Integrated reliability centered distribution system planning — Cable routing and switch placement

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Citation for the original published paper (version of record):

Duvnjak Zarkovic, S., Shayesteh, E., Hilber, P. (2021)

Integrated reliability centered distribution system planning — Cable routing and switch placement

Energy Reports, 7: 3099-3115

https://doi.org/10.1016/j.egyr.2021.05.045

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Research paper

Integrated reliability centered distribution system planning — Cable routing and switch placement

Sanja Duvnjak Žarković

^a,

^∗ , Ebrahim Shayesteh

^b

, Patrik Hilber

^a

aDepartment of Electromagnetic Engineering, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden

bSwedish National Grid, Stockholm, Sweden

a r t i c l e i n f o

Article history:

Received 16 March 2021

Received in revised form 6 May 2021 Accepted 24 May 2021

Available online xxxx

Keywords:

Cable routing

Distribution system planning Mixed-integer programming Network configuration Optimal switch placement Power system reliability Sectionalizers Tie switches

a b s t r a c t

Distribution utilities aim to operate and plan their network in a secure and economical way. The prime focus of this work is to assist utilities by developing a new integrated approach which considers the impacts of system reliability in distribution system planning (DSP). This approach merges different problems together and solves them in a two-stage process, as follows: 1. cable routing and optimal location and number of switching devices (circuit breakers and reclosers); 2. optimal location and number of tie switches. Moreover, the possibility of installing different cable options, with different prices and capacities, is included. The optimization algorithm is designed using mixed-integer programming (MIP). The developed algorithm analytically evaluates relationships between different components in the system and dynamically updates reliability indices, failure rate and restoration time, of every node in the system. This approach has been tested on two distribution systems. Despite the complexity and the exhaustiveness of the problem, MIP converges and provides the optimal solution for every studied scenario. The results show that an integrated approach enables utilities to obtain more comprehensive solutions. Moreover, by understanding the impact of parameter variation enables utilities to categorize their priorities in the decision making process and optimally invest in distribution network with respect to reliability.

© 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

The importance of electricity in everyday life and demands to improve the reliability of distribution systems force utilities to operate and plan their networks in a more secure and eco- nomical manner. Distribution system planning (DSP) is one of the major activities of distribution utilities to deal with reliability enhancement.

DSP is a decision making process that is usually carried out through the reinforcement, reconstruction or installation of new components in the distribution power system, with the aim to provide a reliable and cost effective service to consumers (Hei- dari and Fotuhi-Firuzabad, 2016). DSP studies can include find- ing the optimal number and location of switching devices, size and location of distribution substation, number of feeders and their routes, etc. Therefore, DSP represents a complex nonlinear problem that involves a large set of variables (both continuous and discrete). To solve DSP problems, mathematical or heuristic optimization methods are used (Sadegheih and Drake, 2008).

∗

Corresponding author.

E-mail addresses: sanjadz@kth.se(S.D. Žarković),

ebrahim.shayesteh@ee.kth.se(E. Shayesteh),hilber@kth.se(P. Hilber).

According to Shahsavari et al. (2015), better reliability may be achieved either by decreasing the customer interruption du- ration per fault or by decreasing the frequency of customer in- terruptions. This work successfully implements both of these achievements.

Namely, the main idea of this paper is to integrate different DSP problems together, i.e. optimal cable routing and switch placement, using a conventional optimization algorithm — mixed integer programming (MIP). The developed approach considers the impact of system reliability on a very detailed level. The reliability measure is the outage cost, that takes into account both failure rate and restoration time of nodes.

Reliability analysis of power systems considers both analytical and simulation approach. It may not always be possible to analyt- ically calculate reliability measurements that are of interest due to e.g. complexity of the system, complexity of the calculations, or some unknown functions (Wallnerström and Hilber, 2014). In general, simulation techniques allow a significantly easier im- plementation of complex connections and model details than analytical approach. Analytical approach is claimed to be exact, while the simulation approach approximative.

However, the focus of this work is a development of an op- timization algorithm, i.e. simulation approach, that analytically

https://doi.org/10.1016/j.egyr.2021.05.045

2352-4847/©2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Nomenclature

Indices

N Number of all nodes in the system

L Number of lines in the system

S Number of available switches

sn Index of the supply node

i , ^j , ^l Node indices

s Switch type index

Parameters

λ

^′

The failure rate of the cable (f/km,yr) λ ^(sn) Initial failure rate of the supply node

(f/yr)

r(i) Initial restoration time of node i (h/f) r

_tie

Improved restoration time due to tie

switch placement (h/f)

P

_L

(i) Active power demand at node i (kW) S

_L

(i) Apparent power demand at node i (kVA) LC (i , ^j) Power transfer capacity limit of the

cable (kVA)

C

_c

The investment and installation cost per km of the cable (

_e

/km)

C

_s

The investment and installation cost of the switch s (

_e

/unit)

C

_t

The investment and installation cost of tie switch(

_e

/unit)

MC

_s

The maintenance cost of the switch s (

_e

/yr)

MC

_t

The maintenance cost of the tie switch (

_e

/yr)

k(i) Average interruption cost rate for inter- rupted power at node i (

_e

/f,kW) c(i) Average interruption cost rate for en-

ergy not supplied at node i (

_e

/kWh) d(i , ^j) Euclidean distance between nodes i and

j (km)

sf Safety factor associated with the cable length needed for installation

f

_s

Scaling factor (

_e

/km)

I Continuous current carrying capacity of the cable (A)

V

_N

Nominal voltage level in feeders (kV)

