A decomposition approach for integrated planning of primary and secondary distribution networks considering distributed generation

Electrical Power and Energy Systems 106 (2019) 146–157

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A decomposition approach for integrated planning of primary and secondary distribution networks considering distributed generation Ricardo A. Hincapie I.a, , Ramon A. Gallego R.a, , Jose R.S. Mantovanib, ⁎

a b

⁎

T

⁎⁎

Department of Electrical Engineering, Technological University of Pereira (Universidad Tecnológica de Pereira), Pereira, Colombia Department of Electrical Engineering Universidade Estadual Paulista Julio de Mesquita Filho, Ilha Solteira, Brazil

ARTICLE INFO

ABSTRACT

Keywords: Bilevel model Distributed generation Distribution system planning Primary and secondary networks Tabu search algorithm

This paper presents a new model for optimal integrated planning of medium and low voltage distribution systems with penetration of distributed generation (DG) in the low voltage network. The proposed bilevel model takes into account in the upper and the lower levels, the medium and low voltage network planning respectively. This approach considers as conflict between these two agents (upper and lower levels), the size and location of the distribution transformers (DT), i.e. the incidence in both networks of the power flow circulating from the primary to the secondary system. The main objective of this approach is to find a joint global solution that establishes a balance to benefit the planning of both networks, by decomposing the problem in two subproblems (or levels). The upper and lower levels involve the costs of installing and upgrading the new and existing elements (branches, DT, substations and DG) and the cost of the energy technical losses. This problem is formulated as a mixed integer non-linear model, and is solved using a tabu search algorithm (TSA). To verify the efficiency of the proposed methodology, three cases of study are compared: (i) traditional integrated planning, (ii) bilevel integrated planning and (iii) bilevel integrated planning with DG in the LV network. The obtained results show the importance of considering both networks in a simultaneous way in the electric distribution system planning, which allows finding answers of lower global costs.

1. Introduction Distribution system planning (DSP) is known as the set of strategies that allow to determine how many, where and when an electric element (or elements) could be installed in the network, in order to satisfy the growing demand in a defined time horizon [1]. Traditionally the DSP has considered the installation of new elements as electrical circuits (medium and low voltage - MV/LV), sources (substations and DTs) and the upgrading of the size of existing elements in both voltage levels [1–21]. Due to the combinatorial nature of the DSP problem (NPcomplete), this has been usually solved separately for both voltage levels (primary and secondary), which has reduced the search space of the problem. According to this, several methodologies and different mathematical models have been used in order to solve this problem. However, the number of investigations in this topic is greater on the primary DSP [1–11] than on the secondary DSP [12–17], and a few consider the integrated planning of both networks [18–21]. In the last years, the interest of the electric sector by the connection

of DGs and energy storage systems (ESS) in the electrical networks has increased considerably, due to the technical and economic benefits [5,6,8,21–24]. Initially the DGs were used to solve operative problems. Subsequently, due to the great obtained impacts, they were incorporated into the DSP problem. Despite the positive impacts that the DGs present, only a work considers these elements in the integrated planning of primary and secondary networks [21]. The main difference between the works mentioned before and the approach proposed in this research consists on a new bilevel mathematical model, which is used to represent in a simultaneous way the integrated planning of both networks (primary and secondary) by decomposing the problem in two subproblems (or levels). Additionally, penetration of DG in the LV network is considered. It was decided to use DGs in this voltage level, because it was desired to observe the impact of a small-scale penetration given that in countries like Colombia, there are laws that encourage this type of connections to the network. A bilevel formulation is a hierarchical optimization model that involves two levels (agents) known as upper (leader) and lower

Corresponding authors at: Department of Electrical Engineering, Universidad Tecnológica de Pereira, Carrera 27 No. 10-02, Pereira, Colombia. Corresponding author at: Department of Electrical Engineering, Universidade Estadual Paulista Julio de Mesquita Filho, Av. Brasil 56 Centro, Ilha Solteira, Brazil. E-mail addresses: [email protected] (R.A. Hincapie I.), [email protected] (R.A. Gallego R.), [email protected] (J.R.S. Mantovani). ⁎

⁎⁎

https://doi.org/10.1016/j.ijepes.2018.09.040 Received 18 October 2017; Received in revised form 14 August 2018; Accepted 24 September 2018 0142-0615/ © 2018 Elsevier Ltd. All rights reserved.

