for optimal expansion planning of transmission network investments

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FOR OPTIMAL EXPANSION PLANNING OF TRANSMISSIONNETWORK INVESTMENTSVICTOR A. LEVI aa School of Engineering Sciences , University of Novi Sad , 21000 Novi Sad, YugoslaviaPublished online: 29 Mar 2007.

To cite this article: VICTOR A. LEVI (1996) FOR OPTIMAL EXPANSION PLANNING OF TRANSMISSION NETWORK INVESTMENTS,Engineering Optimization, 26:4, 251-270, DOI: 10.1080/03052159608941121

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Eng. Opt., 1996, Vol. 26, pp. 251-270 0 1996 OPA (Overseas Publishers Association) Reprints available directly from the publisher Amsterdam B.V. Published in The Netherlands under Photocopying permitted by license only license by Gordon and Breach Science Publishers SA

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A NEW BRANCH-AND-BOUND MODEL FOR OPTIMAL EXPANSION PLANNING OF TRANSMISSION NETWORK INVESTMENTS

VICTOR A. LEV1

School of Engineering Sciences, University of Noui Sad 21000 Noui Sad, Yugoslavia

(Received 9 March 1995: i r r finaljorm 10 October 1995)

This paper presents a new model and a solution methodology for optimal planning of high voltage transmission network investments. The objective of this problem is to minimize capital costs needed for new network elements while meeting imposed operation constraints, and it is formulated via a linear mixed integer model. Then a new branch-and-bound algorithm is proposed as the problem solution. The fundamental idea ofthis algorithm is to reduce the number of discrete variables contained in the original set of unknowns. The proposed methodology was verified on several test examples, as well as on the real transmission network of the eastern part of former Yugoslavia.

KEY WORDS: Power systems, transmission network expansion planning, investment problem, linear mixed integer programming, decomposition/aggregation method.

1 INTRODUCTION

The transmission network expansion planning problem consists of two mutually related tasks, namely the investment and operation subproblems.' The topic of this paper is the solution of the investment subproblem. It gives the type, location and timing of the most economic additions (reinforcements) to an existing transmission network, while meeting the imposed operation constraints. Since this problem is essentially of a discrete nature, it can be defined via a mixed integer programming formulation. In this case two principal difficulties appear. The first one is due to the nonlinearity of the model and the second one due to the high number of integer variables. The first difficulty can be overcome either by using an approximate lin- earized modeP3, or by reformulating the model with the aid of additional integer variable^^,^. The second difficulty is the consequence of the high dimensionality of the set of integer variables in the case of real-sized transmission networks. Unfortu- nately, no efficient solution has been suggested so far to overcome this difficulty.

The new formulation of the transmission network investment problem presented in this paper is compatible with the overall transmission network expansion methodology described in Ref. [l]. The proposed model represents also a direct imporvement of the formulation suggested in Ref. [ 6 ] , because it gives more accurate description of transmission network investments. In the proposed model, the

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objective function and the imposed operation constraints are defined within the frame of linear mixed integer programming. However, this model was discovered to bc inconvcnient for practical use with standard solution procedures due to the large number of integer variables. This feature required development of a new branch- and-bound solution methodology based on substituting the initially specified investment model with an equivalent, much simpler, formulation. This equivalent model offers the possibility of applying the decomposition method proposed in Refs. [1,6] which is essentially the application of the superposition theorem for linear optimization models. The fundamental achievement obtained by applying this procedure is the possibility to substantially reduce the original set of integer variables in a "very reliable" way. The development of an appropriate branch-and-bound algorithm was needed for such a solution strategy, requiring the introduction of a new separation of the region of feasible solutions. Then, a software package suitable for expansion planning of real transmission networks was developed. It was verified on several test examples, as well as on the high voltage transmission network of the eastern part of former Yugoslavia.

2 TRANSMlSSlON NETWORK MODEL

Active and reactive power flows in a high voltage transmission network are described by thc well known load flow model7. However, when dealing with expansion planning of a transmission network, the load flow model turns out to be too complex, so that certain simiplifications need to be introduced. The underlying assumptions lead to the so-called "direct current" (DC) load flow model7, representing the basis of the present transmission network investment problem. The starting equation of the DC,load flow gives the relation between the avtive power flow in a branch (k,nl) and the corresponding node voltage phase angle difference (Fig. 1):

where l,-,, is the active power flow in branch (k ,m), b,-, is the susceptance of branch (k,m) and 0,,0, are the voltage phase angles in nodes "k" and "m" respectively. In the next step, the active power balances are established in all but one network

Figure I Active power flow in branch (k ,m) .

