Multi‐criteria optimisation approach to increase the delivered power in radial distribution networks

Abstract

This study proposes a new methodology to increase the power delivered to any load point in a radial distribution network, through the identification of new investments in order to improve the repair time. This research work is innovative and consists in proposing a full optimisation model based on mixed‐integer non‐linear programming considering the Pareto front technique. The goal is to achieve a reduction in repair times of the distribution networks components, while minimising the costs of that reduction as well as non‐supplied energy costs. The optimisation model considers the distribution network technical constraints, the substation transformer taps, and it is able to choose the capacitor banks size. A case study based on a 33‐bus distribution network is presented in order to illustrate in detail the application of the proposed methodology.

Nomenclature

individual objective

satisfaction level of solution with respect to the ith objective

desire reference level of the ith objective

maximum of individual objective

minimum of individual objective

Φ

set of non‐dominated solutions or plans

r₂

lower bound for real expected repair time (h)

r₃

upper bound for real expected repair time (h)

average of repair times (h)

t_α/2

t‐distribution

n

quantity of repair times

s

standard deviation

μ

real expected repair time (h)

λ₂

lower bound for interruption rate (interruptions per year)

λ₃

upper bound for interruption rate (interruptions per year)

chi‐square distribution

λ

interruption rate (interruption per year)

m

action type

ij

component from bus i to bus j

Z₁

investment objective function

Z₂

non‐supplied energy (NSE) objective function

I.R.R.

internal rate of return (%)

N_E

total number of system components

S_ij

power flow in component (line) ij

λ_ij

interruption rate of component ijth (interruption per year)

r_ij

repair time of component ijth (h)

N_m

total number of actions for repair time reduction

cost of repair time reduction in monetary units (m.u.) for component ijth with action m

X_{ij, m}

decision variable {0,1} for component ijth with action m to reduce repair time

annual cost of repair time reduction in monetary units (m.u.) for component ijth with action m

NSE cost in monetary units (m.u.) for component ij

final repair time for component ij

NSE^Final

obtained NSE (kVAh/year)

NSE^Max

threshold for NSE (kVAh/year)

CRF

capital recovery factor

BNF

Monetary savings

d_r

discount rate

t

lifetime project

generated active power at bus i

active load in bus i in p.u.

reactive load in bus i in p.u.

P_i(v, δ)

active power injections in bus i in p.u.

Q_i(v, δ)

reactive power injections in bus i in p.u.

a

bank capacity

reactive power output by capacitor banks in bus j

W_{j, a}

binary decision variable (0, 1) to install a bank capacity (a) in bus j

reactive power (a) installed in bus i

lower limit of generated active power in bus i

upper limit of generated active power in bus i

lower limit of generated reactive power in bus i

upper limit of generated reactive power in bus i

lower limit of voltage magnitude in bus i

V_i

voltage magnitude in bus i

upper limit of voltage magnitude in bus i

S_ij(v, δ)

apparent power flow in line ij in p.u.

rating power limit of line ij

tap^min

lower limit of transformer tap

tap

transformer tap

tap^max

upper limit of transformer tap

Δr_ij

repair time variation for component ijth using a repair time reduction action

m.u.

monetary units

1 Introduction

Over the past few decades lots of efforts have been devoted to the distribution systems reliability assessment. Reliability is the ability to deliver electricity to all delivery points within acceptable levels of quality, in the desired amount and at the minimum cost. Obviously, these are conflicting goals, because increasing the quality and quantity of the energy provided to customers will necessarily increase the investment cost in networks, as well as the operation cost [1]. Network planners and operators must find the adequate balance taking into account the uncertainties of future conditions. The reliability criteria can be deterministic and/or probabilistic, however, in both cases a consistent database and an exhaustive statistical analysis of all the available information are needed such as interruption rates (λ) and average repair times (r) of distribution system components. Moreover, maximising the reliability in distribution power systems involves minimising the unserved energy, and therefore load curtailment.

There are two types of uncertainties in distribution power systems: randomness and fuzziness [2–5]. It is well known that interruption frequencies or interruption probabilities of distribution overhead lines are related to weather conditions [2, 3, 5–7]. Weather conditions are usually described in fuzzy words. Each weather condition (rain, wind, storm etc.) can be categorised into different degrees (such as heavy, medium or light rain) [2–5]. This classification is obviously vague or fuzzy. Interruption frequencies or interruption probabilities of distribution lines are also caused by environments (such as tree falling or animal activities) and operational conditions (such as load levels). It is very difficult to distinguish exactly the effects of these conditions on the outage data of individual components using a probability model, since there are little or any statistics available. Some utilities do not have enough statistical records, but they may have a good judgment on the range of outage parameters (such as repair time). A fuzzy approach allows obtaining adequate models for all these cases [2–5].