T Time horizon under study (yr)

dr Annual discount rate (%)

lg Load growth rate (%)

Binary Variables

x(i , ^j) Equal to 1 if there is a direct link from node i to node j, and 0 otherwise y

_s

(i , ^j) Equal to 1 if there is a switch s directly

between nodes i and j, and 0 otherwise q(i , ^j) Equal to 1 if there is a direct link from node i to j without a switch, and 0 otherwise

z(i , ^j) Equal to 1 if there is a direct link from node i to j with a switch, and 0 otherwise

q(i) Equal to 1 if there is at least one direct link from node i without a switch, and 0 otherwise

x

^′

(i , ^j) Equal to 1 if there is a link from node i to node j (i and j do not need to be directly linked, but they belong to the same feeder), and 0 otherwise

y

^′_s

(i , ^j) Equal to 1 if there is a switch s between nodes i and j (i and j do not need to be directly linked, but they belong to the same feeder), and 0 otherwise

q

^′

(i , ^j) Equal to 1 if there is a link from node i to j without a switch, and 0 otherwise z

^′

(i , ^j) Equal to 1 if there is a link from node i

to j with a switch, and 0 otherwise t(i , ^j) Equal to 1 if there is a tie switch from

node i to j, and 0 otherwise

t

^′

(i , ^j) Equal to 1 if there is a tie switch from node j to i, and 0 otherwise

Continuous Variables

λ

N

(i) The final failure rate of node i (f/yr) λ

^′_N

⁽ⁱ⁾ The auxiliary failure rate of node i (f/yr) λ

^′′_N

⁽ⁱ⁾ The auxiliary failure rate of node i (f/yr) λ

L

(i , ^j) The failure rate of the cable between

nodes i and j (f/yr)

r

_N

(i) The final updated restoration time of node i (h/f)

r(i , ^j) Restoration time of node j due to fault at node i (h/f)

U(i) Unavailability of node i (h/yr)

Aux(i , ^j) Auxiliary variable that checks if nodes i and j belong to the same feeder LF (i , ^j) Power transfer between nodes i and j

(kW)

V

_L

(i) Voltage at bus node i (kV)

D Distance between nodes connected by a tie switch (km)

C

^IC

Cable investment and installation cost (

_e

)

C

^IS

Switch investment and installation cost (

_e

)

C

^MS

Switch maintenance cost (

_e

)

C

^OC

The average customer outage cost (

_e

) C

^IT

Tie switch investment and installation

cost (

_e

)

C

^MT

Tie switch maintenance cost (

_e

)

evaluates relationships between different components and dy- namically updates failure rate and restoration time of every node in the system, while simultaneously deciding on the cable out- line and optimal switch placement. The aim of such approach is to obtain exact values of reliability indices, rather than just approximations. However, computation time is considered to be the biggest drawback of such simulations.

The algorithm is designed in two stages. This is also known as a stage-by-stage planning, where a single stage is planned at a time, followed by the planning of the next stage (Ehsan and Yang, 2019). This type of planning differs from the single- stage planning, where the planning is simply performed for one stage or multi-stage planning that carries out the planning of all stages at the same time and hence, minimizes the total costs over the planning horizon. Although total costs over the planning

3100

horizon are not minimized, stage-by-stage approach minimizes the investments and costs in each stage.

In the first stage, cable routing and optimal placement of two types of switching devices (circuit breakers and reclosers) are done simultaneously. Both of these problems are highly com- plex, combinatorial and non-linear. Solving them together brings additional complexity and challenge. Cable routing is based on whether to build new cables/routes between nodes (whose po- sitions are fixed), or to keep existing ones, taking into account cables’ capacity and maintaining the radiality of the network.

Moreover, the developed cable routing considers different phases in DSP, from keeping the cable layout as it is, up to building the system from scratch. On the other hand, switching devices positioned on suitable places throughout the network can signif- icantly improve the reliability of the power system — by isolating faulty segment(s) and thereby keeping the upstream sections healthy (Sahoo et al., 2012). The novelty in the proposed opti- mization model is the calculation of the outage cost which is directly linked to the proposed cable layout and switch place- ment. Moreover, based on the suggested outline of the network, the failure rate as well as the outage cost of every node is dy- namically updated. With the obtained results from stage 1, stage 2 focuses on optimal placement and number of tie switches. Tie switches are used to provide supply to some feeder branches due to faults occurring in an upstream branch (Sahoo et al., 2012).

The algorithm identifies nodes whose restoration time can be improved and places tie switches between them. By improving the restoration time, outage cost of every node is improved as well. Therefore, it is very important that tie switch placement is done after the outline of the network in stage 1 is obtained, in order to recognize where the tie switch would be the most beneficial.

1.1. Literature review

This subsection reviews relevant work related to optimal feeder routing, optimal switch placement or both (see Table 1).

Authors in Mehrtash et al. (2019) propose a novel graph-based model for the resilient feeder routing problem using geographical information system (GIS). The uncertainty of rooftop solar gener- ations and demand forecasting errors are taken into account, and a stochastic programming-based solution algorithm is developed.

Economic resilience metric that is not related to any specific disruption, but rather its consequences is introduced. In Ahmed et al. (2017), a stochastic optimization algorithm for the planning of distribution system feeders is provided. The algorithm finds the optimal feeder routing considering the stochastic variations of load demands (e.g., electric vehicle charging stations) as well as renewable-based distributed generators (DGs) (e.g., photovoltaic and wind DGs). The main optimization problem is decomposed into a master problem and a sub-problem. The master prob- lem is formulated using a genetic algorithm (GA) that generates different feeder routing scenarios, and the sub-problem is used to solve the optimal power flow problem for each GA scenario.

