub- periods of the total period of analysis, is constructed. (ii) the ‘dynamic subproblem’ is solved. Based on the set of economically selected network configurations, the optimal sequence of configurations for the total analysis period is determined using dynamic program- ming techniques.

Load arid other system data are accurately taken into account in the proposed method. The most common alternatives of distribution system reinforcement are modelled and optimised. The proposed method can be applied very efficiently on real size systems as shown by the applications presented herein.

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The objective of the optimal distribution system plan- ning problem is to minimise the total cost during the total study period considered. The constraints of this optimisation problem are as follows: (i) line and trans- former current capacities, (ii) voltage drop at load points, (ii~i) demandisupply balance (for the existing loads and the new ones). and (iv) radial operation. The elementary system of Figs. 1 and 2 is used to clarify the principles of the method proposed.

Principles of the proposed method

Fig. 1 Simple distribution network: existing network

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Fig. 2 tives

Simple distribution network: reinforcement and extension alternu-

The reinforcement and extension options, which can be used to feed the increased existing loads (e.g. loads 1,2,3) and the new ones (e.g. load 4) during the study period (e.g. 5 years), are illustrated in Fig. 2. Practi- cally, these options are: (i) reinforcement of the existing lines and substations, (ii) installation of shunt capaci- tors, (iii) installation of voltage regulators on the main lines, (iv) construction of new lines (e.g. 2-3 or 3 4 ) , or (v) construction of new substations (e.g. I1 or 111) and lines (e.g. 11-2).

In the proposed method it is considered that the expansion alternatives are known beforehand. Also, that the load data (load patterns describing load varia- tions per load type, growth rates of the existing and new loads etc.) are also given.

The problem of optimal planning can be solved by decoupling it in the following subproblems: a Static subproblem: The most economical network configurations that satisfy the technical constraints, for several different subperiods of the study period, are determined. First, the most economical configurations that satisfy the technical constraints for the first year are determined. Similarly, a number of economical con- figurations are determined for the 2nd, the 3rd etc., years. Multiyear subperiods may be also considered. These subsets of selected configurations, denoted as A,, if they have been constructed for the sequential years from n to m, are illustrated in Fig. 3. The union of these subsets is the set A of economical configura- tions.

Fig.3 Configurations subsets A,_m: they resent low total cost for the seauential vears n-m, while thev satisfv tL ouerational constraints for " _

t h s e year; Union of subsets is the set of economical configurations A

b Dynamic subproblem: Given the set A of economical configurations, the optimal sequence of configurations for the study period is defined.

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Prior to the solution of the dynamic subproblem, it is desirable to classify the selected configurations in sub- sets of technically accepted operation B,. Each subset B, includes the configurations that can feed all the loads until year m, without operational limits viola- tions. It is clear that A , C B,, while it is probable that A, n Bm+k # @. If there is an increase of the loads during the study period, it is Bm+,, C B,. The set of selected configurations, classified in the subsets Bm illustrated in Fig. 4, is the result of the static subprob- lem solution.

Fig. 4 B,,, of technicaily accepted operation Members of B, satisfy the operational constraints until year m

Clussij'kation ofthe selected configurations of set A to the subsets

3 Static subproblem solution

3. I Modelling of expansion candidates as branch exchange operations

3.1.1 Modelling of reinforcement candidates: The solution of the static subproblem is based on the modelling of all network reinforcement candidates as branch exchange alternatives, as follows: (i) The alteration of the characteristics of an existing branch (e.g. change of the type of line conductors or the size of transformers) can be easily modelled by an adequate branch exchange. The branch with the new characteristics is installed, and the existing branch (par- allel to the new one) is removed, so that radiality is maintained. (ii) The construction of new lines is modelled by the installation of the new line, and the removal of an existing one on the loop formed (e.g. in Fig. 2, if line 3-2 is installed, one of the lines 1-1, 1-2 or 1-3 should be removed or set out of operation). (iii) The installation (or change of size) of shunt capac- itors can be also modelled by an adequate branch exchange: If a new capacitor is installed, the corre- sponding branch (with the adequate technical and eco- nomical characteristics) is installed. A pseudobranch, that is, a branch that is not affecting the technical and economical evaluations, is removed. The change of the size of a capacitor is modelled by the exchange of the state of the corresponding branches. (iv) Likewise, the installation of a new voltage regula- tor is modelled by the installation of the corresponding branch, and the removal of a pseudobranch (or a branch modelling an existing regulator). (v) The construction of a new substation is modelled by the installation of two branches, corresponding to the transformer (e.g. I1 in Fig. 2) and the line connecting the new substation to the existing network (e.g. 11-2). A branch on the loop formed is removed (e.g. 1-1 or l- 2).