Unlike transmission systems, which are looped, distribution networks are usually made up by radial feeders. The most important consequence of radial feeders is that many customers can be affected by the interruption of a single component. The adequacy is represented by reliability indices such as average interruption rate λ, average outage duration r, and the annual outage duration U. These reliability indices are load point indices and they are of crucial importance for reliability studies. The average outage duration and the interruption rate can be reduced by network configurations, suitable location of substation, feeder length, and by increasing fault avoidance and corrective repair measures. These measures try to modify the interruption rate and repair time of each segment, improving the system reliability. Repair time and interruption rate modifications may require additional efforts which are associated with additional expenditures.

A technique for optimal reliability design in electrical distribution system was developed by Chang and Wu in [8]. This technique solved a non‐linear optimisation problem based on polynomial‐time algorithm.

A value based on a probabilistic approach to design urban distribution systems for determination of the optimal section length for the main feeder, number and placement of feeder ties, feeder and transformer loadings is proposed in [9]. A genetic algorithm approach for reliability design of a distribution system is presented in [10]. The objective function contains the investment cost, and the system interruption cost. System interruption rate and average outage duration at load point are considered as constraints. Louit et al. [11] presented an algorithm for determination of optimal interval for major maintenance actions in electrical distribution networks. Comparative case studies for value‐based distribution system reliability planning are present in [12]. An algorithm for optimum location of distributed generation (DG) in a distribution system based on reliability considerations was proposed in [13]. Arya et al. [14] proposed a methodology for reliability enhancement of radial distribution system by determining optimal values of repair times, and interruption rates of each section. A set of composite models for distribution system reliability evaluation that may be applied to non‐radial type networks was proposed in [15]. In [16], a three‐stage method for planning a power distribution system is proposed in which the substation optimisation, the number of feeders with their active route, and the node reliability optimisation are included. Chandramohan et al. [17] presented the minimisation of radial distribution system operating cost in the regulated electricity market via reconfiguration. Arya et al. [18] presented an analytical methodology for reliability evaluation, and enhancement of distribution system having DG. In [19], Ferreira and Bretas used a non‐linear binary programming model for reliability optimisation of distribution systems. In [20], the authors proposed an evolutionary algorithm for distribution feeder reconfiguration. The problem consists of minimising the power loss, operation cost of DG and non‐supplied energy (NSE) simultaneously. The considered constraints including the radial structure of the network, line thermal limits, transformer capacities and bus voltages are within their admissible ranges in this approach. Canizes et al. [5] presented a methodology which aims to increase the probability of delivering power to any load point of distribution networks by identifying new investments in distribution components not considering the network technical constraints. The proposed methodology uses a fuzzy set approach to estimate the outage parameters, and the investments aims to reduce the components interruption rates and repair times.

The references cited above do not have considered a way to increase the delivered power considering investments actions in order to reduce the repair time of the distribution network components, and all the technical network constraints.

Hence, this paper proposes a new and innovative methodology for increasing the delivered power to any load point of the distribution network, by identifying new investments in distribution components, while minimising the costs of those investments, as well as the cost of the NSE (multi‐objective problem). The investments aim to reduce the components repair time. The repair time reduction can be obtained by increasing the operation personnel, upgrading the automation system, improving the communication system and so on. The proposed methodology is a weighted AC optimisation model based on mixed‐integer non‐linear programming (MINLP) using the Pareto front technique [21, 22]. A case study based on 33‐bus distribution test network [23] is used to demonstrate the proposed methodology.

Since the presented problem is a multi‐objective optimisation problem, it requires a multi‐objective method for solving. This paper utilises a Pareto‐based approach, which can obtain a set of optimal solutions instead of one [24].

This paper is organised as follows: Section 2 introduces the fuzzy set membership models for outage parameters. Section 3 presents the proposed methodology to increase the delivered power by reducing the NSE in radial distribution network. Section 4 presents the case study and the discussion of the obtained results. Finally, the most relevant conclusions are duly drawn in Section 6.

2 Fuzzy set membership models for outage parameters

Each outage model is characterised by the following parameters – interruption frequency and repair time; interruption rate and repair rate; transition rates between multiple states; unavailability or forced outage rate; and standard deviation.

These parameters are the input data in risk evaluation, and they can be estimated from historical interruption statistics.

Most of the data collection systems provide the interruption frequency or the unavailability and repair time. Usually, the interruption rate and the repair rate are not directly collected, but they can be calculated from the interruption frequency and repair time [2–5].

