1

Reliability Assessment of DistributionSystems including Microgrids

Stine Fleischer Myhre∗, Olav Bjarte Fosso∗, Oddbjørn Gjerde†, Poul Einar Heegaard‡∗Department of Electric Power Engineering, NTNU

†SINTEF Energy Research‡Dept. of Information Security and Communication Technology, NTNU

Trondheim, Norway{stine.f.myhre, olav.fosso, poul.heegaard}@ntnu.no; oddbjorn.gjerde@sintef.no

Abstract— The electrical distribution system is movingtoward a more decentralized, complex, and dynamicsystem. The system is experiencing a higher penetrationof distributed energy resources, flexible resources, andactive end-users, leading to an active distribution system.If these components are used actively, they can have apositive effect on the distribution system’s reliability. Thispaper aims to investigate how a microgrid with renewableenergy sources and energy storage might influence thereliability of electricity supply in a radially operated dis-tribution system. The reliability is investigated from boththe distribution system perspective and the microgridperspective with the application of different scenarios.A reliability assessment method for modern distributionsystems based on Monte Carlo simulations is presented.The method includes load flow calculations to capturethe behavior of the system and system components.The model is tested on the IEEE 33-bus network. Theresult is confirmed through statistical testing showingthe statistical significance in providing support from themicrogrid on the distribution system’s reliability.

Keywords—Active Distribution Systems, DistributionSystem Reliability, Microgrid, Monte Carlo.

I. INTRODUCTION

The electrical power system is under constant devel-opment, and in the upcoming years, a lot of changeswill take place in the distribution system [1], [2]. Thepower system is moving from being somewhat hierar-chical with the one-way transfer of energy to a moredecentralized system where the system boundaries willbe more invisible. The biggest changes can be seen inthe distribution system, which will be more decentral-ized, complex, and dynamic with higher penetration ofdistributed energy resources and active end-users [3],[4], [5]. This will transform the distribution systeminto an active system with bidirectional power flowsthat will change the system behavior.

From a reliability perspective, this opens up newpossible solutions. With more flexible resources inthe system and integration of distributed generation(DG), there is a potential to increase the systemreliability. Some research has been done to investi-gate the potential reliability improvement with thesetechnologies integrated into distribution systems. Ref.[6] provides an overview of how flexible resources

may impact the security of electricity supply (SoS)and includes a review of methods and indicatorsto quantify the impact. A review on power systemflexibility is provided in [7]. In addition, flexibilityis investigated concerning power system security. Asurvey of reliability assessment techniques for moderndistribution systems is assessed in [8]. Here, multiplemodeling techniques used for assessing distributionsystem reliability are investigated concerning activecomponents in the system. The availability and impactof active components, such as DGs [9], [10], energystorage units [11], and systems with a combination ofthese [12], [13], have been investigated. The literatureillustrates that DGs and flexible resources might havea positive impact on the reliability of distributionsystems if administrated wisely.

Microgrids are an increasingly studied concept re-lated to the improvement of system reliability [14].The main attribute of microgrids is the possibilityto operate in two different modes: 1) Grid-connectedmode and 2) Islanded mode. This gives the opportunityof the microgrid to operate in an islanded mode duringoutages in the main system. If the microgrid and thedistribution system work together, there is a potentialof improved reliability related to unintentional outagesin the system. By islanding parts of the distributionsystem with the microgrid, the microgrid could be theprovider of these distribution system load points. Somestudies have investigated the possibility of islandingparts of the distribution system during failure events.In [15] and [16], an analytical approach is used to findthe contribution from the microgrid to the distributionsystem reliability. The contribution of the microgridis determined by possible operating conditions of themicrogrid and the DGs inside the microgrid, and isevaluated based on the ratio between demand and sup-ply. A simulation-based approach with Monte Carlohas been used in [17] for intentional islanding ofthe distribution system. However, the paper does notconsider the active participation of flexible resourcesand DGs to restore supply. Furthermore, a sequentialMonte Carlo method is used in [18] to evaluate theimpact from local and mobile generation units. Here,

arX

iv:2

111.