Electrical Power and Energy Systems 106 (2019) 146–157

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Nomenclature

NG i, g

Parameters

CijEP ,p CijES ,c

CiESS ,s CiND ,d CiNG ,g CijNP ,p CijNS ,c

CiNSS ,s Iijmax ,c Iijmax ,p nL Rijp Rijs

Sicu ,d Si,fed Simax ,d Simax ,g Simax ,s SiSD ,l Vimax Vimax abcn Vimin Vimin abcn

NP ij, p

fixed cost to expand the capacity of an existing primary feeder between nodes i j , type p ["$"] fixed cost to expand the capacity of an existing secondary circuit between nodes i j , type c ["$"] fixed cost to expand the capacity of an existing substation at node i, type s ["$"] fixed cost of a new DT at node i, type d ["$"] fixed cost of a new DG at node i, type g ["$"] fixed cost of a new primary feeder between nodes i j , type p ["$"] fixed cost of a new secondary circuit between nodes i j , type c ["$"] fixed cost of a new substation at node i, type s ["$"] maximum current limit of a secondary wire type c [A] maximum current limit of a primary wire type p [A] number of levels of the load duration curve resistance of the primary feeder between nodes i j , type p [ ] resistance of the secondary circuit between nodes i j , type s[ ] power losses in the copper of a DT at node i, type d power losses in the iron of a DT at node i, type d maximum power limit of a DT type d [kVA] maximum power limit of a DG type g [kVA] maximum power limit of a substation type s [MVA] secondary demand at node i, for a load level l [kVA] maximum voltage limit at node i [kV] maximum voltage limit at secondary node i, on phases a, b, c , and neutral [kV] minimum voltage limit at node i [kV] minimum voltage limit at secondary node i, on phases a, b, c , and neutral [kV]

NS ij, c

NSS i, s

Iij, l Iijabcn ,l SiDT ,l SiG, l SiS, l Vi, l Viabcn ,l Sets DT

EP

ES ESS

ip

is ND

NG NL NP

NS NSS PF

PN

SC

Variables EP ij, p

ES ij, c ESS i, s

ND i, d

d binary decision variable to install a new DG at node i, type g binary decision variable to install a new primary feeder between nodes i j , type p binary decision variable to install a new secondary circuit between nodes i j , type c binary decision variable to install a new substation at node i, type s current flow on primary branch i j , for a load level l [A] current flow on secondary branch i j , on phases a, b, c , and neutral, for a load level l [A] power injected to a DT at node i, for a load level l [kVA] power injected by a DG at node i, for a load level l [kVA] power injected by a substation at node i, for a load level l [MVA] bus voltage at primary node i, for a load level l [kV] bus voltage at secondary node i, on phases a, b, c , and neutral, for a load level l [kV]

SN

SS

binary decision variable to expand the capacity of an existing primary feeder between nodes i j , type p binary decision variable to expand the capacity of an existing secondary circuit between nodes i j , type c binary decision variable to expand the capacity of an existing substation at node i, type s binary decision variable to install a new DT at node i, type

TD

TG

TP

TS TSS

(follower). In this kind of problem, each level is an optimization model composed by an objective function and its respective set of constrains. One of its main characteristics is the conflict between these two agents, where the decisions of one affect the decisions of the other [25]. In the proposed methodology in this work, the upper level is the planning of medium voltage distribution system (leader) and the lower level is the planning of the low voltage distribution system (follower). Generally, the distribution networks of the medium and low voltage belong to the same owner, and it is possible to think that there is no conflict between the participating agents (actually only one agent). However, the conflict in this work is treated according to the location and dimensioning of the DT, which affects the power flows that circulate from the MV to the LV network. This situation is reflected in the technical and economic aspects in the planning of both systems. A suitable location of a DT for a secondary network may be inappropriate for the medium voltage network, causing an inadequate sizing of their electric elements. On the other hand, the location of the DT (from the point of view of the MV system), can affect the technical

set set set set set set set set set set set set set set set set set set set set set set

of of of of of of of of of of of of of of of of of of of of of of

new and existing DT existing primary feeders existing secondary circuits existing substations nodes connected with node i of primary network nodes connected with node i of secondary circuit new DT new DG the number of levels of the load duration curve new primary feeders new secondary circuits new substations new and existing primary feeders nodes of primary networks new and existing secondary circuits nodes of secondary circuits new and existing substations types of DT types of DG types of primary feeders types of secondary circuits types of substations.

losses at the LV network, which would increase the costs of the project. Similarly, the size of DT impacts directly the technical aspects of both networks. A nominal value of a DT, imposes technical requirements in the low voltage network in order to comply aspects of voltage regulation and low technical losses. To avoid these situations, the bilevel model proposed considers the interaction of the two networks in a simultaneous way, allowing to find a joint global solution that guarantees an equilibrium in the planning between the two networks. In a bilevel optimization model, the leader realizes a first movement anticipating the decision of the follower [25–30]. After that, the follower takes a decision based on the movement of the leader. In Fig. 1, the movements employed in this work between both levels are observed. In this figure, the leader (primary DSP) proposes a location and size for the DT (Simax , d ) and the follower (secondary DSP) reacts to this strategy. In other words, the secondary network is planned using the movement of the leader, which causes different values of the required power to feed the LV load. Once the optimization problem of the LV is solved, the follower returns to the leader the power quantity used for 147

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dimensioning of the DTs.