In order to solve the midterm and long-term optimal feeder routing problem, Taghizadegan et al. (2019) proposes a stochastic multistage expansion planning method. The proposed method is solved using particle swarm optimization algorithm which finally converges to a solution with minimum costs, comprised of feeder’s installation costs, power losses cost, cost of active purchased power from power market, and reliability costs. In Ku- mar et al. (2014), six different graph theory based approaches are applied for optimal feeder routing. However, the reliabil- ity assessment, that includes calculation of indices SAIFI, SAIDI, CAIDI, ASAI and ASUI, is done in the second phase, after obtaining the network layout. The study in Ghadiri et al. (2017) evaluates

the economic efficiency of hybrid AC/DC distribution systems compared to conventional AC ones via genetic algorithm. The pro- posed methodology determines the optimal AC/DC distribution substation location and size and AC/DC feeder routing, as well as length and capacity of AC/DC feeders in both low voltage and medium voltage sides. The cost function comprises of investment and operational costs, including costs of construction, reliability, and loss. In Ugranlı (2018), a multistage (dynamic) distribution network expansion planning is proposed. The intermittent nature of load, solar, and wind power are modeled using several time blocks in the proposed method. The objective function of the proposed distribution network expansion problem consists of the installation costs of substations, feeders, costs of energy pur- chased from substations and cost of unserved energy. The overall problem is formulated as a mixed-integer linear programming.

In the recent years, cable routing problem alone was exten- sively applied on wind farms. Assuming that the site and the best turbine positions have been identified, finding the optimal cable connection among turbines, while minimizing the total cable cost is implemented through a mixed mixed-integer linear pro- gramming (MILP) in Fischetti and Pisinger (2019, 2018a, 2017), Fischetti and Monaci (2016), Fischetti and Pisinger (2018b,c), a hop-indexed integer programming in Bauer and Lysgaard (2015), a fast heuristic specifically designed for wind farm cable routing problem in Gritzbach et al. (2018) and particle swarm opti- mization algorithm in Jin et al. (2019). In Cazzaro et al. (2020), authors describe, implement and test five different metaheuristic schemes for cable routing problem within an offshore wind farm:

Simulated Annealing, Tabu Search, Variable Neighborhood Search, Ants Algorithm, and Genetic Algorithm. Moreover, a construction heuristic, called Sweep is described, that typically finds an initial high-quality solution in a very short computing time. It has been shown that Variable Neighborhood Search obtains the best overall performance, while Tabu Search is the second best heuristic.

Nevertheless, the previous papers that dealt with cable routing within a wind farm failed to identify the reliability of the analyzed system. However, in Žarković et al. (2021), beside minimizing cable installation cost in the onshore wind farm, the developed MILP algorithm aims to minimize the cost of lost energy produc- tion, which is considered as the measure of reliability. Moreover, based on the suggested outline of the network, the failure rate as well as the cost of lost energy production for every wind turbine within the farm is updated. The algorithm also takes into account the possibility of installing cables with different capacities and different costs. However, the algorithm assumes that switching equipment are a part of the turbine supply and do not take into account optimal switch placement.

On the other hand, papers that deal with optimal switch place-

ment almost always consider reliability assessment. For example,

the solution framework that considers allocation of sectionalizing

and tie switches of different capacities, with manual or automatic

operation schemes, is proposed in de Assis et al. (2014). The

approach minimizes the costs of allocation and energy not sup-

plied, under reliability and flow capacity constraints. The solution

framework is based on genetic and memetic algorithm concepts

with a structured population. The algorithm defines four different

cases for assessing t

_kl

(the expected duration of interruption in

section k caused by failures in section l). In Ghoreishi et al. (2013),

a genetic algorithm based method is proposed to determine accu-

rate location of sectionalizers and tie points in order to minimize

the cost of energy not supplied and the cost of switches and lines

installation. This placement is aimed at restoration of maximum

loads with utmost importance in a distribution network. Authors

in Shahbazian et al. (2020) present an innovative high-accuracy

linear conversion for a MILP approach that aims to optimally

deploy manual and automatic switches in the distribution system

Table 1

Taxonomy of the reviewed work.

Reference Optimal cable/feeder routing Optimal switch placement Optimization Algorithm used for Problem solving