The repair time is a parameter that can be directly collected by a data collection system and the expected repair time confidence interval is given by

(1)

(2)

(3)

The real expected repair time (μ) is placed between r₂ and r₃, the lower and higher bounds respectively, as it can be seen in (3). With the point and interval estimates, a triangle membership function of repair time r (in hours per interruption) can be easily created [2–5].

The interruption rate λ and interruption frequency f are numeric with very close values for power system components, although they are conceptually different. f and λ are often replaced by each other in practical engineering calculations of power system risk evaluation. Equations (4)–(6) are used to estimate the two bounds of the interruption rate

(4)

(5)

(6)

Considering (4)–(6), a triangle membership function of interruption rate can be easily created. The forced outage rate or unavailability of system components (U) can be determined using the interruption frequency (interruption rate) and repair time [2–5]

(7)

The membership function of unavailability can be obtained from λ and r. For this, the interval calculation rules for a given membership function grade must be used [2–5].

The use of the centroid of the function obtained by the gravity technique is an appropriated approach, and it is very similar to the weighted mean used in the probability theory, used for the defuzzification of the outage parameters. This technique finds the balance point by calculating the weighted mean of the membership functions.

3 Multi‐objective optimisation

When different incommensurable objectives with conflicting/supporting relations or without any mathematical relation with each other are considered, an appropriated tool like multi‐objective optimisation must be used in order to deal with this problem. Usually it is not possible to obtain an optimum at which all such objectives are optimised. Thus, the Pareto front method can be used to characterise the solutions from the multi‐objective problem. Pareto front optimal method is also known as non‐inferiority or non‐dominancy method [22].

3.1 Satisfying decision making

The decision maker's judgment to select the final solution has a subjective imprecise nature, thus a method to support its decision can be necessary. Fuzzy satisfying method can be used to select the preferred solution among non‐dominated solutions obtained in optimisation stage. To the decision maker it will be asked to determine its imprecise goals for each objective. As a result, the final solution will be found. Since the trade‐offs between each objective are determined in the first stage, one can expect a much reasonable judgment from decision maker comparing to prior methods [25].

After non‐dominated set determination, it is desirable to obtain a flexible and realistic solution that represents a trade‐off between different objectives [26]. Fuzzy satisfying method due to simplicity and similarity to human reasoning is of great interest. Fuzzy sets are defined by membership functions, which represent the degree of membership in a fuzzy set using values from 0 to 1. Fig. 1 presents a linear type membership function, which will be used in this work.

Fig. 1

Linear type membership function (adapted from [25])

The value of the membership function indicates to what extend a solution is satisfying the objective Z_i. Decision maker is fully satisfied with objective value of if and not satisfied if . Equation (8) presents the model used for all objectives

(8)

Once defined the membership functions, decision makers need to choose the desirable reference level of each objective. To obtain the final solution a method called ‘minimax’ [27], have been applied. Thus, the following optimisation problem is used:

(9)

4 Proposed methodology

The following proposed method when compared with the existence works, considers a technique to increase the delivered power, i.e. reducing the NSE. The technique deals with investments action in order to reduce the repair time of the distribution network components, and all the technical network constraints based on deterministic formulation. Thus, the proposed methodology minimises the investment cost, as well as the NSE cost in radial distribution networks. This methodology aims to maximise the power availability in each load point of this network type. A scheme of the proposed methodology is shown in Fig. 2.

Fig. 2

Diagram of the proposed methodology

The proposed methodology has five main aspects, which are presented in more detail as follows.

4.1 Database

A consistent database creation and an exhaustive statistical analysis of all available historical data (such as repair times, number of interruptions, and number of repairs) are an important issue for the proposed methodology.

4.2 Target for NSE

Network operator goal is to achieve a desire NSE for the radial distribution network. The NSE is given by the following equation:

(10)

4.3 Fuzzy logic functions for r, λ and U

Several effects such as weather conditions, environmental, and operational conditions are very difficult to distinguish precisely on the outage data of individual components using a probability model, since there are little or any statistics available. Usually, utilities do not have enough statistical records of outage parameters. As a result, the fuzzy set approach allows obtaining adequate models. Equations (1)–(6) allow determining the membership functions for repair times and interruption rates of all distribution system components. Hence, membership function of unavailability can be calculated by (7).

4.4 Weighted AC optimisation based on MINLP

Considering investment minimisation and NSE cost minimisation the problem turns into a multi‐objective optimisation problem which is more complicated rather than a single objective of the optimal power flow (OPF) problem. Also, the multi‐objective optimisation problem has a greater search space than the single objective problem.