• Proposal of specialized tabu search algorithm for solving the master •

Fig. 1. Bilevel approach of the integrated DSP problem.

•

each DT (SiDT , l ). In this point, the medium voltage network can be planned. A bilevel problem can be transformed into a single level problem, if the problem of the lower level is continuous and convex. To this end, it can be used: (i) the KKT conditions or (ii) the incorporation of the primal and dual constraints and the strong duality condition of each lower level problem into the upper level. In this paper, both levels are formulated as a mixed integer non-linear model and thus they are nonconvex. As a consequence, the bilevel problem proposed is solved using a metaheuristic technique. This way, a TSA is applied because it has been successfully employed in problems with similar mathematical complexity [9,15,16]. In Table 1 a comparison of the listed references is presented. In this table, the terms used mean: MVP (MV planning), LVP (LV planning), EPS (Electric Power Systems), CO (Classical Optimization), H (Heuristic), and MH (Metaheuristic). To the best of our knowledge, little attention has been paid to the low voltage distribution electric networks compared to medium voltage systems, and even less attention has been given to the integrated planning of medium and low voltage networks (see Table 1). This way, the major contributions of this work are:

This paper is organized as follows. The mathematical formulation of the problem is presented in Section 2. Section 3describes how the new bilevel approach is solved. Numerical results are shown in Section 4, and conclusions are presented in Section 5. 2. Problem formulation The mathematical formulation of the bilevel problem is presented in (1)–(17), where the primary and secondary networks are represented by one-phase and three-phase models, respectively. The upper level (primary DSP) given by (1)–(8), has an objective function to minimize the investment and operative costs. The objective function of (1) is the present value of six terms. Terms 1 and 2 are the costs of installing new primary feeders and the upgrading of existing primary feeders, respectively. Terms 3 and 4 are the costs of installing new substations and the upgrading of existing substations, respectively. Term 5 is the cost of installing new DTs. Term 6 is the operative cost of the network (energy technical losses in primary feeders). The set of constrains is presented in (2)–(8). Eq. (2) considers the nodal balance given by Kirchhoff’s laws. Eqs. (3) and (4) are the operative limits of the primary feeders and substations, respectively. Eq. (5) is the voltage limit in all nodes of the primary network. Eqs. (6)–(8) ensure that only one type of wire, substation or DT can be chosen to be installed in the same place, respectively. The lower level (secondary DSP) given by (9)–(17), has an objective function to minimize the investment and operational costs. The objective function of (9) is the present value of five terms. Terms 1 and 2 are the costs of installing new secondary circuits and the upgrading of existing

• Proposal of an optimization technique based on a mixed integer •

problem and slaves subproblems that are solved iteratively. The solution scheme of the bilevel problem throught tabu search is illustrated in Fig. 1. A new coding scheme to represent both problems. The difference in relation to other papers from the literature is that the DTs are considered in the coding of the primary PSD. This scheme allows generating the proposal that must solve the problem of the lower level depending of the capacity and location of these elements. As a consequence of this, the secondary PSD coding only considers the location and size of network sections and DGs, facilitating their solution. A new test distribution system. This system is described in detail for both voltage levels, and its application and the obtained results in this paper can be a point of reference for future research.

non-linear bilevel model, where difference in relation to other mathematical models from literature is simultaneously to consider the integrated planning of MV and LV electric power distribution systems. Decomposition of the original problem in a master problem and n slave subproblems, throught the hypothesis that there is a conflict of economic interests in the actions of investments between the medium and low voltage networks, due to the position and the

Table 1 Comparison of the listed references. Ref.

MVP

LVP

MVP/LVP

Use of GD/ESS

Bilevel use in EPS

Objective function Mono

[1,7] [2–4,11] [5,6] [8] [9] [10] [12–16] [17] [18] [19,20] [21] [22] [23,24] [26,27] [29,30] [28]