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NETWORK EXPANSION PLANNING 253

nodes by considering the set of relations (I), giving the D C load flow in matrix notation:

G - D = B.0 (2)

where G is the vector of active power generations in power plants, D is the vector of active power consumptions (loads) in network nodes, B is the network node susceptance matrix, and 0 is the vector of node voltage phase angles.

When a transmission network operation problem is considered, the D C load flow is a simple linear model with the vector 0 being the only unknown variables. However, transmission investments planning is a more difficult task, since an existing transmission network contains new network elements characterized by corresponding susceptances. Then, besides the vector 0, the system node susceptance matrix B becomes also unknown, leading to a non-linear model (2). Since this model is very inconvenient for application in the case of large real-sized transmission networks, a further simplification of the DC load flow aimed for transmission investments planning is proposed in this paper. The main idea is to retain the linearity of the model, and this is achieved by using the path-capacity approachs for the description of new network elements. In this way, the new network elements are modelled with the aid of power flows only, so that the use of unknown susceptances is avoided. With that respect, the load flow model has the form:

G - D = B . 0 - A . f (3)

where A is the node-branch incidence matrix, relating active power flows in new network elements to all network nodes (note that the new elements can be found in existing network branches as well as in new rights-of-way), and f is the vector of active power flows in new network elements.

In the load flow model (3), the existing network is described via the D C load flow, while the path-capacity approach is used for the prospective transmission network (i.e. new network elements). Since the active power balance equation in the so-called "references bus" is not included in the model (3), the equality of the overall system production and consumption need to be added to this formulation:

where I denotes the unit vector. When performing transmission investment planning, the essential operation re-

quirements that need to be met are the constraints associated with allowable gener- ator productions and maximum transmission capacities of individual network branches. The former set of constraints can be simply expressed as follows:

G m < G < G M (5)

where GM and GM denote vectors of minimum and maximum generations, respectively. Since the DC load flow and the path-capacity models are applied for describing the existing and the prospective transmission networks, respectively, the maximum transmission capacities of elements of the existing network are expressed as:

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while the maximum transmission capacities of new network elements are given by:

where JI is the vector of node voltage phase angle differences across existing network elements (i.e. $,-, = 0, +Om), and I$, r a r e the vectors of maximum allowable node voltage phase angle differences and active power flows, respectively.

The relations (3)-(7) define the transmission network model that is the basis of the transmission network investment problem. The latter is discussed in the next section of the paper.

3 TRANSMISSION NETWORK INVESTMENT PROBLEM

The investment problem of transmission expansion planning is usually defined as a static, minimum cost optimization model, subject to prespecified operation constraints. This model is sequentially solved for all years of the planning period. The following linear mixed integer programming model is proposed for the solution of this problem:

sub.ject to:

where

T is the vector or capital costs of one new standard network element in all system branches (existing and new right-of-ways),

Y is the vector of integer investment variables indicating the number of new network elements in all system brances (its individual elements are designated by 3-,, (k, +S),

/l is the set of all branches in the existing network and new right-of-ways connecting nodes (k,m), where the construction of additional network elements is allowed,

f1 is the diagonal matrix of maximum transmission capacities of one new standard element within all branches ( k , m)€P,

Y E , is the maximum number of network elements, that can be constructed in a branch (k, m ) ~ g .

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In this formulation of the problem, the objective is minimization of capital costs for new network elements, which are expressed with the aid of integer investment variables Y(Eq. (8a)). The set of constraints encompasses the following operation limits: node load balance equations (8b); system balance equation (8c); minimum and maximum production limits (8d); maximum transmission capacities of elements of the existing network (8e) and of new network elements (8f); and integrity constraints on integer investment varaibles (8g), defining the maximum number of new network elements Y E m that can be added in a branch ( k , m). The unknowns of the optimization model (8) are the vector of discrete variables Y and the vectors of continuous variables 0, G and f (note that the vector JI can be simply expressed through the variables 0).