The proposed methodology can give to the planner the possibility to decide what objective is better for him depending on the pursued goal. The Pareto front technique can be a useful tool helping the planner to choose the more convenient preference. The Pareto front technique is generally used to solve conflicted objective function [28]. In the proposed methodology, the goal is to improve reliability, minimising the NSE, which is improved by investments in the network. Thus, conflictive objectives are present (maximise reliability and minimise investments).

An MINLP using the weighted Pareto front method is developed as it can be seen in (11)–(26). This mathematical model is applied in order to identify the distribution network components, in which investments allow increasing the delivered power to every load point in the considered network. The weighted technique is used to obtain the non‐dominated solutions. A set of weights randomly generated are used to construct a set of feasible optimisation problems.

The investment aims to achieve a reduction in repair times of distribution networks, while minimising the costs of that reduction as well as the minimisation of the NSE costs. Hence, a multi‐objective problem can be developed.

The repair time reduction can be achieved, for instance: by increasing the operation personnel, by upgrading the automation system, by communication system, among others.

5 Problem formulation

The distribution network planning problem based on OPF model must be able to evaluate the load distribution among substation, distributed generators and feeders. The result should meet demand and technical requirements (a feasible result).

In this paper, the optimisation problem is modelled as an MINLP. The model presented an objective function that represents the costs of future investments in order to reduce the repair time (minimisation of investment – Z₁) as well as the reliability related costs, i.e. NSE costs (NSE costs – Z₂). Also, it ensures the possibility of the optimal capacitors sizing in the network supporting the nodes voltage (injecting reactive power) and considers the substation transformer taps. Moreover, when this method is compared with the state‐of‐the‐art it considers a multi‐objective problem subjected to all technical constraints, as well as the constraints related with the repair time reductions using a deterministic process. Furthermore, the fuzziness uncertainty associated to repair times and failure rates is considered. Thus, this model represents an advantage regarding to the state‐of‐the‐art.

Thus, the multi‐objective function to be minimised is

(11)

(12)

Effective methods like ε constraint technique and normal boundary intersection method can be used for solving the multi‐objective problem. In this work, the authors have the objective to use a simple and popular method for solving the multi‐objective problem. Thus, the weighted method (13) was chosen

(13)

The minimisation of the objective function (13) is subjected to the technical constraints (power balance, line capacity limits, DG capacity limits, substation capacity limits, voltage magnitude and angle, transformer taps) and logical constraints (integer decision variables – capacitors size and investment option).

Thus, the model is subjected to:

• Active and reactive power flow equations

  (14)
  (15)

• Selection of a unique value in each capacitor bank

  (16)

• Reactive power output by capacitors

  (17)

• Upper and lower power output limits (active and reactive powers) of the substation and distributed generator units

  (18)
  (19)

• Bus voltage magnitude limits

  (20)

• Bus angle limits

  (21)

• Capacity limits of distribution lines/cables

  (22)

• Transformer taps limits

  (23)

• Final repair time equation

  (24)

• NSE threshold

  (25)

• Only one investment option (action) can be picked for repair time

  (26)

To obtain the values of Δr_ij, the network operator must evaluate studies that aim to improve the reliability of distribution network. The reliability increase can be achieved by reducing the components repair time. The goal of these studies is to know which reduction of repair time is possible to obtain for each component. By selecting the given repair time reduction actions, the network operator have the associated cost for each component.

6 Case study

Tests were conducted in the 33‐bus distribution test network adapted from [23]. This test system is a hypothetical 12.66 kV system with two feeders, one substation, 33 buses, and 32 load points. The total substation load for this case study is 4549 kVA. Some adaptations were made in this network – two DGs units with 500 kW, 250 kVAr in buses 11 and 27; four capacitors sizes with 150, 300, 450, and 600 kVAr can be installed in buses 5 and 10; the substation transformer has a continuous value tap (±5%).

The 33‐bus distribution network (Fig. 3) does not have an outage database associated, thus the authors created an outage database (Table 1) in order to apply the proposed methodology.