X X X X X X

X X

X X X X

X X X X

X X X

X X X X X X

X X X X X

X X X

148

X X X

Multi

Solution technique CO

H X

X X

X

X X X

X X X

X X X

X X

MH

X X X

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secondary circuits, respectively. Term 3 is the cost of installing new DGs. Term 4 is the operative cost of the network (energy technical losses in secondary circuits). Term 5 is the operative cost associated to the energy technical losses in the DTs. The set of constrains is presented in (10)–(17). Eq. (10) considers the nodal balance given by Kirchhoff’s laws. Eq. (11) is the operative limit of secondary circuits. Eq. (12) is the operative limit of the DTs. Eq. (13) determines the power injected in each DT. Eq. (14) is the operative limit of the DGs. Eq. (15) is the voltage limit in all nodes of the secondary network. Eq. (16) ensures that only one type of wire can be installed between two nodes. Eq. (17) guarantees that only one type of DG can be installed in a secondary node. ij i

min =

p

NP

s

NSS

CijNP ,p

TP

CiNSS ,s

TSS

NP ij, p

+

NSS i, s

+

i

ND

d

TD

ij

PF

p

TP

nL

k1

l=1

CijEP ,p

ij

EP

p

TP

i

ESS

s

TSS

CiND ,d

( c

NG i, g g

ESS i, s +

ND i, d +

k2 Iij2, l Rijp (

NP ij, p

+

EP ij, p )

(1)

s. t . SiS, l = SiDT ,l +

Vi, l j

NP ij, p

(

ip

p

EP ij, p ) Iij, l

+

i

PN

;

l

NL

TP

(2)

Iij, l

Iijmax ,p

ij

PF

;

SiS, l

Simax ,s

i

SS

; l

Vimin

p

s

Vimax

Vi, l

(

NP ij, p

(

NSS ij, s

; s

ij

(3)

TP

(4)

TSS

PF

+

ESS ij, s )

1

i

1

i

SS

DT

ij

c

NS

TS

(8)

CijNS ,c

i nL l=1

min = k1

ij

i

+

c

TS

DT

d

ij

c

ES

TS

CijES ,c

ES ij, c +

+

ES ij, c )]+

NG CiNG , g i, g + TG

g

NG

SC

nL l=1

k1

NS ij, c

2 k2 Rijc × [|Iijabcn ,l | (

NS ij, c

k2 Si,fed + Sicu ,d

TD

SiDT ,l

2

Simax ,d

(9)

s. t . SD SiDT SiG, l + , l = Si, l

Viabcn ,l j

SN

; l

Iijmax ,c

ij

SiDT ,l

Simax ,d

i

+ SiSD ,l SiG, l

Simax ,g

Vimin abcn

Viabcn ,l

c

SC

DT

+

ES abcn ij, c ) Iij, l

NL

3.1. Codification

; d

TD

(12)

In order to encode properly the primary DSP, a new codification scheme is proposed (see Fig. 3). This vector is divided into three parts. The first part considers the location and size of the existing and new substations (size n1 + n2 ); the second part contains the location and size of the existing and new primary feeders (size n3 + n4 ); and the third part involves the location and capacity of the DTs (size n5). In a similar way, the secondary DSP is encoded using a vector that contains the information of the low voltage network (see Fig. 4). This vector is divided into three parts. The first part contains the location and size of the existing and new secondary circuits (size m1 + m2 ). The

2 Rijc [|Iijabcn ,l | ] l = 1 ij

; i

Vimax abcn

NL

i

(17)

(11)

nL

; l

NG

TS

+ SC

i

(16)

; c

NL

; l

Simax ,d

NG

i

TS

; l

2

SiDT ,l ij

i

NS ij, c

NL

Iijabcn ,l

fe cu SiDT , l = Si, d + Si, d

(

is

1

TG

SC

(10)

(7)

TD

ij

As it was mentioned in Section 2, the optimization model is a mixedinteger nonlinear programming model, and to solve it, a TSA is used. This algorithm was proposed by Glover [31], and it is based on the concept of the artificial intelligence. The TSA begins in a point of good quality in the solution space (i.e. feasible and low objective function). This point is known as initial configuration and can be obtained using a constructive heuristic algorithm. Once this configuration is evaluated, a set of topologies are selected using different strategies. These topologies are called neighbors (or neighborhood), and because of the large amount they are, can be reduced in a defined number (reduced neighborhood). The strategies to obtain the reduced neighborhood are known as neighborhood structure and depend on each particular problem. From this set of topologies (neighbors), the best configuration (lower objective function) is selected. These strategies are repeated until a predefined stop criteria is reached. The process described before is called local search. The TSA can start from a different initial configuration, and repeat the local search strategy; this process is known as global search. The best solution is the configuration with lower cost found during the whole process. The pseudocode of the TSA is presented in Fig. 2. The main aspects of the TSA used in this work, are presented next.