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In conclusion, all the reinforcement alternatives can be modelled by the installation of one or two new branch&, and the removal (or set out of operation) of one existing branch on the loop formed. Other costs (line departure from HV/MV substation, substation land and buildings etc.) can be related to these of the installeg or removed branches.

3.1.2 hodelling of extension candidates: New loads &ay appear during the study period, and they should lbe connected to the existing network. Fast heu- ristic algorithms can be applied to detect a path from a new load to a network node. Such an algorithm can be based on the exploitation of the tree with root at the point of the new load, until a network point is reached. Starting from the root, the new branch candidates con- nected to this node are specified; continuing from these branchps the paths following connection candidates are specifiqd up to a point of the existing network. This connection path can be further studied by considering any other connection alternatives as reinforcement can- didates.

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3.2 Branch exchange method (BEM) As it qas been analysed, all network reinforcement and extension candidates can be modelled by corresponding branch exchange operations. Following this, the solu- tion of the static subproblem can be based on a modifi- cation of the switch exchange method (SEM), used for the odtimal reconfiguration of distribution networks [7, 81. This modification is denoted as branch exchange methoql (BEM).

3.2. I iObjective function of BEM: The objective functidn of BEM includes the capital cost, together with tde operation and maintenance (O/M) costs, apart from the energy losses cost included in SEM. The cost of removal of existing branches is also included. More analytically, the total cost change, for each branch exchange alternative, includes the following: (i) Changes in the cost of energy losses. The formula establighed by Civanlar, et al. in [9] can be used for the fast calculation of the power losses changes caused from Bach branch exchange alternative. Changes in energy: losses, for variable loads, can be calculated by the prjoper summation, as described in [7]. A special case isi the installation of capacitors, in which the cur- rent flbws change on all the branches to the path from the capacitor position to the HV/MV substation. In this case, changes in power and energy losses must be calculated for these branches [8]. (ii) TJe fixed annual cost of the installed branches, and the aacompanying equipment. This cost includes the capital cost and the O M cost, which are considered to be copstant. The capital cost is taken into account using fixed annuities. (ii) The cost of removal of an existing branch. This cost depends on the number of remaining years of the expected useful life of the corresponding equipment: If the remaining years of useful life are many, the removed equipment should be reinstalled, if this is pos- sible. In this case the cost of removal includes the removal and installation labour costs; the removed equipment should not be reinstalled if the remaining years are only afew, or if this is not a practical alterna- tive (e.g. in the case of conductors). In this case the cost of removal includes: (a) the annuities for the

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remaining years of useful life (reduced to the year of removal), (6) the cost of the earlier payment of the removal labour cost, and (c) the profit from the earlier receiving of the remaining value. The increase of the remaining value (due to the earlier removal) is usually small and is not taken into account.

The cost of removal of an existing piece of equip- ment is the minimum of the above costs, depending on the remaining years of useful life.

A special case concerns equipment that is removed in the nth year of the study period, while it is not present at the initial configuration (in year 0), that is it has been installed during the years from 1 to n-1. Taking into account that the study period is substantially smaller than the expected useful life of all pieces of equipment (30 to 40 years), we accept that the cost of removal of this equipment, is calculated using the first way, that is, it is expected that it will be reinstalled.

3.2.2 Algorithm of BEM: The algorithm of BEM for cost reduction is similar to this of SEM for losses reduction. An extension is needed so that operations requiring the installation of two branches (i.e. in the case of a new substation and a connecting line) are taken into account. Using the following algorithm, all single branch exchange alternatives (considering one branch coming into operation and one coming out) are examined, for all the different combinations of HV/MV substations. (i) Starting from an initial configuration that feeds all the loads with no limits violations, the installation of a new branch (line, capacitor or voltage regulator) and the removal of another on the loop formed is exam- ined. The alternative that leads to the maximum cost reduction without operational limits violations, is defined. The set of these actions is called step of the method. Sequential steps, until there is no alternative leading to cost reduction, provide an economical and technically accepted radial configuration. Up to this point the algorithm of SEM is used. (ii) All the combinations of HViMV substations are examined successively (step I), and the adequate con- figurations of minimum cost are defined. Since the cost of removal of an initial (existing in year 0) substation is usually high, the combinations that require such a removal, may not be examined. (iii) All the sets of sequential years from n to m are examined successively (following steps 1 and 2).