Mehrtash et al.(2019) ✓ Mixed-integer second-order

cone programming

Ahmed et al.(2017) ✓ Genetic algorithm

Taghizadegan et al.(2019) ✓ Particle swarm optimization

Kumar et al.(2014) ✓ Graph theory based approach

(Ghadiri et al.,2017) ✓ Genetic algorithm

(Ugranlı,2018) ✓ MILP

Fischetti and Pisinger(2019) ✓ MILP

Fischetti and Pisinger(2018a) ✓ MILP

Fischetti and Pisinger(2017) ✓ MILP

Fischetti and Monaci(2016) ✓ MILP

Fischetti and Pisinger(2018b) ✓ MILP

Fischetti and Pisinger(2018c) ✓ MILP

Bauer and Lysgaard(2015) ✓ A hop-indexed integer

programming

Gritzbach et al.(2018) ✓ A fast heuristic

Jin et al.(2019) ✓ Particle swarm optimization

Cazzaro et al.(2020) ✓ Simulated Annealing, Tabu

Search, Variable Neighborhood Search, Ants Algorithm, Genetic Algorithm, Sweep

Žarković et al.(2021) ✓ MILP

de Assis et al.(2014) ✓ Genetic and memetic algorithm

Ghoreishi et al.(2013) ✓ Genetic algorithm

Shahbazian et al.(2020) ✓ MILP

Conti et al.(2017) ✓ Genetic algorithm

Karimi et al.(2021) ✓ Genetic algorithm

Farajollahi et al.(2017) ✓ MIP

Farajollahi et al.(2018) ✓ MIP

Popović et al.(2017) ✓ MILP

Sahoo et al.(2012) ✓ ✓ Particle swarm optimization

Heidari et al.(2015) ✓ Genetic algorithm

Heidari and Fotuhi-Firuzabad(2018) ✓ ✓ Genetic algorithm

to improve system reliability. The objective is to minimize SAIDI, customer interruption cost, investment as well as maintenance costs. Authors in Conti et al. (2017) present a way to provide the optimal number, type, and position of the switches that must be installed each year to achieve both the targeted annual reliabil- ity improvement and the cheapest investment cost during the overall regulatory period. The proposed approach is formulated as a multiobjective optimization problem and it is solved using a genetic algorithm. A model that considers the failure probability of the sectionalizing switches in their optimal placement issue is proposed in Karimi et al. (2021). The model aims to mini- mize switch installation cost and the interruption costs incurred through interrupted users. This study uses the discrete Markov chain model with the aim of obtaining the malfunction possibility under various states. Afterwards, the placement problem is solved with the aim of obtaining the global optimal solution with genetic algorithm. Similarly, the model in Farajollahi et al. (2017) aims to minimize the total costs of sectionalizing switches as well as customers interruption cost, while integrating malfunction probability of switches. The proposed model is formulated as a mixed-integer programming. Same authors propose in Farajol- lahi et al. (2018) a MIP mathematical model to optimally place automation system devices within distribution networks, such as fault indicators and remote controlled switches. The model estab- lishes a trade-off between service reliability improvements and

the relevant costs. Authors in Popović et al. (2017) present a MILP based approach for determining the optimal number, type and lo- cation of automation devices, such as remotely controlled circuit breakers/reclosers, sectionalizing switches, remotely supervised fault passage indicators. Simultaneously, they determine the new (optimal) locations of the automation devices that already exist in the network. In determining the most effective network au- tomation scenario, the proposed approach takes into account the outage cost of consumers/producers due to momentary, short- term, and long-term interruptions, the commonly used network reliability indices (SAIFI, SAIDI, MAIFI, and ASIDI) as well as the cost of automation devices and the cost of crews.

Solving and optimizing each one of these problems contributes greatly to the reliability improvement. Nevertheless, it is impor- tant to emphasize that DSP problems are interconnected and they all affect each other. Therefore, going towards the holistic approach and solving them together, despite all the challenges, can result in a more appropriate and realistic DSP model.

As already mentioned, the main idea of this paper is to in- tegrate different DSP problems together — optimal cable rout- ing and switch placement. This integrated problem, including feeder routing and number and locations of sectionalizing and tie switches, is tackled in Sahoo et al. (2012) as a multi-objective planning approach using particle swarm optimization. However, here the system reliability is expressed through contingency- load-loss index (CLLI). CLLI is a reliability indicator, that is defined

3102

as the ratio of average non-delivered load due to failure of all branches, considered one at a time, to the total load. DSP model proposed in Heidari et al. (2015) assumes that the network under study is a distribution automation system (DAS), which includes automatic voltage and VAR control (AVVC) functions and auto- matic fault management (AFM). However, it does not propose switch placement, only reinforcement of lines and substations.

Genetic algorithm is employed in solving the mixed-integer non- linear programming problem. The same authors did a consecutive study (Heidari and Fotuhi-Firuzabad, 2018) with a novel inte- grated planning model in which network capacity expansion and DAS implementation are performed simultaneously. They pointed out the fact that the large number and variety of the decision variables are the most important challenges for the development of integrated planning method. To overcome this challenge, they combined two separate genetic algorithms, for network capacity expansion and DAS implementation.

It is interesting to notice that the above approaches deal with the complexity of integrated DSP problems using meta-heuristic algorithms. These algorithms have been developed in order to replace conventional optimization methods, which are very in- efficient regarding computational time and real-time control in a large distribution system. Although the use of metaheuristic algorithms makes the number of computation faster and more manageable, these algorithms do not represent the best solution in every implementation (Žarković et al., 2018). Since DSP is a planning and not a real-time problem, the idea of getting more optimal instead of fast results is highly preferable.

Moreover, the reliability of the system is in the most cases evaluated by assessing only restoration time, and neglecting the change in the failure rate, or just taking the static values of failure rates.

1.2. Contributions

The main contributions of this paper are:

• Developing an integrated DSP model, that involves optimal cable routing and switch placement.

• Using conventional mixed-integer programming as a solu- tion algorithm in order to emphasize the importance of getting more optimal instead of fast results.

• Solving the DSP problem in two-stage process: 1. cable routing and optimal location and number of switching de- vices (circuit breakers and reclosers); 2. optimal location and number of tie switches.

• Performing analytical reliability assessment of the proposed network through simulation approach, which is considered as the novelty in the proposed algorithm. The reliability measure is the customer outage cost, that takes into account both failure rate and restoration time of nodes, that are dynamically updated throughout the process.

– The purpose of stage 1 is to simultaneously minimize

cable installation cost, switch installation and mainte- nance cost and customer outage cost, and dynamically update failure rate of every node, while deciding on the network outline — cable routes and places of switches.

– In stage 2 restoration time of every node is dynamically

updated based on the tie switch placement.

• Introducing willingness to rebuild cables in order to address different phases during a DSP process.