The inherent drawback of the investment model (8) caused by the desire to deal with a linear model, is the use of the path-capacity approach when modelling new network elements. However, the greatest difficulty with this model is the presence of a great number of integer variables, since each network branch requires two of them (one in each direction). This means, that modelling of a modest-sized high voltage transmission network requires several hundreds of integer varaibles. As cited in Ref.[9], then there is a very high probability that the optimal solution of the model (8) cannot be obtained when using standard solution procedures. This feature required the development of a new branch-and-bound procedure taking into account the structure of the model (8) and enabling the achievement of the optimal solution in the vast range of real-life situations. This topic will be discussed in the next section.

4 NEW BRANCH-AND-BOUND SOLUTION PROCEDURE

The application of the branch-and-bound algorithm to the investment model (8), can be summarized in following four stepsg.lo:

1. Solve the model (8) without the integrity constraints (8g). 2. Taking into account the previously determined non-integer optimal solution,

divide the region of feasible solutions of the previous problem into two subregions. This is performed by adding new constraints on integer variables Y,- , (branching).

3. Solve the investment model (8) without the integrity constraints (8g) over the previously defined subregions of feasible solutions. If the local optimal solution so obtained is worse (more expensive) then the currently reached (integer) optimal solution, stop the calculation for the considered subregion (bounding by bounds). The same should be done if the local optimal solution is of integer type for all elements of the vector Y (bounding by integrity).

4. Return, either to step 2 or to step 3, depending on whether the branching or the bounding is considered in the next solution stage.

The illustration of this algorithm for a two-dimensional problem is depicted in Figure 2a, left. The initial optimal solution of the problem @a)-(8f) is marked as point 0. Since the optimal value of the integer variable Y:,-,, is non-integer, the

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Figure2 Separation of the region of feasible solutions a) The basic branch -and -bound Algorithm; b) The proposed branch -and -bound algorithm.

infeasible region [Y,",-,,I -{[Yi,-,,I + I ) (the symbol [.I denotes the greatest integer value, less than or equal to the quantity inside the brackets) is excluded from further analyses. In the next stage, the left subregion is chosen for consideration and it is assumed that the obtained local optimal solution is the integer one. Next, the same process is performed for the right subregion, etc. This procedure is schemati- cally represented by the graph approach given in Figure 2a, right.

In steps 1 and 3 of the above branch-and-bound algorithm, the investment model @a)-(89, as well as the sequence of linear programs obtained by extending the initial program @a)-(80 with additional constraints on the integer varibles Yare considered. The pictorial problem representation in Figure 2a. makes it obvious that it is necessary to discuss three typical representatives of the complete set of linear programs: 1. The initial linear program @a)-(8f); 2. The linear program extended with the constraint Y,,-,8< [Yi,-,J and 3. The linear program where the constraint Y,i-,i> {[Y&,J + I ) is added. These optimization problems are dealt with in the following parts of this section.

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Initial Investment Model Without Integrity Constraints

After the relaxation of the integrity constraints (8g) (which now take the form: 0 < Y < YM, where Y is the vector whose elements are Yf-,), the following substitution should be entered into the initial investment model (8):

y 'M = I'.~ M (9)

Eqs. (8a)-(80 and relation (9) give the following optimization model:

subject to

where K denotes the vector of specific capital costs per unit active power flow1, while the constraint (104 represents the combination of previous constraints (81) and (8g) expressed by a single inequality.

While comparing the reformulated investment model (10) and the linear investment model proposed in Ref. [6], a great degree of similarity can be noticed. But, the basic difference between two models is the presence of the continuous investment variables Y' in relations (10a) and (104, which are missing in the model suggested in Ref. [6]. However, it can be shown that these two optimization models are equivalent at the optimum point (the mathematical proof is given in Appendix A), i.e.:

where superscript (") denotes the optimal value of the indicated variable. The relation (A.5) has a very clear physical meaning: the minimum active power flows f over new network elements are defined by the constraints (10b)-(lOe), and the continuous investment variables Y' are also "driven" to these minimum values by minimizing the objective function (10a). In this way, in the initial stage of the branch-and-bound solution procedure, one is faced with the model proposed in Ref. [6]:

subject to: - B . O + G + A . f = D

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where f is introduced in accordance with Eq. (1 le). It is obvious that the number of variables is significantly reduced, since the vector

of continuous investment variables Y' is suppressed from the model ( I I). However, the essential advantage of the optimization model (11) over the model (10) is the possibility of applying the same decomposition procedure for its solution, as proposed in Refs. [1,6]. This is performed by dividing the initial stage of the branch- and-bound procedure into two steps: first, the load flow pattern that determines the maximum transmission capabilities of the existing network is established by using the minimum load curtailment (MLC) model1; second, the prospective expanding network and the superimposed power flows are calculated with the aid of the "linear programming marginal network" (LPMN) model6, enabling the determination of the best network reinforcements. The first step of the decomposition procedure makes possible the fundamental idea of the proposed branch-and-bound procedure: to extract the set of significant integer investment variables in a "reliable" way, and then to solve all other stages with minimum effort. The MLC and LPMN models are briefly elaborated in sequel.