0767

4v1

[ee

ss.S

Y]

15

Nov

202

1

2

parts of the system will be islanded, with some unitsoperating as microgrids. In [19], the reliability of adistribution system with support from a microgridincluding combined heat and power supply systemis assessed. However, the microgrid does not includerenewable energy resources (RES). In addition, someresearch related to the reliability of multi-microgridshas been assessed by [20], [21], and [22]. These papersdo not consider the interaction between the microgridand the distribution system, nor is the reliability of themicrogrid investigated to any significant amount.

For microgrid support to be functioning, an interac-tion between the distribution system operator (DSO)and the microgrid owner must be in place. The typeof interaction is dependent on who owns the microgrid.In [23], [24], and [25], this theme is discussed. Inaddition, regulatory frameworks related to the respon-sibility of islanded load points need to be in place.The regulatory interaction frameworks will not beconsidered in this paper, but are nevertheless importantaspects to consider for the application of such support.

This paper will investigate the reliability of elec-tricity supply from both the distribution system andmicrogrid perspective. The microgrid and the distribu-tion system will therefore be distinguished based onsystem boundaries. This means that the distributionsystem and the microgrid will have clear and well-defined boundaries. The aim of the paper is to present areliability assessment method applied on an interestingcase study with results evaluated based on statisticaltesting. The presentation of the method focuses on thereliability analysis of a distribution system with theintegration of a microgrid, and the method is presentedin more detail in [26]. The contributions of this paperare:

• A simulation approach for calculating the relia-bility in a radially operated distribution systemwith the integration of an active microgrid.

• A case study where the reliability of both thedistribution system and the microgrid is tested onthe IEEE 33-bus network with different scenariosfor microgrid support.

• The evaluation of results based on statistical test-ing with a sensitivity analysis.

The rest of this paper is organized as follows:section II introduces the concept of SoS along withthe definitions, the four aspects of SoS, and the newnetwork configuration. In section III, the method forhow to calculate reliability in a distribution systemwith a microgrid is introduced and illustrated. Thecase study conducted on the IEEE 33-bus networkis discussed in section IV. The results of the casestudy are illustrated and discussed in section V beforethe paper is concluded and future work discussed insection VI.

II. SECURITY OF ELECTRICITY SUPPLY IN THEFUTURE DISTRIBUTION SYSTEM

The European Commission defines the SoS as theability of an electricity system to supply final cus-tomers with electricity [27], while the Norwegian Wa-ter Resources and Energy Directorate (NVE) defines itas the ability of an electricity system to continuouslydeliver electric power of a given quality to the end-user [28]. In this matter, the SoS can be seen as aproperty belonging to the power system where theoutcome is measured based on the delivered power tothe customers both regarding the quality of the powerand the amount of delivered power. In [29], a frame-work to assess SoS in power systems is developed.Furthermore, the paper presents twelve dimensionsthat are critical for SoS in power systems. SoS canfurther be divided into four important aspects. NVE[28] and the Norwegian regulation of the quality ofsupply [30] defines the four SoS aspects as follows:

1) Power capacity—The power systems’ ability tocover the instantaneous load in the system.Power capacity is characterized by the availablecapacity of installed power production in thegrid.

2) Energy availability—The power systems’ abilityto cover the electricity demand in the power sys-tem. There is an energy shortage in the system ifthe generation units in the system generate lessenergy than the electricity demand in the system.

3) Power quality—The power systems’ ability todeliver power of a given voltage quality.

4) Reliability of supply—The power systems’ abil-ity to deliver electric power to the end-user. Thereliability of supply is measured in frequencyand duration of outages in the electrical powerdelivery.

In a traditional power system, the transmission sys-tem operator (TSO) is responsibility for the powercapacity and energy availability, whereas the powerquality and the reliability of supply are regulated bythe distribution system operator (DSO). However, ina modernized power system, the boundaries of theresponsibility for the different SoS aspects will bechanged. With more generation in the distributionsystem, and the distribution system becoming moresimilar to the transmission system, more responsibil-ity might fall on the DSO. In addition, with moreintegration of microgrids that can operate as self-sufficient grids, all the SoS aspects need to be secured.It will therefore be necessary to investigate the SoSdifferently in future power systems. However, in thispaper, the emphasis will be on the reliability of supply.