• Taking into account the possibility of installing different cable options, with different prices and capacities.

• Performing an exhaustive sensitivity analysis and parameter variation in order to obtain more comprehensive solution and help distribution utilities to categorize their priorities in the decision making process.

• Testing the developed approach on two distribution sys- tems, while taking into account multiple case scenarios.

2. Proposed formulation

2.1. Stage 1 - cable routing and switch placement

The optimization model in the first stage is designed to in- vestigate the best possible combination of node connections and switch positions. The main optimization objective is to minimize the total cost, as formulated in (1).

min

x(i,j),ys(i,j)

(

C

^IC

+ C

^IS

+ C

^MS

+ C

^OC

)

(1)

C

^IC

represents investment and installation cost of the potential cable route between nodes.

C

^IC

=

N

∑

i=₁ N

∑

j=₁

x(i , ^j) · C

_c

· d(i , ^j) · sf (2)

C

^IS

represents investment and installation cost of a switch installed between two connected nodes.

C

^IS

=

N

∑

i=1 N

∑

j=1 S

∑

s=1

y

_s

(i , ^j) · C

_s

(3)

C

^MS

is the maintenance cost of the particular installed switch.

C

^MS

=

T

∑

t=₁

1 (1 + dr)

^t

·

(

N

∑

i=₁ N

∑

j=₁ S

∑

s=₁

y

_s

(i , ^j) · MC

_s

)

(4)

C

^OC

is an outage cost that reflects the impacts of interruption duration, failure rate of nodes, load variation and customer dam- age function (Hilber, 2008). In the proposed optimization model, outage cost is the measure of reliability, since it is directly linked to the proposed configuration and switch placement. Based on the suggested outline of the network, the failure rate as well as the outage cost of every node is updated. The lower the outage cost, the better the reliability.

C

^OC

=

T

∑

t=₁

( 1 + lg 1 + dr

)

t

· (

N

∑

i=₁

λ

N

(i) · L

_a

(i) · (k(i) + c(i) · r(i)) )

(5)

The binary decision variables are described as follows:

x(i , ^j) =

{ 1 , if there is a direct link from node i to node j 0 , ^otherwise

y_s(i

,

^j)

= {

1

,

if there is a switch s directly between nodes i and j 0

,

^otherwise

In this model, it is possible to install different switch types.

Number of possible switch options is S (S ∈ { 1 , ² ...} ^{) which}

means that there are S binary decision variables, corresponding to

S switch types with different costs and features. It is important to

notice that these binary variables exclude one another, i.e. if one

of them is 1, all others are 0 for the same pair of nodes (i, j). In

this paper two types of switches are considered: circuit breaker

and recloser with radial topology. Recloser, as well as circuit

breaker, is able to interrupt current flow after a fault is detected

and to isolate a faulted part from a healthy part, interrupting all

downstream customers (Brown and Spare, 2004). No additional restoration is possible until the fault is repaired.

S

∑

s=₁

y

_s

(i , ^j) = y

_cb

(i , ^j) + y

_rc

(i , ^{j) (6)} Derived binary variables, q(i , ^{j) and z(i} , j), are described by (7) and (8).

q(i

,

^j)

= {

1

,

if there is a direct link from i to j without a switch 0

,

^otherwise

z(i , ^j) =

{ 1 , if there is a direct link from i to j with a switch 0 , ^otherwise

q(i , ^j) = x(i , ^j) · (1 −

S

∑

s=₁

y

_s

(i , ^{j)) (7)}

z(i , ^j) = x(i , ^j) ·

S

∑

s=₁

y

_s

(i , ^{j) (8)}

The optimization model is subjected to the following con- straints:

λ

L

(i , ^j) = λ

^′

· d(i , ^j) · sf (9)

λ

^′′N

(sn) = λ ^{(sn) (10)}

x(i , ^j) · λ

^′′_N

^(j) ≥ x(i , ^j) · ( λ

^′′_N

⁽ⁱ⁾ + λ

L

(i , ^{j)) (11)} λ

^′N

(i) ≥ λ

^′′N

(i) +

N

∑

j=₁

q(i , ^j) · λ

L

(i , ^{j) (12)}

q(i) · λ

^′_N

⁽ⁱ⁾ ≥

N

∑

j=₁

q(i , ^j) · λ

^′_N

^{(j) (13)}

q(i) ≥ q(i , ^{j) (14)}

z(i , ^j) · λ

^′_N

⁽ⁱ⁾ = z(i , ^j) · ( λ

^′_N

^(j) − λ

L

(i , ^{j)) (15)}

λ

N

(sn) = λ

^′′N

(sn) (16)

λ

N

(j) =

N

∑

i=₁

x(i , ^j) · ( λ

^′_N

⁽ⁱ⁾ +

S

∑

s=₁

y

_s

(i , ^j) · λ

L

(i , ^j)) , ∀ ^j ̸= sn (17)

x(i , ⁱ⁾ = 0 (18)

N

∑

i=₁

x(i , ^j) = 1 , ∀ ^j ̸= sn (19)

x(i , ^j) + x(j , ⁱ⁾ ≤ 1 , ⁱ ̸= j (20)

N

∑

j=₁

x(sn , ^j) ≥ 1 (21)

S

∑

s=₁

y

_s

≤ x(i , ^{j) (22)}

S

∑

s=₁

y

_s

(i , ⁱ⁾ = 0 (23)

S

∑

s=₁

y

_s

(i , ^j) ≤ 1 (24)

y

_cb

(sn , ^j) = x(sn , ^{j) (25)}

y

_cb

(i , ^j) = 0 , ∀ ⁱ ̸= sn (26)

LC =

√

3V

_N

I (27)

N

∑

l=1

LF (l , ⁱ⁾ = S

_L

(i) · (1 + lg)

^T

+

N

∑

j=1

LF (i , ^j) , ∀ ⁱ ̸= sn (28)

LF (i , ^j) ≤ x(i , ^j) · LC (i , ^j) · 0 . ^{5 (29)}

V

_min

≤ V

_L

(i) ≤ V

_max

(30)

Failure rate of every cable depends on the distance between nodes i and j, and it is described by (9). Eqs. (10)–(17) describe the process of calculating the failure rate of each node. The concept of the suggested algorithm is to analyze the network, move up- and downstream through it, and update failure rate of every node based on the proposed cable layout and switch placement.