Mini~nurn Loud Curtuilment Model

This model takes into account only the security aspects of the existing network operation problem'.6. The objective of this model is the minimization of the overall curtailed demand, while respecting generation and branch loading limits within the existing network1. When using the DC load flow, the MLC model is specified by the following linear programming problem:

min z = _ I T . _ R

subject to - B . B + G + R = D

where R is the vector of curtailed loads. The MLC model is solved by using the cficient relaxation procedure (reduced basis approach) and the dual simplex algorithm with bounded variables'.

In the optimal point of the MLC model (12), (f3",Go,JI0,Ro), all operation constraints are satisfied, but the delivered demands are only on the level the existing network can supply ( D o = D - Ro). This means that the maximum usefulness of the existing network is obtained in this way, and that a planning engineer gets a very good insight into the transmission network problems. Now, the well known fact

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from the transmission investments planning practice can be easily verified: among the large number of possible network reinforcements, only a "few" branches need to be considered, since the transmission overloads are in most cases of a local nature. Thus, the MLC model enables the planning engineer to define the branches that are "good" reinforcement candidates and should be retained in the forthcoming stages of the branch-and-bound procedure. This, in turn, makes possible the extraction of the set of "significant" variables (elements of the vector f, o r equivalently, elements of the vector Y') in the early stage of the solution procedure, giving an opportunity to reach the optimal solution in the case of real-sized transmission networks. The information on the initial loading of all system branches obtained by the MLC model (12), is used for this operation. Only the active power flows f , - , (or the corresponding investment variables Y;-,) of "saturated" branches where the initial relative loading is above a prespecified limit (defined by the user as a percentage of branch transmission capacities), are then set in the LPMN model and in all further solution stages. The proper choice of this limit value is further discussed in the next section of the paper. It should be noted that this step represents the break-point in the overall solution methodology, since from now on, only the superimposed power flows and corresponding "marginal network" models are dealt with.

T o facilitate the following of the proposed branch-and-bound algorithm, a simple 4-node test system is presented in Figure 3. The solution of the MLC model (Fig. 3a) shows that it is not possible to deliver 0.18 p.u. of the consumption in node No.3, since power flows off ,-, and f,-, have reached the limit values.

Linear Programming Marginal Network Model

This model is derived by introducing the optimal solution of the MLC model (12) (0°, G ",I)", Ro) into the transformed investment model ( I 1). In that case, generation and transmission capabilities of the existing network are appropriately modified, and only the power flows that are the consequence of the non-supplied (curtailed) loads _Ro are taken into account. The LPMN model appears in the form6:

subject to: -B-AO+AG1+A.f = R " - G m + G o

where A designates a quantity (AO,AG1,A+) superimposed to the initial value of the corresponding variable (Om, Go,+"), and AG' is the vector of transformed additional generations (AG' = AG - G m + Go).

The LPMN model (13) can be solved by applying the procedure already used for the solution of the MLC model (12). When this solution is found, the initial continu-

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MLC MODEL LPMN MODEL

RLPMN MODEL LPMN

0.18 fl!,=0.10 Y,,=I.O RLPMN

Figure 3 Application of the proposed branch-and-bound algorithm to the 4-node test system. a) MLC model; b) LPMN model; c) CLPMN model; d) RLPMN model.

ous optimal solution of the branch-and-bound procedure is obtained, which in its turn defines the starting optimal integer solution: all non-integer investment variables Y;-, are rounded off to the nearest integer values greater than or equal to the optimal values. This starting integer optimum is efficient and useful for the "bounding by bounds" of other solution stages.

The application of the LPMN model to the 4-node test system is shown in Figure 3b. Since the maximum transmission capacity of one new standard element is limited to f: -, = 0.1 p.u., it is necessary to construct Y, -, = 1.8 new lines in the branch 1 - 3.