A. The reliability evaluation of the future distributionsystems with microgrids

In a traditional radially operated distribution systemwith one-way transfer of power as seen in Fig. 1, a

3

failure of, for example, a line in the network will resultin all the downstream load points being isolated fromthe main power source unless there is an alternativesupply route in the network. In Fig. 2, a failure hasoccurred on a line in the distribution system and beenisolated. This results in the load points, highlighted inthe red dotted box, downstream of the failed line to beisolated from the rest of the distribution system. Sincethere is no alternative supply route or any generationunit in this area, these load points will not be supplieduntil the failure is repaired and the line connected.

Circuit BreakerDisconnector

Figure 1: Traditional radially operated distributionsystem

Circuit BreakerDisconnectorFailure

Isolated and not supplied load points

Figure 2: Traditional radially operated distributionsystem with failure

However, if there is a flexible resource such as amicrogrid present in the network, the situation mightbe changed. The aim is to use the flexibility in thesystem to restore or to help the network during periodswith, e.g., low supply, failures in the network, andcapacity problems. A microgrid can support the systemin two different modes: 1) isolated operation and 2)grid-connected operation or supportive operation.

1) Isolated operation: If a failure occurs on, forexample, a line in the distribution system as seen inFig. 3, the downstream load points will be isolatedfrom the overlaying network and traditionally theseload points will not be supplied. However, if a mi-crogrid or another flexible resource is present in thisisolated part of the network, at least parts of the supplycan be restored. The microgrid will then operate inan island mode during the outage time of the faultycomponent. When the failure is repaired, the isolatedpart can be disconnected from the overlying grid again.

In this operation, the microgrid must be able towithhold the balance in the system along with thefrequency and the voltage levels in the isolated partof the network. This means that the microgrid is incharge of regulating all the SoS aspects during theisolated operation.

Circuit BreakerDisconnectorMicrogrid

Failure

LP Restored

Figure 3: Supply restored with microgrid in islandmode with isolated part of the distribution system

2) Supportive operation: Usually, in distributionsystems, some alternative supply chains might existthat can be used in case of redirection of power orfailure in the network. Then the line can be connectedand help supply loads in the system in areas that wouldhave been isolated during the faulty period. However,some of these lines have lower capacity limits andmight end up being congested, or they might end upcreating bottlenecks in the network. Local productionin the distribution network will prevent congestion andbottlenecks. Fig. 4 illustrates how a microgrid cansupport the system during a fault in the system. Theload points could have been restored with only theconnection of the alternative line, but the microgrid isnow able to support the system with local production.

Circuit BreakerDisconnectorMicrogrid

Failure

LP Restored

Figure 4: Supply restored with alternative supply chainand microgrid support

III. RELIABILITY ANALYSIS

A. Reliability evaluation in traditional distributionsystems

In general, there are two ways of analyzing thereliability in the distribution system, namely throughanalytical approaches or Monte Carlo simulation [31].The analytical approaches use simplified models withaveraged quantities to estimate the reliability indices.Monte Carlo simulations, however, allow for a more

4

detailed representation of the reliability indices bygenerating reliability indices through numerous in-dependent instances with stochastic variations. Thenumerous instances result in a detailed distribution ofreliability indices. Analytical models are more genericand are often, therefore, more applicable for many dif-ferent cases, whereas Monte Carlo simulation modelswill give a more detailed analysis of the system to bemodeled.

RELRAD is an example of an analytical approachdeveloped for calculating the reliability in a radial op-erated distribution system [32]. The RELRAD method-ology calculates the individual load point reliability in-dices analytically based on the fault contribution fromall the network components and their consequence on agiven load point. However, such an approach does notconsider the active components in the network, suchas microgrids, DGs, and batteries, and since it usesaverage values for load and generation, the electricalconsequence and the behavior of the network are notconsidered. Therefore, simulation is a more suitablemethod to calculate the reliability in an active distri-bution system including microgrids, flexible resources,and DG.