Briefly, the algorithm proceeds by searching all the line, capacitor and voltage regulator network reinforce- ment ‘policies’, for all the substation policies. The con- figuration of minimum cost is defined for each substation policy, for each set of years from n to m.

3.3 BEM for load balancing, analytical BEM (ABEM) When the construction of a subset An-m is examined, an initial configuration satisfying all the operational con- straints for the year (m) has to be defined, as a starting point for the application of BEM.

The determination of a technically accepted starting Configuration for a specific year can be achieved using a modification of SEM for load balancing [7]. This is based on the experience that network reconfiguration for load balancing, improves all operational indices

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(losses, voltage quality, maximum voltage drop, mean branch current flows). Using branch exchange opera- tions, for the reduction of load balancing index (LBI, [7]), an accepted initial configuration can be defined for a specific year. After each branch exchange, the opera- tional constraints are examined; if they are satisfied, the application of the method stops.

At the end of the static subproblem solution a set of economical configurations should be constructed. This set must include more than one configuration for each substation policy, for each set of sequential years.

This can be achieved by using an extension of the Analytical SEM (ASEM in [7]). At each step of the corresponding analytical branch exchange method (ABEM), the A4 technically accepted configurations with the lowest cost are stored and used successively as starting points at the next step of the method. The M more economical configurations, for each substation policy, are finally defined, for each set A,,,.

3.4 subproblem solution The general algorithm for the construction and classifi- cation of the economical configurations set, can be briefly described as follows: (i) Start: definition of the existing configuration (in year O), the load growth rates and the new forecasted loads, and the reinforcement and extension alternatives. Definition of the sets of sequential years (from n to nz) that the economical configurations subsets (An-m) will be constructed for. Continue with the first set of sequential years, and the first substations combination. (ii) Determination of an initial configuration that feeds all the loads of year (m), without operational limits vio- lations: (a) Determination of a radial configuration that feeds all the loads of the year. (b) Reinforcement of the network if voltage or cur- rent flow limits are violated. If a configuration that satisfies all operational constraints can not be found, go to step (iv) (next substation policy).

(iii) Application of ABEM for the determination of the A4 more economical (and technically accepted) configu- rations (for the given substation policy, for the sequen- tial years n-m). (iv) If there is another substations combination, go to step (ii). (v) If there is another set of sequential years n-m, go to step (ii). (vi) At this point the set of selected configurations A has been constructed, as the union of the subsets An-,n (Fig. 3). Classify the selected configurations to the sub- sets B, (Fig. 4).

General algorithm of the static

4 Dynamic subproblem solution

The dynamic subproblem can be modelled as a dynamic programming problem, since it is required to determine an optimal policy, consisting of optimal deci- sions in sequential stages. Using dynamic programming terminology [lo]. we can define the following: (i) Stage: Each year of the study period (ii) State: The state in each stage k (year k) is defined as the initial configuration at the beginning of the year.

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(iii) Decision: The choice of a configuration for the stage-year. Decision 6, is the state of the next stage k + 1. (iv) Constraints: The decision-configuration in stage k must satisfy the operational constraints for year k. (v) Policy: A set of decisions for all the stages. (vi) Optimal policy: The policy that minimises the total cost.

The objective function of the problem is

c = x{[capital cost](bk) + [O/IM cost](s,) N

,=l + [losses costI(6k) + [removal cost](6k-I,6k)} (1)

where N the number of years of the study period. This function is cumulative of the cost in each stage.

Reliability cost is not taken into account since it does not vary much for different configurations, while extended data for the protection co-ordination are required for its calculation [8].

The dynamic programming solution requires the validity of the Markovian property: given the current state in each stage, an optimal policy for the remaining stages should be independent of the policy adopted in previous stages. This is valid, since: (i) OiM and losses costs are related only to the decision

(ii) If the capital cost for a piece of equipment is calcu- lated using fixed annuities, it can be easily proven that the nominal value of each annuity is independent of the year of investment. (iii) According to the analysis presented in Section 3.2, the cost of removal in year k of equipment installed during the years 1 to k-1, is independent of the year of installation.

The cost in each stage (given the state c3k-l, and the decision Sk) includes the optimal cost of the next stages, resulting from the decision 6/c(Cl+1 (a,>): Ck(bk-I,bk) = {[capital cost](bk) + [O/M cost](bk)

+ [losses cost](bk) + [removal cost] (&- I , 6k) + Cl,, ( 6 k ) )

sk.