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5 INVESTMENT MODELS EXTENDED WITH CONSTRAINTS ON INVESTMENTS VARIABLES

T o simplify the presentation, the separation of the region of feasible solutions is done on the initial investment model (lo), but it is generally valid. According to the basic branch-and-bound algorithm (Fig. 2a.), if the investment variable Y;,-,, is chosen for branching, two new linear programs have to be analyzed, each of them having one of the following additional constraints:

Y ; ; - , ; 2 7 ; l . { [ Y ; , - m l + 11 (14'3)

After adding the constraint (l4a) or (l4b) to the investment model (lo), the fol!ow- ing essential requirement is set: the already introduced decomposition principle must be applied in all other solution stages in order to enable the use of the reduced set of integer variables. Unfortunately, in the case of the constraint (14b) it is not possible to transform the obtained linear program in the form similar to the model (I I) (see Appendix B). For this reason, the new separation of the region of feasible solutions presented in Figure 2b. was derived. In this case, the inequality constraint (14b) is replaced by the following set of equality constraints:

In this way, the analysis of the right-hand side subregion of Figure 2a. is substituted with the consideration of a few straight lines, where the investment variable Y,,-,, has specified values (Figure 2b). The proposed separation of the region of feasible solutions is the combination of the basic branch-and-bound algorithmg (equation (14)) and the Land-Doig a l g ~ r i t h m ' ~ (equations (15)). In the following stage of the branch-and-bound procedure, only the linear program with the first equality constraint of set (15) is studied (in addition to the linear program extended with equation (14a)). The consideration of the rest of linear programs with constraints (15) depends on the progress of the solution procedurelo. It should be pointed out that the maximum number of new network elements within branches is usually very low, so that the applicaltion of the proposed separation principle does not add much to the branching of the decision tree (Fig. 2b.).

Investment model extended with construint (140)

This model is solved by adding the equation (14a) to the investment model (10) and by considering the equivalence of the models (10) and (1 1). This leads to the following optimization problem:

Subject to: -B.0 + G + A . f = D

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where the only difference from the model (1 1) is the decrease of the transmission capacity of new elements in branch k i - m i Eq.(l6g). By using the same decomposition procedure as in case of the investment model (II), the optimization problem (16) can be decomposed into the MLC model (12), and the constrained linear programming marginal network (CLPMN) model, the latter being the only one to be considered in this solution step. Its form is:

min z = KT.f

Subject to: - B . A O + A G ' + A . f = R " - G m + G o

h,-,,<fl.[ y;.-",,l (17)

Eq.(17) reveal that we are dealing with superimposed power flows, and that the corresponding marginal network has limited transmission capacity in branch k , - In,. Thus, the name of the model. The optimal solution of the LPMN model (13) is a dual feasible basic solution for the CLPMN model (17), so that additional dual simplex iterations should be run in order to solve the CLPMN model (17).

The 4-node test system, where the construction of only one new element in branch 1-3 is allowed, is shown in Figure 3c. The power flow through this element is thus limited to the maximum value (0.1 p.u.), requiring the construciton of 0.8 new elements (i.e. power flow equal to 0.08 p.u.) in branch 3 -4.

Investment Model E.ufended with the First Constraint ( 1 5 )

This model can be solved by adding the first equation (15) to the investment model (lo), o r equivalently, by replacing this equality in relations (10a) and (10f). The consideraiton of the equivalence of models (10) and (11) leads to the following optimization problem:

min z = (K')?'.fl (1 8 4

Subject to: - B . B + G + A . f = D

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where K', f', Y O M and T' denote corresponding vectors without the element ki - mi. The model (18) represents the solution of the investment problem ( l l ) , where ICY:,-,,I + 1) new network elements are already built in the branch ki-mi (see equation (15)), and the additional reinforcement of the same branch is not allowed. This is the reason why capital costs regarding new elements in branch ki - mi cannot be found in the objective [unction (18a), and why the corresponding transmission capacity is limited according to Eq. (l8g). The same decomposition principle can be applied to the model (18), producing the reinforced linear programming marginal network (RLPMN) model which is the only one to be solved in this step. Its form is as follows:

min z = (K')T.f'

Subject to: -B.AO + AG' + A . f = R" - Gm + Go

The principal difference between the RLPMN model (19) and the CLPMN model (17) is the presence of already constructed {[Yii_,J + 1) new elements in the former one (this is the reason for its name). The optimal solution of the LPMN model (13) is a primal feasible basic solution for the RLPMN model (19), and it is necessary to apply additional primal simplex iterations, in order to solve the RLPMN model (1 9).