B. Reliability evaluation in modern distribution sys-tems

To calculate the reliability in future smart andflexible distribution systems, an appropriate method isneeded. The traditional approaches will not be ableto utilize the full potential of the active componentsin the distribution system and a form of simulationapproach will be more advantageous. Since the activecomponents interact with the distribution system basedon multiple independent factors, such as failures andweather conditions, the simulation of their impacton the distribution system must cover all possiblescenarios. A Monte Carlo simulation approach, whichmakes use of random sampling of input, is well suitedfor this type of problem.

In this relation, RELSAD, a reliability assessmenttool for modern distribution systems was developed.RELSAD is a sequential Monte Carlo simulationmodel. The method is built on object-oriented pro-gramming as a Python package. A description ofRELSAD is given in [26].

In RELSAD, a power system - Ps is first created be-fore distribution systems - Ds and microgrids - Ms arecreated inside the Ps. After this, the electrical systemcomponents can be created and added to associatednetwork layers. The electrical components are lines -l, buses - b, disconnectors - d, circuit breakers - cb,generation units - pu, and batteries - pb. In addition, aload - pd can be assigned to each bus.

The incremental procedure of the model is illus-trated in Algorithm 1. After a Ps is created withassociated systems and components, the incrementalprocedure will set the load and generation at the

system buses for the current time increment. Then thefailure status of each component is drawn based onrandom sampling and the failure rate of the differentcomponents. If any component is in a failed state,the Ds and/or Ms with the failed components willbe divided into sub-systems, and a load flow andoptimization problem will be solved for each sub-system. When this is performed, the history variablesof the components are updated. The history variablescan then further be used to evaluate the reliability ofthe Ps, the different subnetworks (the Ds and Ms), andthe individual load points. The rest of this section aimsto describe the different procedures of the reliabilityevaluation in greater detail.

Algorithm 1: Increment procedure

1 Set bus pd and pu for the current timeincrement;

2 Draw component fail status;3 if Failure in Ps (in the systems, Ds and Ms)

then4 Find sub-systems;5 foreach sub-system do6 Update pb demand (charge or discharge

rate);7 Run load flow;8 Run optimization problem;9 end

10 Update history variables;11 end

C. Microgrid reliability

The reliability evaluation of a microgrid is de-pendent on the operation mode of the microgrid. Ifthe microgrid is in island mode, the microgrid willfunction as an independent system. Whereas, if themicrogrid is connected to the overlying distributionsystem during the outage period, the microgrid will bea part of the distribution system and the systems needto interact. In [33], new metrics for the reliability ofmicrogrids in island mode are proposed.

In this study, the aim is to investigate the impact ofthe microgrid as well as the reliability of both systems.Since the microgrid primarily will be connected tothe distribution system, island mode is only achievedduring outages in one of the systems. This will thenmake the regular reliability indices well suited forevaluating the reliability in the microgrid.

Algorithm 2 illustrates the controller procedure forthe microgrid. First, the sectioning time of the cb isupdated if needed. This is done in case the systemwas in a sectioning time in the previous time step.Then, if a new outage happens in one of the systems,Ds and Ms, the circuit breaker of the Ms will open.In the Ms, all lines will be checked for failures. If theline containing the circuit breaker is failed, then the

5

circuit breaker should remain open regardless of theMs mode. If not, the algorithm will check in whichmode the Ms should operate in. If the Ms is supposedto operate in island mode during failures in the Ds, theprocedure will check for failed lines in the Ds. In caseof no failed lines in the Ds, the failed sections in theMs are disconnected and the circuit breaker closed. Ifthe microgrid is in a supportive mode, where it can beislanded with parts of the Ds buses, the failed sectionsin the Ms are disconnected before the circuit breakeris closed.

This will result in the Ms being in one of three statesduring an outage in the Ps: 1) in island mode, 2) inisland mode with parts of the Ds, or 3) connectedto the Ds with the possibility of supply from theoverlying network.