Table 1. Outage data for 33‐bus distribution network

Bus out	Bus in	Mean time to repair – r, h	Number of interruptions	Time period, years	Average interruption rate – λ, interruptions/year	Unavailability – U, h/year
Lines or cables
33	1	30.200	3	10	0.300	9.060
1	2	26.700	11	10	1.100	29.370
1	18	25.100	9	10	0.900	22.590
2	3	11.900	21	10	2.100	24.990
2	22	9.500	21	10	2.100	19.950
3	4	12.100	12	10	1.200	14.520
4	5	25.600	14	10	1.400	35.840
5	6	8.600	17	10	1.700	14.620
5	25	11.700	21	10	2.100	24.570
6	7	29.300	11	10	1.100	32.230
7	8	9.700	29	10	2.900	28.130
8	9	12.100	12	10	1.200	14.520
9	10	9.200	26	10	2.600	23.920
10	11	12.100	21	10	2.100	25.410
11	12	10.100	12	10	1.200	12.120
12	13	8.300	8	10	0.800	6.640
13	14	13.800	8	10	0.800	11.040
14	15	29.800	12	10	1.200	35.760
15	16	8.900	30	10	3.000	26.700
16	17	20.300	17	10	1.700	34.510
18	19	12. 000	15	10	1.500	18.000
19	20	25.500	14	10	1.400	35.700
20	21	25.000	11	10	1.100	27.500
22	23	11.000	21	10	2.100	23.100
23	24	9.400	17	10	1.700	15.980
25	26	9.300	14	10	1.400	13.020
26	27	9.300	16	10	1.600	14.880
27	28	27.000	21	10	2.100	56.700
28	29	20.600	25	10	2.500	51.500
29	30	11.800	16	10	1.600	18.880
30	31	9.500	26	10	2.600	24.700
31	32	9.300	14	10	1.400	13.020

Fig. 3

Single‐line 33‐bus distribution network (adapted from [23])

For this case study only lines and cables were considered for the analysis, and it is assumed that the substation and DG units have 100% of availability.

The actions that the system operator can apply to reduce the repair time are the increase the operation personal (IOP), automation system upgrade (ASU), and communications upgrade (CUp).

Each one of these actions has a cost and results in a repair time reduction for each system component.

The NSE cost is 2 mμ/kVAh. A set of 1000 random weights is determined. A 10‐year lifetime project with a 1.75% discount rate, which leads to a 0.110 capital recovery factor, is considered for the NSE cost and investment in repair time reduction cost.

GAMS, with DICOPT solver, has been used to develop the weighted AC optimisation that uses MINLP. The MATLAB software has been linked with GAMS in order to solve the model to all 1000 weights. The way used to link GAMS to MATLAB can be found in [29]. For this case study, a computer with one processor Intel Xeon E3‐1225 3.20 GHz with four cores, 4 GB of random‐access‐memory (RAM), and Windows 8 Professional 64‐bit operating system was used.

The initial network data considered in this case study are presented in Table 2. The initial losses are obtained running a power flow for distribution networks based on the algorithm of Thukaram et al. [30]. Tables 3 and 4 present the network data and reduction times per action with cost, respectively.

Table 2. Initial data for 33‐bus distribution network

Network active load, kW	3715.000
Network reactive load, kVAr	2300.000
Initial NSE, kVAh/year	545060.000
Initial NSE cost, m.u./year	1090121.000

Table 3. Network data

Bus out	Bus in	R, Ω	X, Ω	Active load, kW	Reactive load, kVar	Apparent load, kVA	Thermal limit, kVA
33	1	0.0922	0.0470	100.0	60.0	116.619	450
1	2	0.4930	0.2511	90.0	40.0	98.489	450
2	3	0.3660	0.1864	120.0	80.0	144.222	450
3	4	0.3811	0.1941	60.0	30.0	67.082	329
4	5	0.8190	0.7070	60.0	20.0	63.246	450
5	6	0.1872	0.6188	200.0	100.0	223.607	329
6	7	0.7114	0.2351	200.0	100.0	223.607	329
7	8	1.0300	0.7400	60.0	20.0	63.246	329
8	9	1.0440	0.7400	60.0	20.0	63.246	329
9	10	0.1966	0.0650	45.0	30.0	54.083	329
10	11	0.3744	0.1238	60.0	35.0	69.462	329
11	12	1.4680	1.1550	60.0	35.0	69.462	329
12	13	0.5416	0.7129	120.0	80.0	144.222	329
13	14	0.5910	0.5260	60.0	10.0	60.828	329
14	15	0.7463	0.5450	60.0	20.0	63.246	229
15	16	1.2890	1.7210	60.0	20.0	63.246	329
16	17	0.7320	0.5740	90.0	40.0	98.489	329
1	18	0.1640	0.1565	90.0	40.0	98.489	329
18	19	1.5042	1.3554	90.0	40.0	98.489	329
19	20	0.4095	0.4784	90.0	40.0	98.489	329
20	21	0.7089	0.9373	90.0	40.0	98.489	450
2	22	0.4512	0.3083	90.0	50.0	102.956	450
22	23	0.8980	0.7091	420.0	200.0	465.188	450
23	24	0.8960	0.7011	420.0	200.0	465.188	450
5	25	0.2030	0.1034	60.0	25.0	65.000	450
25	26	0.2842	0.1447	60.0	25.0	65.000	329
26	27	1.0590	0.9337	60.0	20.0	63.246	329
27	28	0.8042	0.7006	120.0	70.0	138.924	329
28	29	0.5075	0.2585	200.0	600.0	632.456	329
29	30	0.9744	0.9630	150.0	70.0	165.529	329
30	31	0.3105	0.3619	210.0	100.0	232.594	329
31	32	0.3410	0.5302	60.0	40.0	72.111	329