Regarding the 4-node test system, the RLPMN model (19) is run after the previously obtained solution via the CLPMN model (Fig. 3d). According to the CLPMN model solution, one zero-cost valued new element in branch 3-4 entered the RLPMN model. The optimum solution of the RLPMN model shows that the power flow through this new element is forced to the maximum value (0.1 p.u.), and that additional 0.8 new elements are needed in branch 1 - 3.

6 COMPUTATION RESULTS

Based on the proposed linear mixed integer investment model and the new branch- and-bound algorithm, the software package was developed and incorporated as an independent entity in the overall transmission expansion planning methodology

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proposed in Ref. [I]. This package has been verified on several test examples, as well as on the high-voltage transmission network of the eastern part of former Yugos- lavia. The 6-node Carver system is used as another test example in this paper, with all data defined in Ref. [I I]. The possibility of generation rescheduling is also taken into account, and the new generation node is initially connected with the existing network at node No. 2. The solution stages of the proposed branch-and-bound procedure are given through the complete decision tree, as shown in Figure 4. The following abbreviaitons are used in this figure: IF-infeasible solution; BB-branching by bounds; BI-branching by integrity.

It is interesting to note that the definition of the initial investment model (8) with seven existing branches and eight new rights-of-way (the maximum number of new network elements in all branches was set to two, except in branches 2-6 (three) and 4-6 (four)), requires the total number of possible solution stages (i.e. vertices in the decision tree in Fig. 4) of 1.01674 x lo1'. After the applicaiton of the MLC model (12) in the first solution stage, if the very flexible limit value of the relative branch loading equal to 50% is taken into account, ten appropriately directed variables/,-, are set in the LPMN (13), CLPMN (17) and RLPMN (19) models. This resulted in the significant reduciton of possible solution stages to 131220, in total. It is obvious from Figure 4. that the investment model is solved in 48 stages and that the optimal

Figure4 Decision tree of the proposed branch-and-bound algorithm when solving Garver's test cx:mplel'.

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solution is the construction of one new element in branch 2-6, two new elements in branch 4-6, and one in branch 3-5 (this is a cheaper solution than the ones obtained in Refs.[1,6]). The initial solution stage of the branch-and-bound procedure is shown in Figure 5a. (MLC and LPMN models), while the optimal solution (RLPMN model) is given in Figure 5b. The latter should be superimposed upon the solution of the MLC model, producing the final, optimal state given in Figure 5b. If these reinforcements are added to Garver's system'' and the MLC model (12) is again run in order to check the optimal solution, the power flows shown in Figure 5c are obtained. It is obvious that it is not possible to deliver 5.7 M W in node No. 2. This is the consequence of the use of the path-capacity approach for the modelling of new network elements. This problem can be easily solved by storing some of the suboptimal solutions during the branch-and-bound procedure. The number of mem- orized suboptimal solutions should be carefully chosen, since it influences the

MLC MODEL 80 150

4

RLPMN MODEL (48) 136.5

100

3

128.8 4

200

LPMN MODEL

RLPMN (48) + MLC MODELS

1

4

Figure Sa and Sb

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V. A. LEV1

MLC MODEL

Figure5 Transmission expansion planning of Carver's system" a) Initial solution; b) Optimal solution; c) Power flows in "optimally" reinforced network.

branching by bounds. The construction of two new elements in branch 3-5, to- gether with three new elements in branch 2-6, is an example of such an approach.

The high-voltage transmission network of the eastern part of former Yugoslavia is used for testing the proposed methodology under real-life circumstances. This system consists of 61 nodes,and 72 branches, and the corresponding figure is given in Ref. [I]. The following conclusion are drawn from a great number of simulations:

1. If the user-defined limit value of the relative branch loading is set to a constant value, the number of integer variables that enter the branch-and-bound solution procedure is primarily dictated by the general load level, and secondarily by the topological structure of the network. This number determines the efficiency of the proposed branch-and-bound algorithm, since the large set of additional constraints on integer variables (15) produces very deep decision tree (Fig. 4.). From this point of view, the application of the proposed methodology is most difficult in the case of heavily loaded networks considered in a year close to the planning horizon. With the flexible limit value of the relative branch loading equal approximately to 60-70%, all of the horizon year planning cases were solved by the proposed methodology.