Algorithm 2: Microgrid controller procedure

1 Update sectioning time;2 if cb is open and sectioning time ≤ 0 or l

recovered from failure then3 Check l in Ms for failure;4 if l with cb not failed then5 if Ms in island (survival) mode then6 if No failed l in Ds then7 Disconnect failed sections;8 Close cb;9 end

10 else11 Disconnect failed sections;12 Close cb;13 end14 end15 end

D. Load flow and optimization problem

To calculate the electrical consequence of a failurein the network when there are generation resourcesavailable, a load flow calculation needs to be per-formed. Since the network is radially operated, aForward-Backward sweep (FBS) approach is used toperform the load flow calculation [34], [35]. Thebenefit of using an FBS approach is that the loadflow components are calculated iteratively and donot need to go through the Jacobian matrix, as inthe Newton-Raphson method. This way convergenceproblems arising from an ill conditioned matrix ofweak networks can be avoided.

The FBS approach calculates the load flow of thesystem by updating the power flow through the back-ward sweep before the buses voltage magnitude andangle are updated in the forward sweep. The methodis based on [36].

Backward sweep: In the backward sweep, the ac-tive and reactive power over the lines is calculated. Theactive and reactive power over line l for iteration k, Pl

and Ql, is calculated by adding the accumulated activeand reactive load, P

′

l and Q′

l, at the downstream busesand the accumulated active and reactive power losses,P ∗l and Q∗l , over the downstream lines including thepower loss over line l and the load at the current bus

Pl = P′

l + P ∗l (1)

Ql = Q′

l +Q∗l (2)

where the accumulated active and reactive load atthe buses are calculated as

P′

l = P loadi +

∑db

P loaddb

(3)

Q′

l = Qloadi +

∑db

Qloaddb

(4)

Here P loadi is the active load at node i, Qload

i isthe reactive load at node i, and P load

dband Qload

dbare

the active and reactive loads at the downstream buses,respectively.

The accumulated active and reactive power lossesare calculated as

P ∗l = P lossl +

∑dl

P lossdl

(5)

Q∗l = Qlossl +

∑dl

Qlossdl

(6)

Here P lossl and Qloss

l are the active and reactivepower loss over line l, respectively, and P loss

dland

Qlossdl

are the active and reactive power loss of thedownstream lines, respectively.

The active and reactive power loss over a line isfurther calculated as

P lossl = Rl

P′2l +Q

′2l

V 2j

(7)

Qlossl = Xl

P′2l +Q

′2l

V 2j

(8)

Where Rl and Xl are the line resistance and reac-tance, respectively, and Vj is the voltage at the endingbus.

Forward sweep: In the forward sweep, the busvoltage magnitudes and angles are updated by usingthe updated active and reactive power from the back-ward sweep. Then the voltage magnitude at bus i inrelation to the voltage magnitude at bus j for iterationk can be calculated by

Vi = Vj − Il(Rl + jXl) =√V 2j − T1 + T2

(9)

where the term T1 and T2 are

T1 = 2(PlRl +QlXl)

6

T2 =(P 2

l +Q2l )(R

2l +X2

l )

V 2j

The voltage angle can be updated as

δi = δj + arctanIm(Vi)

Re(Vi)(10)

where δj is the voltage angle of bus j, Im(Vi) isthe imaginary part of Vi, and Re(Vi) is the real partof Vi.

Optimization problem: Since the system linesand the generation resources have limited capacity,an optimization problem is introduced to minimizethe total shedded load in the network based on theprice. The price is based on the Cost of energy notsupplied (CENS) [37]. The CENS scheme, in Norway,is intended to give the DSOs incentive to build andoperate the network with a focus on a socio-economicand reliable operation of the network. Similar schemescan be seen in [38], [39].