(2) If the cost resulting from each decision SIC has been

calculated for each state the optimal decision 6 ;(dk-J and the optimal cost C*, (dk-l), can be defined. It is noticed that there is only one state in stage 1 (the initial configuration in year 0). Also, that in the last stage (year N) the optimal cost of the next stages is zero (C>+l (6,) = 0). The decision in each stage (k) can take values from the subset Bk of configurations that present technically accepted operation for year k. The construction of the subsets Bk reduces the decision alternatives in each stage.

Since the dynamic subproblem has been modelled in terms of dynamic programming, the optimal policy providing the minimum total cost can be defined, by the backward and forward examination of the stages Dol.

5 Applications

5. I Applications system, planning data The proposed method is applied for the optimisation of the expansion of the typical overhead system illustrated

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in Fig. 5. The system consists of three distribution lines, departing from two HViMV substations. The 134 nodes system is presented under scale (the length of the main lines is: I-a6 36km, I-b8 43km, 11x7 44km). The existing system is illustrated by the continuous lines, while the 43 reinforcement and extension candidates are illustrated by dashed lines (there are several type or size alternatives for each construction or installation decision).

HV/MV substation

- -__ new line _ _ _ _ =f new capacitor

,,*$ new voltage .__ regulator

.- - - - -. conductor type change 3' new HV/MV substatlon

Fig. 5 tions system

Existing con$guvation and expansion candidutes of the apphca-

The expansion candidates include: (i) construction of new lines for the connection of new loads (e.g. line a6-a6-2) (ii) capacitors installation (e.g. at node c6) (iii) voltage regulators installation (e.g. at node c4) (iv) construction of interconnections for load transfers (e.g. line a6-1-b8-9) (v) reinforcement (conductor type change) of existing lines (e.g. of line c k 5 ) (vi) construction of new departures from the existing substations (e.g. line I-b6) (vii) construction of new HV/MV substations and con- necting lines to the existing network (e.g. at position

The positions of capacitors and voltage regulators installation are selected, based on the application of adequate static optimisation methods [8].

Load characteristics and variation are defined using five load types (residential, industrial, etc.), and two load patterns for each load type (for two seasons, defined by 12 points per pattern). The rate of increase of the loads is defined separately for each load type, and each year of the study period (the mean increase rate is 8.5%). The new loads are considered to be known (by size, type and year of installation which is indicated in parenthesis in Fig. 5).

The minimum voltage limit is considered to be 90% of the nominal. The discount rate is 8%. The study period is 10 years.

IV) .

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The following applications are presented: optimal system planning using all the expansion alternatives, sensitivity analyses, and examination of the contribu- tion of capacitors and voltage regulators.

5.2 Optimal system planning using all the expansion alternatives During the: solution of the static subproblem, the 12 most economical configurations for each substations policy are stored. Subsets A,, of economical configu- rations for one year are created, that is, the most eco- nomical configurations for more than one year (belonging to the subsets etc., Fig. 3) are not determined. Storing more configurations for each sub- stations policy does not affect the results. Static sub- problem solution results to the determination of 57 1 economical configurations, classified to the subsets B, of technically accepted operation (Fig. 4): 80 configura- tions belong to subset BIo, 94 to B9, 118, 118, 190, 300, 344, 382, 476, 571 to subsets B8 to B,, respectively.

The optimal expansion of the system, as defined after the solution of the dynamic subproblem, is illustrated using Fig. 6. In this Figure the configuration of the system at the end of the study period is presented, while the optimal new constructions are illustrated using bold characters (all the conductors are of the ACSR type, copper equivalent). The year of each con- struction is indicated in parenthesis (all the construc- tions are supposed to be made at the beginning of each year).

HV/MV substation

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HV/MV substation

600 CkVA(1) reconnection

(6) Fig. 6 New construcr:ions are illustrated in bold characters Year of construction is indicated in parenthesis

Optimal system expansion for the study period

The main proposed action is the construction of a new HVIMV substation (at site IV) in year 6. Capaci- tors are also installed on the three lines and their size is altered daring the study period. An interconnection is also constructed for the transfer of loads from line B to line A, in the fifth year. New lines are constructed for the connection of the new loads, the corresponding year. The total cost for the study period is 6.877 million ecu. 7.2% of this, is due to the energy losses.