The algorithm starts by initializing failure rate of the supply node (10). Afterwards, the algorithm moves downstream through the network, with the assumption that there is a switch between every two connected nodes, i and j. That means the failure rate of every node λ

^′′_N

(j) is affected only by its upstream part of the network (failure rate of upstream node λ

^′′N

(i) and upstream cable λ

L

(i , j)). Any failure that happens downstream of that node is automatically isolated by the switch. Consequently, every down- stream node j has a higher failure rate than its upstream node i; hence the auxiliary failure rate of every node λ

^′′_N

is calculated (11). Once the algorithm reaches the bottom of the network, it starts moving upstream. By doing that, now it actually checks where did optimization propose switches and updates failure rate of every node ( λ

^′_N

). Any line not secured by the switch automatically affects failure rates of its upstream nodes (12), (13).

Eq. (14) is a constraint assuring that the binary variables are in consistence with one another. Eq. (15) assures that every node i protected with the switch is not affected by the failure rate of its downstream node j. Once the algorithm reaches the top, it updates the failure rate of the supply node (16) and moves again downstream for the final check and update (17).

The proposed optimization algorithm involves multiplication of binary and continuous variables, as well as multiplication of two binary variables, making the problem nonlinear. According to Klanšek (2015), mixed integer linear programming (MILP) holds advantage over mixed integer nonlinear programming (MINLP) from the view point of solution time and result quality.

Moreover, even though MINLP can find acceptable solutions for nonlinear problems, it is significantly more complex and has not yet reached the state of maturity and reliability as MILP. There- fore, to solve whole optimization problem as MILP, linearization methods presented in Coelho (2013) are used.

Eqs. (18), (19), (20) represent constraints that are ensuring radial supply to each node and hence only one upstream node, without any loops made (Ahmadi and Martí, 2015). Eq. (18) ensures that there is no connection leading from and to the same node. In (19) it is assured that every node (except supply node) needs to have only one upstream connection. This paper considers directional connections, meaning that x(i , ^{j) and x(j} , ⁱ⁾ are not the same. If there is a connection from node i to node j (i being the upstream node), (20) prevents possible connection from j to i, i.e. creating a loop in the system between nodes i and j. The supply node is assumed to be capable of serving the power demand to all sink nodes. The link to the supply node, i.e. at least one connection downstream of the supply node, is confirmed by (21). Eqs. (22) and (23) represent switch placement constraints that prevent placing the switch between two nodes that are not connected. It is assumed that the switch installed between nodes i and j is positioned just below the node i, i.e. at the beginning of the section. Thus, the switch is able to isolate any fault that happens below node i. Eq. (24) ensures that S binary variables exclude one another, i.e. if one of them is 1, all others are 0 for the same pair of nodes (i, j). Eqs. (25) and (26) define

3104

the rules for placing the circuit breaker. Circuit breaker should be installed always and only at the beginning of every feeder, i.e. on the lines that are directly connected to the supply node.

The power transfer capacity of the cables is calculated according to (27), where V denotes the voltage level in the feeders and I denotes the current capacity of the cables. Eq. (28) represents power balance at node i. In other words, (28) assures that the input power flow to node i (i.e. the summation of power flow entering the node i from the upstream node l) is equal to the demand at node i in the last year of the planning horizon plus the output power flow from node i (i.e. the summation of power flow exiting to the downstream node(s) j). Eq. (29), on the other hand, guarantees that the power flow at each line is smaller than the power capacity of the line. As suggested by the utility practice, a safety margin of 50% usability of the current capacity is adopted for normal operation conditions (Babu, 2017). Eq. (30) represents constraint that keeps the voltage of every node within 5% of the nominal value.

The optimization ensures the delivery of demanded load to all nodes, radial connectivity and power transfer boundaries of the cables.

2.2. Stage 2 - tie switch placement

The main purpose of Stage 2 is to choose the best positions of tie switches. Tie switch should connect two nodes that belong to different feeders, taking into account the distance between nodes as well. The tie switch’s function is to alter network topology in order to provide alternate routes for supplying power to loads when necessary (Sritakaew and Sangswang, 2006). Tie switch helps in improving restoration time of nodes that can be isolated from the fault. This affects outage cost, i.e. reliability of the system. Optimal tie switch placement is highly dependent on the network outline (cable routing and switch placement) obtained in stage 1. Therefore, output parameters of stage 1 are input parameters of stage 2.