(6)

TSS

1

3. Solution methodology

(5)

NL

TP

ND i, d d

p

; l

PN

1

;

NL

NL

i

EP ij, p )

+

l

ES ij, c )

+

The bi-level mathematical formulation is a mixed integer non-linear model. This model is non-linear due to the multiplication of variables in the equations of Kirchhoff’s laws and the square of the current in the operative costs in both objective functions. Additionally, the model is mixed integer because it has integer and continuous variables (decision variables, current magnitudes, voltage levels, etc.). Both objective functions are expressed in present value using the factor k1. The constant k2 involves the energy price and the number of hours of each level of the load duration curve. In order to evaluate the upper level problem it is necessary to know the location, capacity and power injected in the DTs. The location and capacity of the DTs are proposed by the upper level (Simax , d ). However, the power flows in the primary network cannot be calculated because the power injected in the DTs are unknown. In other words, the terms 1, 2, 3, 4 and 6 in the objective function, and the set of constrains need to know the power injected in each DT. Once the capacity and location of the DTs is proposed by the upper level, the lower level problem can be solved. Note that the proposed decisions in the upper level problem, are parameters in the lower level problem; i.e. the location and capacity of the DTs are fixed when the secondary DSP is realized. After the lower optimization problem is solved, the power injected in each DT is obtained (SiDT , l ), and then, the upper level problem can be solved.

EP ij, p +

CiESS ,s

NS ij, c

TS

c

SC

DT

; l

; g SN

TS

; l

NL

(13) (14)

TG

NL

(15) 149

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Fig. 2. Pseudocode of the TSA. Fig. 3. Codification scheme for primary DSP.

Fig. 4. Codification scheme for secondary DSP.

secondary circuits, phase balance, branch exchange and installing of DGs. Note that the DTs are not used as neighborhood criteria because these are the link between both networks in the bilevel model proposed in this paper. The size and location of the DT proposed by the upper level, must be the same when the lower level is solved. In a similar way, when the lower level returns to the upper level the power injected in a DT, this value must be the same when the upper level is solved.

second part considers the location of the loads in each phase (size m3). The third part involves the location and capacity of the DGs (size m4 ). The codification employed for the primary and secondary DSP use integer numbers, where each number (for all the elements) is associated to different capacities. A zero indicates that the respective element is not proposed to be installed. In part two in Fig. 4, the numbers 1, 2 and 3 are associated to the load connection to the phases a, b and c, respectively. The relationship of the coupling variables between both levels are explained in Section 3.6.

3.4. Evaluation of the configurations

3.2. Initial configuration

During the procedure, the proposed configurations are evaluated using the method of penalties (i.e. the unfeasible configurations are allowed but their respective objective functions are penalized). The sum of the objective function plus the penalty costs of the respective violated constrains are called fitness function (Ffit ). They are obtained as follows:

The initial configuration for both networks (medium and low voltage) is obtained using a constructive heuristic algorithm [16–19]. This algorithm begins from the existing source (substations or DT), and in each step, a new branch (primary feeder or secondary circuit) is connected to the system. When a branch is linked, the operative limits are verified (voltage regulation and capacities of the elements). This strategy stops when all demand nodes have been connected to the network. It is important to highlight that only feasible topologies in the initial configuration are allowed.

FfitMV = Eq. (1) + fpVMV ( FfitLV = Eq. (9) + fpVLV (

VMV ) VLV )

+ fpIMV (

+ fpILV (

IMV )

ILV )

+ fpSMV (

+ fpSLV (

SMV )

SLV )

(18) (19)

Factors fpV , fpI and fpS are associated to the penalties for violation of voltage limits, and overloads in branches and sources (DT or substations). Terms with subscripts MV and LV refer to MV and LV networks, respectively. These factors multiply the value which each restriction was violated ( ). If any constraint is not violated, delta is zero. The units of these factors ensure that each term is expressed in monetary units.

3.3. Neighborhood criteria The neighborhood structure of the upper level (primary DSP) uses the next criteria: the upgrading of existing substations and branches, location of new substations and branch exchange. For the lower level (the secondary DSP) the criteria are: the upgrading of existing 150

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Reading database

Propose location and sizing of DT

NO

Begin the DSP for the LV networks Stop criterion?

Generate the initial configuration of the LV networks (Section 3.2)

YES

Execute the TSA (Sections 3.3 - 3.5)

Execute the TSA (Sections 3.3 - 3.5)

Determine injected power to the DT

Generate the initial configuration of the MV network (Section 3.2)

End and print results

Fig. 5. Flowchart of the proposed approach.

3.5. Stop criterion

(iv) solve the upper level. Note that in this work, a global iteration is similar to the global search in a traditional TSA. The general methodology ends when the stop criterion of global iterations is reached. The flowchart of the proposed approach is presented in Fig. 5.

For both levels, the local search stops when the local incumbent solution is not improved for a predefined number of iterations or a fixed number of iterations is reached. Similarly, the global iteration stops when a predefined number of iterations is reached or when the global incumbent solution is not improved for a predefined number of iterations.