Table 4. Reduction times per action and costs

Bus out	Bus in	IOP, h	ASU, h	CUp, h	IOP, m.u.	ASU, m.u.	CUp, m.u.
1	2	8.00	16.00	21.40	76,000	114,000	152,000
1	18	7.50	15.10	20.10	6400	9600	12,800
2	3	3.60	7.10	9.50	56,400	84,600	112,800
2	22	2.90	5.70	7.60	18,000	27,000	36,000
3	4	3.60	7.30	9.70	53,600	80,400	107,200
4	5	7.70	15.30	20.50	52,400	78,600	104,800
5	6	2.60	5.20	6.90	20,400	30,600	40,800
5	25	3.50	7.00	9.40	37,600	56,400	75,200
6	7	8.80	17.60	23.40	16,400	24,600	32,800
7	8	2.90	5.80	7.70	12,400	18,600	24,800
8	9	3.60	7.20	9.70	11,600	17,400	23,200
9	10	2.70	5.50	7.30	10,800	16,200	21,600
10	11	3.60	7.30	9.70	9600	14,400	19,200
11	12	3.00	6.10	8.10	8000	12,000	16,000
12	13	2.50	5.00	6.70	6800	10,200	13,600
13	14	4.10	8.30	11.00	3600	5400	7200
14	15	8.90	17.90	23.80	3200	4800	6400
15	16	2.70	5.30	7.10	2400	3600	4800
16	17	6.10	12.20	16.30	1600	2400	3200
18	19	3.60	7.20	9.60	4800	7200	9600
19	20	7.70	15.30	20.40	3200	4800	6400
20	21	7.50	15.00	20.00	1600	2400	3200
22	23	3.30	6.60	8.80	16,000	24,000	32,000
23	24	2.80	5.60	7.50	8000	12,000	16,000
25	26	2.80	5.60	7.50	36,400	54,600	72,800
26	27	2.80	5.60	7.50	35,600	53,400	71,200
27	28	8.10	16.20	21.60	34,800	52,200	69,600
28	29	6.20	12.40	16.50	32,400	48,600	64,800
29	30	3.50	7.10	9.40	8400	12,600	16,800
30	31	2.90	5.70	7.60	5600	8400	11,200
31	32	2.80	5.60	7.40	1600	2400	3200
33	1	9.10	18.10	24.20	84,800	127,200	169,600

Twenty‐nine non‐dominated solutions or plans were obtained by the weighted AC optimisation. The algorithm took around 4954 s (1.38 h) to compute the set of 1000 weighted solutions. Table 5 presents the obtained non‐dominated solutions, and the economic evaluation for each of those solutions. The signal ‘minus’ in the values of NSE benefit and final benefit represents the benefit for system operator. It is important to note that without any investment the value of NSE cost at the end of 10 years (lifetime project) would be 9,920,103 m.u.