The minimum annual voltage of the system, and the annual energy losses, are illustrated in Fig. 7, for the total study period. It is clear that the installation of the capacitors and the construction of the interconnection (which operates only for one year) are intended to

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postpone the construction of the new substation, by keeping the minimum voltage of the system over 90% until the 6th year.

,\”

E

E

5- ._ C .- >

98

x

0 1 2 3 1 5 6 7 8 9 1 0 year

Fig. 7 __ minimum voltage, % 0-0 energy losses, MWh

Operation indices changes during the study period

The proposed method has been coded using C lan- guage, and the computer program is executed on a low cost PC (IBM compatible, with a 486 processor at 80MHz). The execution time for this application is about 3.5h, 85% of which is required for the construc- tion of the set of economical configurations (static sub- problem solution). The solution of the dynamic subproblem is much faster.

5.3 Sensitivity analyses These analyses can be performed by solving only the dynamic subproblem. This is due to the fact that small changes in the data of the problem are not expected to affect strongly the selection of the economical configu- rations. This is validated through several applications, and a general rule of thumb can be expressed: the big- ger the changes required for sensitivity analyses (high data uncertainty), the wider the economical configura- tions set that should be constructed. For this reason, at the first application the 12 best line, capacitor and volt- age regulator policies have been stored for each substa- tion policy, although the same results are obtained by storing the nine best configurations.

If the discount rate value is 12% (instead of 8%) it is expected that some investments will be postponed. This is confirmed after year 7, since the investments until year 6 are the minimum possible that satisfy the opera- tional limits. The capacitors are removed aftei- the installation of the ncw HViMV substation (year 6) and they are reinstalled in year 9.

If the annual growth rate of the small industrial loads is 11% (instead of 8.9% mean value in the first application), the installation of the new substation is required in year S (one year earlier). The secondary actions (capacitors installation and load transfers) are also required one year earlier.

5.4 Examination o f the contribution o f capacitors and voltage regulators If the installation of capacitors is not taken into account, the new substation is again constructed in year 6 (at the same site IV). Until this year, line B is reinforced in the 2nd year, by constructing a new departure from substation I that feeds part of the loads of line B (new line I-bS, branch b 4 b 5 is set out of operation). After the installation of the new substation (year 6) the same part of line B is connected to the new substation. A voltage regulator is installed on line C (at node c4, in year 1). This is removed in the 7th year, and the loads after node cS are connected to the new substation. The total cost of this planning scenario is 7.068 million ecu.

Furthermore, if the installation of voltage regulators is also not taken into account, the new substation (IV) should be constructed in the second year. No further reinforcement actions are suggested. The total cost of this solution is 7.358 million ECU, that is 7% higher than this of the optimal expansion employing all rein- forcement alternatives.

It can be concluded that the suggested solution is seriously affected by the modelling and optimisation of the installation of capacitors and voltage regulators.

6 Conclusions

A new heuristic method for the optimisation of MV distribution systems in short and medium range plan- ning is proposed in this paper. This method is based on decoupling the problem to the static and dynamic sub- problems, that is, (i) the construction of an extended set of economical (and technically accepted) configura- tions, that might be selected to be included into the optimal solution, and (ii) the determination of the opti- mal sequence of configurations for the total study period, based on the set of economical configurations.

The static subproblem is solved using branch exchange methods, that are based on the corresponding methods for the optimisation of MV networks opera- tion (for losses reduction and load balancing). The dynamic subproblem is solved using dynamic program- ming.

The proposed method is demonstrated in the paper using a typical distribution network. The main conclu- sions, and benefits of the method, as they are revealed from several applications, are the following: (i) Detailed economical and technical modelling of the system equipment and loads is provided by the method. For example, load variation is taken into account, by using load patterns for different load types and sea- sons. (ii) All operational constraints are also modelled in detail. Energy losses are not linearised. (iii) The proposed method models and optimises all the expansion alternatives of the system, like capacitors and voltage regulators installation, lines and substa- tions construction. (iv) The method can be used for extended sensitivity analyses, since only the dynamic subproblem may be solved in these cases. (v) The proposed method solves the problem of distri- bution system optimal planning within acceptable com- putational time for systems of realistic size, although it has been modelled using conventional computer pro- gramming techniques.

The applicability of the method could be substan- tially improved, if it is modelled taking advantage of parallel processing abilities. For example, during the construction of the set of economical configurations, the examination of one set of sequential ycars is fully independent from this of any othcr year scts.

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pp. 631-638

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