4. Numerical results 4.1. System description To validate the proposed methodology, the real distribution system of Fig. 6 is used. In this figure, existing and proposed branches are represented by solid and dotted lines respectively. Existing and proposed substations are represented by squares. The black points are primary nodes, the white circles are secondary nodes, and the white circles with a black point inside are nodes shared by both networks. To supply the 138 new secondary demand nodes are proposed: 33 new DTs, 15 new DGs, and 147 new secondary circuits. The primary distribution network has 48 existing nodes, 1 existing substation (type 2) and 51 existing feeders (type 3). To supply the new power demand, 60 potential new feeders and 1 new substation could be installed. Additionally, 5 types of substations, 8 types of wires, 8 types of DTs, and 4 types of DGs are considered (see Table 2). Candidate nodes to install DTs and DGs are presented in Table 3. The full system database can be found in [32]. The nominal voltage of this system is 13.2 kV and 404 V for primary and secondary networks, respectively. The maximum voltage regulation for primary and secondary systems is 10% and 5%, respectively. The planning horizon is 20 years. The load duration curve is discretized in three load levels of 100%, 60%, and 30% of peak demand, with durations of 1000, 6760 and 1000 h respectively. The discount rate is 10% and the energy cost is 0.15 USD/kwh. The penalty factors are 1000, 1500, 4000, 150, 100, and 1000 for fpVMV , fpIMV , fpSMV , fpVLV , fpILV , and fpSLV , respectively. The stop criterion for the local search is 50 iterations or 15 iterations if the best solution found is not improved. The number of global iterations is 100. In order to analyze the benefits of the proposed methodology, three different cases are studied: (1) traditional planning, (2) bilevel integrated planning and (3) bilevel integrated planning with DG in the LV network. In case 1, the distribution planning considers medium and low voltage networks as two independent systems and it is used for some utilities in Colombia [33]. Cases 2 and 3 apply the proposed methodology in Section 3. The difference between them is the presence of DGs in the LV network in case 3. In this case, the DGs are modeled as PQnodes and their possible locations are based on the expert criterion of the planner. The three cases have the same design aspects and they were written in Matlab. The algorithm used in case 1 is described as

3.6. General methodology As it was mentioned before, the procedure begins when the upper level proposes the location and sizing of DTs (Simax , d ). These values are considered parameters to the lower level, and with this information, the secondaries DSP can be solved. Then, an initial configuration for each DSP is obtained as it was explained in Section 3.2 and their objective function are determined using Eq. (9). After that, the local search begins with the creation of the neighborhood (Section 3.3) from the initial configuration. A reduced neighborhood is obtained, and its configurations are evaluated using the fitness functions described in Section 3.4. In this way, the best configuration (lower fitness function) is selected and comes to be the new configuration. These steps continue executing themselves and the local search ends when the stop criteria is reached (see Section 3.5). The best objective function is stored and the power injected to the DTs (power demand, and power technical losses in DTs and secondary circuits) is returned to the upper level (SiDT , l ). It is important to emphasize that when solving the lower level, really are planned n secondary systems simultaneously. With the location, sizing and power injected in the DTs, the upper level (primary DSP) can be solved. Likewise to the lower level, the same steps to solve the upper level are employed. Initially, an initial configuration is obtained as described in Section 3.2 and its objective function is determined using Eq. (1). Next, the reduced neighborhood is obtained using the criteria presented in Section 3.3 and these configurations are evaluated using the fitness function as illustrated in Section 3.4. Later, the TSA selects the configuration with the lower objective function, and in this point, a new neighborhood is determined. The procedure ends when the stop criterion is reached. The feasible configuration found in the local search with the lowest cost, is selected as the best solution. The incumbent is the sum of both objective functions (the best solutions found in the upper and lower levels). Once this value is obtained and stored, a new global iteration starts. This global iteration is composed of four steps: (i) propose the new location and sizing of DTs, (ii) solve the lower level, (iii) determine the power injected in the DTs and 151

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Fig. 6. Integrated distribution system.

follows:

Table 3 Candidate nodes to install DT and DG.

i. Generate a list of proposed DTs. ii. Select a DT and identify branches connected to it. iii. Connect a new branch and verify if its end node has been selected. If it has been connected, go to the step vi. Otherwise, go to the next step. iv. Assign the size of the conductors and verify the operative conditions. If the operative limits of the LV network are not violated, check the branch and its end node as connected, and go to the step vi. Otherwise, go to the next step. v. Disconnect branch. vi. If there are more branches associated to the node, go to the step iii. Otherwise, go to the next step. vii. Verify the LV networks without connections. If there are disconnected nodes, go to the step ii. Otherwise, go to the next step. viii. Assign the size to the DT and generate a list of primary nodes to connect DTs. ix. Select a DT to be connected to the MV network. x. Choose and connect the primary node closer to the DT. xi. Assign the size of the conductors and verify the operative conditions. If the operative limits of the MV network are not violated, check the DT as connected and go to the next step. Otherwise, select the next closer node to the DT and go to the step x. xii. If there are DTs without connect to the MV network, go to the step ix. Otherwise, stop and print results.