Table 5. Non‐dominated solutions or plans

Plan	Investment cost, m.u.	NSE cost, m.u.	NSE benefit, m.u.	Final benefit, m.u.	Payback, years	I.R.R., %
1	5,495,616	3,634,406	−6,285,698	−790,082	7.96	14.38
2	5,515,920	3,622,558	−6,297,546	−781,626	7.97	14.17
3	5,524,380	3,619,442	−6,300,662	−776,282	7.98	14.05
4	5,531,148	3,611,348	−6,308,755	−777,607	7.98	14.06
5	5,551,452	3,596,553	−6,323,550	−772,098	7.99	13.91
6	5,566,680	3,585,344	−6,334,760	−768,080	8.00	13.80
7	5,600,520	3,563,654	−6,356,450	−755,930	8.02	13.50
8	5,620,824	3,553,364	−6,366,739	−745,915	8.03	13.27
9	5,627,592	3,550,057	−6,370,046	−742,454	8.04	13.19
10	5,839,092	3,435,065	−6,485,038	−645,946	8.19	11.06
11	5,859,396	3,424,776	−6,495,328	−635,932	8.21	10.85
12	5,866,164	3,421,469	−6,498,635	−632,471	8.21	10.78
13	6,043,824	3,339,754	−6,580,349	−536,525	8.36	8.88
14	6,077,664	3,318,066	−6,602,037	−524,373	8.38	8.63
15	6,097,968	3,307,777	−6,612,326	−514,358	8.39	8.43
16	6,104,736	3,304,470	−6,615,634	−510,898	8.40	8.37
17	6,740,928	3,012,588	−6,907,516	−166,588	8.88	2.47
18	6,999,804	2,910,635	−7,009,469	−9665	9.09	0.14
19	7,086,096	2,877,304	−7,042,800	43,296	9.16	−0.61
20	7,201,152	2,835,734	−7,084,370	116,782	9.25	−1.62
21	7,358,508	2,778,242	−7,141,862	216,646	9.38	−2.94
22	7,410,960	2,759,358	−7,160,746	250,214	9.42	−3.38
23	8,091,144	2,573,350	−7,346,753	744,391	10.02	−9.20
24	8,317,872	2,511,792	−7,408,311	909,561	10.22	−10.94
25	9,752,688	2,137,593	−7,782,510	1,970,178	11.40	−20.20
26	10,355,040	1,982,614	−7,937,490	2,417,550	11.87	−23.35
27	10,970,928	1,837,392	−8,082,711	2,888,217	12.35	−26.33
28	11,349,936	1,788,978	−8,131,126	3,218,810	12.70	−28.36
29	11,512,368	1,782,840	−8,137,264	3,375,104	12.87	−29.32

Fig. 4 presents the set of non‐dominated solutions (plans), which are plotted as NSE cost versus investment cost.

Fig. 4

Non‐dominated solutions or plans for the 33‐bus distribution system

The fuzzy satisfying decision method is used to select the preferred solution among non‐dominated solutions obtained in optimisation stage. The minimum and maximum values as well as the desirable reference levels are presented in Table 6.

Table 6. Input values for fuzzy satisfying decision method

	Investment	NSE
maximum cost, m.u.	11,512,500	9,920,103
minimum cost, m.u.	0	1,782,902
reference level	0.2	0.7

The maximum cost value for investment objective was determined considering the weight of investment equal to one and the weight of NSE equal to zero. The minimum value is considered equal to zero, i.e. without any investment in the network. Regarding to the NSE the maximum value is equal to the NSE value for the 10 years lifetime project without any investment. The minimum value is that one which corresponds to the weight equal to zero.

To specify the reference levels for each objective the trade‐offs shown in Fig. 4 can help the decision maker to choose reasonable reference values. For instance, in the investment objective, it is considered the last value where it is yet possible to obtain an I.R.R. positive, i.e. 6,999,804 m.u. (plan 18) and also the minimum value of investment in the network, i.e, 5,495,616 m.u. Thus, it is obtained(6, 999, 804 − 5, 495, 616)/(6, 999, 804 − 0) = 0.21. In the NSE objective, it is considered the maximum cost value (3,634,406 m.u.), which corresponds to the minimum value of investment in the trade‐off curve, and also the minimum cost (1,782,840 m.u.) in the same curve. So, it is obtained(9, 920, 103 − 3, 634, 406)/(9, 920, 103 − 1, 782, 840) = 0.77. Taking into account these values the reference levels for each objective are set to the ones presented in Table 4.

Applying the optimisation problem from (9) it is selected the solution of plan 1 (see Table 5).

Plan 1 presents the higher benefit (monetary units – m.u.) for the system operator (790,082 m.u.). The objective function value is of 9,130,022 m.u. for 10 years of project lifetime. Considering this plan, the payback presents a value of 7.96 years and the internal rate of return a value of 14.38%.

The values for final repair times can be found in Table 7. The zero in unavailability reduction means that the respective components do not have any investment action (in repair time).