2. Every analyzed network can be solved with the aid of the proposed methodology, if the limit value of the relative branch loading is properly adjusted. This means that by raising the value of this parameter, it is possible to reduce the number of integer variabels and to finish the solution procedure even in case of problematic networks. Generally, this parameter should be set at the highest possible value, but

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NETWORK EXPANSION PLANNING

the optimal solution should be the same as in the case when this parameter is equal to 0%. During simulations on the Yogoslav power system, the change of "optimal" solutions was recorded at the lowest limit value of 81%, indicating that it is not allowed to take the limit value above this quantity. However, it is suggested to test every analyzed network against different limit values of the relative branch loading, and to find the threshold value representing the best trade-off between solution accuracy and efficiency. It seems to the author that the proper choice of the limit value of the relative branch loading generally lies in the range 65-75%.

3. The optimum integer solution obtained after the application of the whole branch-and-bound procedure can be significantly differnet from the continuous optimal solution already available after the initial solution stage. This is especially true when the horizon year planning is in the process, since there are no "small pieces" of network reinforcements in the optimal integer solution. This is very important, because an adequate planning target for the network now exists. However, when applying the sequential, year-by-year expansion planning policy, the optimal solution is in many cases obtained already in the first stage, after the applicaiton of the L P M N model (only one new network element was needed for the construction).

7 CONCLUSIONS

This paper has been devoted to the derivation of a new model and a solution methodology for optimal investment decisions within the frame of transmission network expansion planning. The investment problem was specified as a linear mixed integer optimization model, whose objective is the minimization of capital costs for new network elements, while meeting imposed operation constraints. Such a formulation of the problem required the introduciton of a great number of integer variables into the optimization model, and this was the principal obstacle to obtain- ing its solution in the case of real-sized transmission networks. Thus, the new branch-and-bound algorithm was developed as a solution methodology. In the first stage of this algorithm, the optimization problem is decomposed into two subproblems, resulting in a significant reduciton of the number of integer variables. Then, in all other solution stages, only superimposed power flows (obtained through marginal network models) should be treated. Finally, the verification of the proposed investment model and the solution methodology was performed on test examples, and on the real high-voltage transmission network of the eastern part of former Yugoslavia.

The proposed investment model represents an independent entity suitable for integration into the overall transmission expansion planning methodology proposed in Ref. [I]. Since this model produces the continuous optimal solution (initial stage), as well as the integer optimal solution (final stage), it is possible to combine these solution variants in the practical use of the model. It seems to the author that the best combination might be the application of the continuous optimal solution in the year by year expansion planning process, while the final optimal solution should be used for planning in greater time intervals (such as the horizon year expansion planning). The reason for this choice is to facilitate and accelerate the planning

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process with the aid of the initial model solution, retaining the accurate objective for the system at the same time.

I . Lcvi, V. A. and Calovic, M. S. (1991) A new decomposition based method for optimal expansion ~ l n n n i n ~ . of lnrce transmission networks. IEEE Trans. on Power Svstems. 6. (3). 937-943. . . . ..

2. k d m s . k . N. a i d Laughton. M. A. (1974) Optimal planning of po'wer networks using mixed integer progr:~mming. Proc. IEE, 121, (2), 139-147.

3. Lee. S.T.. and Hicks. K.L. and Hnvilicza. E. 11974) Transmission exoansion bv branch-and-bound . , . . ~~

integer pr.ogramming'with optimal &-capacity curves. IEEE Trans. on Po& ~pplrratus and Sys- rems, PAS-93, (5). 1390- 14M).

4. Gilles. M. L. (1986) Optimum HVAC-transmission expansion planning-A new formulation. IEEE Tram ott Power Svsfems. PWRS- I . 11). 48-56 . , ,,

5. Seifu. A,. Ss~lon. and iist, G. (1989) Optimization of transmission line planning including security constraints. IEEE Trons. on Power Sysrents, 4, (4), 1507-1513.

6. Levi, V. A. and Calovic, M. S. (1993) A linear programming based decomposition method for optimal planning of transmission network reinforcements. IEE Proceedings- Part C , 140, (6), 516-522.