Element

Nodes

DT

2, 8, 11, 16, 30, 33, 37, 45, 48, 51, 56, 59, 64, 80, 83, 87, 91, 94, 97, 104, 106, 109, 111, 113, 116, 118, 122, 124, 127, 129, 132, 135, 137

DG

6, 23, 38, 43, 54, 62, 71, 88, 95, 103, 110, 117, 130, 131, 136

4.2. Obtained results The algorithm was implemented in Matlab (2013), using a PC Intel®Core i7-4770 16 GB RAM. The CPU time for Cases 1, 2 and 3 is 1620 s, 3060 s and 3360 s, respectively. The comparison of the obtained results are shown in Tables 4 and 5 in present value, where the term ETL means energy technical losses. Figs. 7–12 show the best solutions found by the three cases of study. To facilitate the visualization of the obtained topologies, the primary and secondary networks of each case, are presented in separate figures. In these figures, the primary and secondary branches are represented by solid lines. In the primary networks of the three cases, the branches have a type 3 wire. In the figures of the secondary networks in the three cases, the number in parentheses is associated to the type of wire for each branch; branches without number, have a type 1 wire. The DTs are represented by black triangles and their type is presented next to this, by an underlined number.

Table 2 Element costs. Substations

Wires

DT

DG

Type

Smax [MVA]

Cost [MUSD]

R [ /km]

Imax [A]

Cost[USD/m]

Smax [kVA]

Cost[USD]

Smax[kW]

Cost[USD]

1 2 3 4 5 6 7 8

7 10 20 30 40 – – –

0.336 0.672 1.344 2.016 2.688 – – –

1.04 0.65 0.52 0.32 0.26 0.18 0.14 0.12

150 180 205 275 305 390 460 600

14 20 26 40 47 57 64 72

30 45 75 112.5 150 225 300 400

3177.58 3953.08 5502.70 7439.72 9376.75 11053.72 16806.00 22408.00

50 75 100 125 – – – –

2500 3750 5000 6250 – – – –

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obtained in case 3, which has penetration of DG. The highest cost was obtained in case 1, showing that the bilevel model decreases the expansion plans of the integrated electric planning. Despite the fact that the case 1 has lower fixed cost than case 2 (1.263 and 1.267 millions of USD, respectively), the operative cost in case 2 (USD 0.521 M) is lower than in case 1 (USD 0.651 M). This is because case 2 proposes branches with larger size, which reduce the energy technical losses in the LV and MV networks, as shown in Table 4. This shows that the bilevel integrated model can find a topology with lower global cost. Observe that case 3 has the lowest fixed and operative costs (1.246 and 0.490 millions of USD, respectively). Note that the penetration of DG in the LV network allows to install DTs and LV circuits with smaller sizes than cases 2 and 3. Also, the MV network is impacted too, which is reflected in the lowest operative cost of the three cases. This confirms that the bilevel model with penetration of DG allows to find lowest global cost, obtaining topologies with lesser energy technical losses, due to the fact that the power circulating through the integrated system decreases. In Case 1, the existing primary circuits have greater chargeability, and therefore the existing substation has a higher chargeability than the proposed substation. For the above, this topology presents technical losses greater than the other two cases, due to the increase in the currents that circulate through the network. The opposite occurs with the topologies of the MV network of Cases 2 and 3, in which circuits with have similar loading, causing lower technical losses. In other words, the bilevel methodology finds solutions that distribute better the circulation of flows in the network. Regarding low voltage networks, in all cases there is a similar amount of DTs (28, 32 and 31 for Cases 1, 2 and 3, respectively). When observing the number of DTs selected with capacities lower than type 4 (112.5 kVA), in Case 1 there are 11 TDs, while in Cases 2 and 3 are proposed 19 and 20 DTs, respectively. This indicates that the bilevel methodology allows the installation of a greater amount of DTs with low capacities, but with better chargeability, allowing a better use of the network elements. Cases 2 and 3 (bilevel methodology) find solutions in the MV network with similar topologies, which shows a tendency despite the use of DGs in the LV networks. The difference between these two cases is shown in the capacities of the DTs selected in the LV networks. These capabilities impact on the investment costs in the MV networks and in