Beside λ

N

(i) that is obtained in stage 1, there are two more input parameters, binary variables x

^′

(i , ^{j) and y}

^′s

(i , j) and they slightly differ from decision variables x(i , ^{j) and y}

s

(i , ^j).

x

^′

(i , ^j) =

⎧

⎪ ⎪

⎨

⎪ ⎪

⎩

1 , if there is a link from node i to node j,

where i and j do not need to be directly linked, they just need to belong to the same feeder 0 , ^otherwise

y

^′_s

(i , ^j) =

⎧

⎪ ⎪

⎨

⎪ ⎪

⎩

1 , if there is a switch s between nodes i and j, where i and j do not need to be directly linked, they just need to belong to the same feeder 0 , ^otherwise

Derived binary variables, q

^′

(i , ^{j) and z}

^′

⁽ⁱ , j), are described by (31) and (32).

q

^′

(i , ^j) =

{ 1 , if there is a link from i to j without a switch 0 , ^otherwise

z

^′

(i , ^j) =

{ 1 , if there is a link from i to j with a switch 0 , ^otherwise

q

^′

(i , ^j) = x

^′

(i , ^j) · (1 −

S

∑

s=₁

y

^′_s

(i , ^{j)) (31)}

z

^′

(i , ^j) = x

^′

(i , ^j) ·

S

∑

s=₁

y

^′_s

(i , ^{j) (32)}

The main objective of stage 2 algorithm is to find nodes which outage cost can be improved (lowered) and connect them through tie switches, taking into account tie switch investment and maintenance cost, as formulated in (33).

min

t(i,^j)

(

C

^IT

+ C

^MT

+ C

^OC

)

(33)

C

^IT

represents investment and installation cost of the tie switch.

C

^IT

=

N

∑

i=₁ N

∑

j=₁

t(i , ^j) · C

_t

(34)

C

^MT

is the maintenance cost of the tie switch.

C

^MT

=

T

∑

t=₁

1 (1 + dr)

^t

·

(

N

∑

i=₁ N

∑

j=₁

t(i , ^j) · MC

_t

)

(35)

Customer outage cost C

^OC

in stage 2 is updated with new restoration time for every node r

_N

(i).

C

^OC

=

T

∑

t=₁

( 1 + lg 1 + dr

)

t

· (

N

∑

i=₁

λ

N

(i) · L

_a

(i) · (k(i) + c(i) · r

_N

(i)) )

(36)

The binary decision variable in stage 2 is described as follows:

t(i , ^j) =

{ 1 , if there is a tie switch from node i to node j 0 , ^otherwise

The optimization is subjected to the following constraints:

t(i , ⁱ⁾ = 0 (37)

t(i , ^j) + t(j , ⁱ⁾ ≤ 1 (38)

t

^′

(i , ^j) = t(j , ^{i) (39)}

t(i , ^j) ≤ (1 − x

^′

(i , ^j)) · (1 − x

^′

(j , ^{i)) (40)} (1 − t(i , ^j)) · Aux(i , ^j) =

N

∑

l=₁

x

^′

(l , ⁱ⁾ · x

^′

(l , ^j) , ∀ ^l ̸= sn (41) (1 − t

^′

(i , ^j)) · Aux(i , ^j) =

N

∑

l=₁

x

^′

(l , ⁱ⁾ · x

^′

(l , ^j) , ∀ ^l ̸= sn (42) U(j) =

N

∑

i=₁

λ

N

(i) · r(i , ^{j) (43)}

r

_N

(j) = ∑ ^U(j)

i∈_Sij

λ

N

(i) (44)

Eqs. (37) and (38) prevent placing tie switch from and to the same node and placing more than one tie switch between two same nodes, respectively. In order to assist updating restoration time of every node, an auxiliary binary variable t

^′

(i , j) is defined in the (39). t

^′

(i , j) is equal to 1 when t(j , i) is equal to 1. Eq. (40) prevents placing tie switch between two nodes that are linked and belong to the same feeder. Aux(i , j) is the auxiliary variable that checks if nodes i and j have common upstream node l (beside supply node). If they do, (41) and (42) prevent placing tie switch between two such nodes.

Important part of stage 2 algorithm is to identify interde-

pendencies between nodes and calculate unavailability of every

node (43) and afterwards to update restoration time of every

node (44). These interdependencies are expressed through r(i , ^j)

- restoration time of node j due to fault at node i. S

_ij

is a set

of all nodes i affecting node j. There are six rules defined in the

algorithm needed to assess r(i , j) is as given in Box I.

r(i , ^j) =

⎧

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

r(i) , ⁱ = j

r(s) , ⁱ = s

q

^′

(j , ⁱ⁾ · r(i) , if there is a link between j and i without a switch (j being the upstream node), supply to node j will be restored after fault at i is cleared

( ∑

N

l=1

t(j , ^l)) · (q

^′

(i , ^j) · r(i) + z

^′

(i , ^j) · r

_tie

) , if there is a tie switch leading from node j, the restora- tion time of node j due to fault at i will be improved only if there is a link with a switch between i and j

(1 − ∑

N

l=₁

t(j , ^l)) · ( ∑

N

l=₁

t

^′

(j , ^l)) · (q

^′

(i , ^j) · r(i) + z

^′

(i , ^j) · r

_tie

) , if there is a tie switch leading to node j, the restoration time of node j due to fault at i will be improved only if there is a link with a switch between i and j

(1 − ∑

N

l=₁

t(j , ^l)) · (1 − ∑

N

l=₁

t

^′

(j , ^l)) · (x

^′

(i , ^j) · r(i)) , if there is no tie switch from and to node j, supply to node j due to fault at upstream node i will always be restored after fault at i is cleared

Box I.

Fig. 1. 5-node system example.

Table 2

Calculation of failure rate for every node — example.