7. Debs. A. S. 11988) Modern Power Svsrems Conrrol and Operation. Kluwer Academic Publishers. Bos- ton, M. A,, USA.'

8. Kennington, J . L. and Helgason, R. V. (1980) Algorirhmsjor Network Programming. John Wiley & Sons. New York. N. Y.. USA.

9. Garfinkel, R. S, and Nemhauser. G . L. (1972) Integer Programminy. John Wiley & Sons, New York, N. Y., USA.

10. Taha, H. A. (1975) Infeycr Proymmminy. Academic Press, New York. N. Y., USA. I I . Garvcr, L. L. (1970) Transmission network estimation using linear programming. IEEE Trans. on

Powvr Apporofus and Systems, PAS-89, (9, 1688-1697.

APPENDIX A-EQUIVALENCE OF INVESTMENT MODELS

Let the optimum of the investment model (10) be defined through the optimal values of the objective function zo and unknown variables Y '"; 8"; Go; I+V and f O , and let any point belonging to the overall feasible region defined by relations (1Ob)-(100) be specified with the aid of quantities: z; Y'; 8; G; 1+4 and f. Then, the following inequality holds:

By taking into account constraint (100, the following relation is also valid:

Y ' " > f O (A21

where the inequality Y'" > f o is or prime importance for further analyses. If the region of feasible solutions bounded by constraints (10b)-(10e) is marked by S, the feasible region of the vector of power flows, f (the first inequality of constraint (100) by SJ, and the feasible region of the vector of investment variables Y' (the second inequality of the equation (100 by SY, following relations are true:

Since every feasible vector f belongs to the section of sets 6 and SJ, which is the subset of the feasible solutions set of vector Y ' - SY, the optimal point f a can be always chosen as one feasible solution of the vector Y', i.e. Y" = f 0 (the one-to-one

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correspondence between vectors Y' and f exists). By incorporating this condition into Eq. (A.2) (only the inequality is considered), and taking into account the non- negativity of vectors Y' and K (i.e. Y' 2 0; K 2 O), the relation (Al) finally gives:

This equaiton represents the contradiction to the initial condition (Al) on the optimal solution zo. In this w a y , it can be concluded that in the optimal point of the investment model (lo), the elements of the vector Y' are equal to the corresponding terms of the vector f, i.e.:

APPENDIX B-EXTENSION O F T H E INVESTMENT MODEL

When extending the investment model (10) with the constraint Y;,-m,>/il, {[Y~t-ml] + 11, the following model is obtained:

min z = KT.Y'

Subject to: - B . O + G + A . f = D

1T.G = 1T.D

G m < G < G M

144 G $ O < f < Y '

O < Y ' G Y ' M

Y ; , - , " , > ~ ~ ~ . { C Y : , - ~ , ~ + 1)

By introducing the following substitution into the model (Bl):

- Y&~~-~~' .{ [Y;- ,"I + 1) Yx:-",, - it is reduced to the form:

min z = ( K ' ) ~ . Y " + K; .[Y&,,+f,'.{[ + 1))

subject to - B . O + G + A . f = D

IT.G = 1T.D

G m < G < G M

lJll G JI O < f < Y "

G y-mi+?il.{[q-m,l + 1)

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o ~ Y ~ ~ < Y ~ ~ * (B3h)

O < Y,'-,,< y<!!,8-~i'.{[ T-,] + 1) (B3i)

where Kc, f , Y " and YOM denote corresponding vectors without the element ki - mi. The last inequality (B3i) is obtained by introducing Equation (B2) into the constraint (Blh) and by associating the upper limit constraint (Blg) for branch k i - mi to the same relation.

The transformation of the investment models considered so far to simpler forms was based purely on Eq. (A5), telling us that the power flows fo through new elements are equal to the transformed continuous investment variables Y '" a t the optimum point. However, Eq. (B3g) makes it clear that at the optimal point of the model (B3), two fundamentally different situations can be recognized:

I . If Y,'Z,, = 0, then fl <Ti1.{[ T - , J + 1).

2. If Y,':Z,,#O, then fl >Ti1.{[ x:-,J + I } .

In the first case, it is obvious that f j f YL,'!,, holds, so that it is not possible to reduce model (B3) into the simpler form without the investment vector Y' (vectors f and Y' can be now different).

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for optimal expansion planning of transmission network investments

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