Table 4 Expansion cost comparison in Millions of USD. Cost

Description

Case 1

Case 2

Case 3

Fixed

Substations MV feeders LV circuits DT DG Total

0.336 0.379 0.288 0.260 – 1.263

0.336 0.410 0.255 0.265 – 1.267

0.336 0.397 0.254 0.225 0.032 1.246

ETL in MV ETL in LV Total

0.413 0.238 0.651

0.337 0.184 0.521

0.307 0.183 0.490

1.915

1.789

1.737

Variable

Total cost

Table 5 Expansion cost of LV and MV networks in Millions of USD. Network

Description

Case 1

Case 2

Case 3

LV

DT DG LV Circuits ETL Total

0.260 – 0.288 0.238 0.786

0.265 – 0.255 0.184 0.705

0.225 0.032 0.254 0.183 0.696

MV

Substations MV feeders ETL Total

0.336 0.379 0.413 1.128

0.336 0.410 0.337 1.084

0.336 0.397 0.307 1.041

Total cost

1.915

1.789

1.737

In the three cases the existing substation was not upgraded and a new type 1 substation was installed. In case 3, the nodes of the proposed DGs are: 23 (type 2), 43 (type 3), 103 (type 2), 110 (type 2), 117 (type 2), 130 (type 3), 131 (type 2) and 136 (type 2). In all the cases the operative limits are within their limits (chargeability of the elements and voltage regulation). The consolidated results from Table 4 show that the lower total cost is obtained in cases 2 and 3, supporting the advantages of using the proposed methodology in this paper. As expected, the lowest cost was

Fig. 7. Case 1 - Primary network. 153

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Fig. 8. Case 1 - Secondary networks.

the technical losses in the LV networks, by circulating flows with lower values in Case 3 respect to the Case 2, due to the use of DGs.

the location and sizing of the DTs allow a better circulation of the power flow between both systems (MV and LV networks), which reduces the technical losses and the investment costs in the elements of the network. Additionally, as expected, Case 3 (bilevel model with DG) presented the best results (lowest costs), impacting the capacities of the selected elements (smaller sizes), and therefore, lower technical losses.

4.3. Final comments In order to validate the efficiency of the proposed methodology, three cases of study were considered. The first case is a traditional methodology used by utilities, and the other two cases use a bilevel methodology; the difference between these two cases is that the last one considers penetration of DG in the LV networks. The three cases were applied to the same distribution system, where the obtained results reflect that a bilevel integrated planning presents a lower cost when compared to a traditional planning. The lower cost is obtained because

5. Conclusions This project presents a new bilevel mathematical model for the integrated planning of electric power distribution systems, which considers the primary and secondary networks as a single system. The objective functions of both problems take into account the costs of

Fig. 9. Case 2 - Primary network. 154

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Fig. 10. Case 2 - Secondary networks.

investment in new elements and upgrading of existing elements, as well as the operating costs associated with energy technical losses in circuits and DTs. The set of constraints considers technical and operational aspects of both systems. Additionally, the penetration of DG in the LV network is included. The proposed bilevel model is developed from the conflict that exists in the planning of both electrical networks, when they are treated together. In this paper, the conflict is related to the location and capacity of the DTs, affecting the power flows that circulate in both networks. According to this, the variables in conflict correspond to the location and sizing of the DTs, and their injected power. The bilevel model considers in the upper level the planning of the MV network and

in the lower level the planning of the LV network. The role of both agents was defined from two aspects: (i) the hierarchical structure that these systems have in an electrical system, depending on the voltage level they have, and (ii) the direction of the power flows. The strategy of dividing the problem into two levels simplifies its solution, allowing reducing the number of variables and constraints used respect to an integrated model. However, the problem is not completely disconnected, since when the bilevel model is solved, there is an impact on the related variables between both problems. In this paper, this impact is given by the DTs, since their capacity, location and maximum demand affect the solution of each level. This aspect allows finding solutions of great quality, due during the process the sensitivity

Fig. 11. Case 3 - Primary network. 155

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Fig. 12. Case 3 - Secondary networks.

of one level with respect to the other is taken into account; in other words, in spite of the fact that each level is solved separately, implicitly the incidence between both levels is considered. Some variants were made to the coding schemes traditionally used in the specialized literature. In the upper level coding scheme the sizing and locations of the DTs were involved, and in the coding scheme of the LV network, secondary network sections, phase balance and distributed generators were considered. This new coding was done in order to facilitate the movements and proposals between both agents. In fact, the obtained results show that the application of the TSA considering the proposed coding scheme is efficient, since it allows finding good quality configurations for the test system used. Future scope of the work are: (i) to consider a three-phase model of the primary network, (ii) to involve different technologies of DGs, (iii) to study the impact of energy storage elements together with DGs in the DSP, and (iv) to analyze the conflict when the utility is not the owner of the DGs.

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