Affecting elements Affected elements

Node 1 Node 2 Node 3 Node 4 Node 5

Node 1 λ1 λ1 λ1 λ1 λ1

Line 1–2 λL12 λL12

Line 1–3 λL13 λL13

Line 2–4 λL24

Line 3–5 λL35 λL35

Summation λN1=∑

λN2=∑

λN3=∑

λN4=∑

λN5=∑

2.3. Simple 5-node system example

A gist of the proposed algorithm in both stages can be ex- plained through a simple example. A 5-node system is considered, where node 1 is the supply node, while 2, 3, 4 and 5 are sink nodes. Initial failure rate of the supply node is given ( λ

1

), as well as the average restoration time of every node (r

₁

, r

₂

, r

₃

, r

₄

, r

₅

).

Each possible connection/line between nodes has also its own failure rate and it is proportional to the distance between nodes (9). A possible network outline proposed by the optimization after running both stages is depicted in Fig. 1.

In the first stage, cable routing and switch placing are opti- mized, and failure rate of every node is updated based on the

proposed outline. Final failure rate of every node is a summation of all elements affecting that node (Table 2).

Values for binary variables x(i , ^{j), y}

s

(i , ^{j), x}

^′

⁽ⁱ , ^{j) and y}

^′_s

⁽ⁱ , ^{j), are} presented in matrices below.

x(i , ^{j) 1 2 3 4 5}

1 1 1

2 1

3 1

4 5

x

^′

(i , ^{j) 1 2 3 4 5}

1 1 1 1 1

2 1

3 1

4 5

y

_s

(i , ^{j) 1 2 3 4 5}

1 1 1

2 1

3 4 5

y

^′_s

(i , ^{j) 1 2 3 4 5}

1 1 1 1 1

2 1

3 4 5

In the second stage tie switch placement is optimized and restoration time of every node is updated (43), (44). Interdepen- dencies between nodes are presented in Table 3, where r(i , ^j) represents restoration time of node j due to fault at node i.

As it can be seen from Table 3, tie switch has affected and improved only restoration time r(2 , 4). Therefore, the idea of the proposed stage 2 algorithm is to identify nodes whose restoration

3106

Fig. 2. Existing layouts.

Table 3

Calculation of restoration time for every node — example.

r(i,^{j) 1 2 3 4 5}

1 r1 r1 r1 r1 r1

2 r2 rtie

3 r3 r3

4 r4

5 r5 r5

time can be improved (based on the network outline) and place tie switches between them.

2.4. Willingness to rebuild cables

Distribution system can go through different phases during a planning process. These phases can include existing system with certain upgrades, heavy reinforcements of existing system or even building a new system from scratch. To address these phases for a cable routing problem, a new term is introduced:

willingness to rebuild cables.

For an existing distribution system that does not yield any cable layout changes, the willingness to rebuild cables is 0%.

On the other hand, to connect nodes completely from scratch implies 100% willingness to rebuild cables. Everything in between can depend on utility’s willingness to change cable layout or rebuild certain cables due to different reasons, such as reliability improvement, capacity constraints, etc.

Therefore, willingness to rebuild cables is considered as a changeable percentage of a cost needed for construction of an already existing cable. In other words, for a potential new line between two nodes the optimization model considers full con- struction cost, while for an existing cable it takes into account only a fraction of the full cost (which depends on the willingness’s value).

3. Case study

3.1. Test systems

The effectiveness of the developed algorithm is tested on two systems. These systems are based on an actual feeder network

from the Stockholm municipality area in Sweden (Babu, 2017).

One reference network consists of one primary substation (supply node) and 52 secondary substations (sink nodes), while the other has one supply and 29 sink nodes. The supply node in both cases is capable of serving the power demand to all sink nodes. Detailed data collection from utility sources include: the design of the grid, load demand for every sink node and geographic coordinates of every node. Based on these coordinates, Euclidean distances between nodes are estimated. To address the real terrain between the nodes, these Euclidean distances are multiplied with a safety factor, sf . According to the utility practice (Žarković et al., 2019), sf is assumed to be 1.7. Graphical representations of both systems are presented in Figs. 2(a) and 2(b).

In these models, nodes are interconnected by a 22 kV under- ground cable network. The current capacity of cables is assumed to be 375 A. The average price for this cable type is obtained from Swedish Energy Market Inspectorate (Energimarknadsin- spektionen) (Energimarknadsinspektionen, 2020). This price cov- ers catalog price for the indicated cable (with specified size and voltage level) but also installation, working and some unforeseen costs, such as blasting, stones, crossing obstacles. All relevant cost parameters are presented in Table 4.

Reliability data for network elements are based on the Elforsk technical report (He, 2007). Failure rate of the cable λ

^′

^{is 0.02} f/(km,yr). Initial failure rate of the supply node λ (sn) is 0.02 f/yr, while the restoration time of every substation r(i) is 5 h/f.

Restoration time improved by placing a tie switch r

_tie

is assumed to be 0.5 h/f. It is assumed that the customer type that pre- vails in the system is households. Therefore, interruption cost rates for energy not supplied c(i) and interrupted power k(i) are considered to be 0.584

_e

/kWh and 0.195

_e

/(f,kW), respec- tively (Energimarknadsinspektionen, 2019). The time horizon T under study is 15 years, the annual discount rate dr is 8%, and the annual load growth rate lr is 2% (Popović et al., 2017).

To address different phases for a cable routing problem, five different scenarios are considered in stage 1: building a system from scratch (100% willingness to rebuild cables), only placing switches and not changing cable layout (0% willingness to rebuild cables), setting willingness to rebuild cables to 5, 10 and 20%.

Setting willingness to a certain percentage forces optimization to