Table 7. Final repair time and final unavailability considering plan 1

Bus out	Bus in	Final repair time – Fr, h	Interruption rate – λ, interruption/year	Final unavailability – U, h/year	Unavailability reduction – U, h/year	Unavailability reduction, %
Lines or cables
33	1	30.200	0.300	9.060	0.000	0.000
1	2	5.300	1.100	5.830	23.540	80.150
1	18	5.000	0.900	4.500	18.090	80.080
2	3	11.900	2.100	24.990	0.000	0.000
2	22	1.900	2.100	3.990	15.960	80.000
3	4	12.100	1.200	14.520	0.000	0.000
4	5	5.100	1.400	7.140	28.700	80.078
5	6	3.400	1.700	5.780	8.840	60.465
5	25	11.700	2.100	24.570	0.000	0.000
6	7	5.900	1.100	6.490	25.740	79.863
7	8	9.700	2.900	28.130	0.000	0.000
8	9	12.100	1.200	14.520	0.000	0.000
9	10	9.200	2.600	23.920	0.000	0.000
10	11	12.100	2.100	25.410	0.000	0.000
11	12	2.000	1.200	2.400	9.720	80.198
12	13	3.300	0.800	2.640	4.000	60.241
13	14	2.800	0.800	2.240	8.800	79.710
14	15	6.000	1.200	7.200	28.560	79.866
15	16	1.800	3.000	5.400	21.300	79.775
16	17	4.000	1.700	6.800	27.710	80.296
18	19	2.400	1.500	3.600	14.400	80.000
19	20	5.100	1.400	7.140	28.560	80.000
20	21	5.000	1.100	5.500	22.000	80.000
22	23	2.200	2.100	4.620	18.480	80.000
23	24	1.900	1.700	3.230	12.750	79.787
25	26	9.300	1.400	13.020	0.000	0.000
26	27	9.300	1.600	14.880	0.000	0.000
27	28	5.400	2.100	11.340	45.360	80.000
28	29	4.100	2.500	10.250	41.250	80.097
29	30	2.400	1.600	3.840	15.040	79.661
30	31	1.900	2.600	4.940	19.760	80.000
31	32	1.900	1.400	2.660	10.360	79.570

Table 8 presents the optimised value for NSE when plan 1 is considered. The proposed methodology leads to a 36.6% reduction in the NSE value. For the considered plan only bus 5 has a capacitor bank with 600 kVAr. Regarding to the substation transformer tap, the obtained value is of 1.001 p.u.

Table 8. Optimised NSE considering plan 1

	Initial value	Threshold value	Obtained value
NSE, kVAh/year	545060.000	200000.000	199690.000

Table 9 shows the chosen actions to be applied in the network components. The symbol ‘✓’ denotes that the respective action was chosen. It is possible to see that the automation system upgrade action was chosen for two components, while communications’ upgrade was chosen for 22 components.

Table 9. Actions for each network component considering plan 1

Bus out	Bus in	IOP	ASU	Cup
33	1	—	—	—
1	2	—	—	✓
1	18	—	—	✓
2	3	—	—	—
2	22	—	—	✓
3	4	—	—	—
4	5	—	—	✓
5	6	—	✓	—
5	25	—	—	—
6	7	—	—	✓
7	8	—	—	—
8	9	—	—	—
9	10	—	—	—
10	11	—	—	—
11	12	—	—	✓
12	13	—	✓	—
13	14	—	—	✓
14	15	—	—	✓
15	16	—	—	✓
16	17	—	—	✓
18	19	—	—	✓
19	20	—	—	✓
20	21	—	—	✓
22	23	—	—	✓
23	24	—	—	✓
25	26	—	—	—
26	27	—	—	—
27	28	—	—	✓
28	29	—	—	✓
29	30	—	—	✓
30	31	—	—	✓
31	32	—	—	✓

7 Conclusions

A new methodology to reduce the NSE in a radial distribution network by identifying new investments is proposed in order to reduce the repair time in network components. To estimate the network outage parameter (which is a very important issue in the proposed method) a fuzzy set approach was used.

As a main contribution, an AC optimisation model based on MINLP was developed considering the Pareto front technique (weighted method) to achieve a reduction in repair times of distribution networks components, while minimising the costs of that reduction, as well as the NSE costs. The optimisation model proposed in this paper identifies the actions to be taken, as well as the components in which the system operator must invest at minimum cost. The model also considers the distribution network technical constraints, and it is able to consider the substation transformer taps and to choose the size of capacitor banks.

Regarding all the obtained plans (non‐dominated solutions) the system operator can carry out an analysis in order to choose a solution preference. Through the obtained results the proposed method proves to be adequate to support the distribution network operator to plan future network investment actions.

8 Acknowledgments

This work was supported by FEDER Funds through COMPETE program and by National Funds through FCT under the projects FCOMP‐01‐0124‐FEDER: UID/EEA/00760/2013, and PTDC/SEN‐ENR/122174/2010, and by the GID‐MicroRede, project no. 34086, co‐funded by COMPETE under FEDER via QREN Programme, and by the SASGER‐MeC, project no. NORTE‐07‐0162‐FEDER‐000101, co‐funded by COMPETE under FEDER Programme. The present work also developed under the EUREKA – ITEA2 Project SEAS with project number 12004.