The optimization problem, in eq. 11, minimizes thetotal load shedded in the network based on the CENSfor shedding that specific load type. This is subjectedto load flow balance and capacity limitations over thepower lines, the load, and the generation. Here Cn

is the cost of shedding load at node n while P sn is

the amount of shedded power at node n. P gj is the

production from generator j. P dk is the load demand

at node k while P li is the power transferred over line

i. γi = 1 if line i is the starting point, -1 if line i isthe ending point. λj = 1 if there is a production unitat node j, otherwise it is 0. µk = 1 if there is a loadon node k, otherwise it is 0.

minimizeP s

n

Ps =

Nn∑n=1

Cn · P sn (11)

subject to:Nl∑i=1

γi · P li =

Ng∑j=1

λj · P gj −

Nn∑k=1

µk · (P dk − P s

k )

minP gj ≤ P

gj ≤ maxP g

j ∀j = 1, . . . , Ng

0 ≤ P sk ≤ P d

k ∀k = 1, . . . , Nn∣∣P li

∣∣ ≤ maxP li ∀i = 1, . . . , Nl

E. Reliability indices

There are different reliability indices used for mea-suring and quantifying the reliability in distributionsystems. They can be divided into customer-orientedindices and load- and production-oriented indices [31].The three basic reliability parameters are the faultfrequency or the average failure rate, λs, the annualaverage outage time, Us, and the average outage time,rs. In a radial network, the three different reliabilityparameters can be calculated as in eq. 12, eq. 13, andeq. 14. Here, λi and ri are the failure rate and outagetime at load point i, respectively. These equations are

used to state the load-point reliability. However, thesedo not tell anything about the electrical consequenceof a failure or the cost related to a failure.

λs =∑

λi (12)

Us =∑

λiri (13)

rs =Us

λs=

∑λiri∑λi

(14)

Load- and Production-oriented indices: The load-and production-oriented indices aim to indicate theelectrical consequence of failures in the system [31].The total Energy not Supplied (ENS) in a system canbe calculated as seen in eq. 15

ENSs = UsPs (15)

The interruption cost for the system can be calcu-lated as seen in eq. 16 [40]. Here, ci is the specificinterruption cost for each customer category at loadpoint i.

CENSs =∑

ENSici (16)

Customer-oriented indices: The customer-orientedindices aim to indicate the reliability of the distributionsystem based on the interruption experienced by thecustomers [31]. In this paper, three important indiceswill be investigated:

1) System Average Interruption Frequency Index

SAIFI =

∑∀i λiNi∑Ni

(17)

where Ni is the total number of customersserved, and

∑∀i λiNi is the total number of

customer interruptions. SAIFI is a measure ofthe frequency of interruptions the customersin the system expect to experience. Anyinterruption seen from the consumer is countedas a fault, regardless of origin.

2) System Average Interruption Duration Index

SAIDI =

∑UiNi∑Ni

(18)

where∑∀i UiNi is total number of customer

interruption durations. SAIDI is a measureof the expected duration of interruptions acustomer is expected to experience.

3) Customer Average Interruption Duration Index

CAIDI =

∑UiNi∑∀i λiNi

=SAIDI

SAIFI(19)

CAIDI is the ratio between SAIDI and SAIFI

and measures the average duration each givencustomer in the system is expected to experience.

7

F. Statistical analysis

1) Statistical difference: Statistical analysis is im-portant to confirm the hypotheses made by researchers.A statistical test checking for equal means can beperformed to quantify whether the result data setsfrom different scenarios differ. This is performed todetermine if the result data sets are significantly dif-ferent or not. There are two classes of hypothesis tests,namely 1) parametric [41] and 2) non-parametric [42].Both classes are bounded by assumptions about thedistribution of the result data sets. Parametric testsare restricted to normally distributed data sets withequal variance, while non-parametric tests have looserrestrictions. For this reason, parametric tests havestronger statistical power, making them the preferredchoice if the data set satisfies its assumptions. Todecide which class of tests to use, an evaluation ofthe distribution of the results data set must be carriedout.

The evaluation of normality can be done by per-forming an Anderson-Darling (AD) test. The ADtest is used to establish if a data set comes from apopulation with a specific statistical distribution [43].The outcome of the test is measured by the deviationA2, between the number of samples and a weightedlogarithmic expression of the data in the data set. Anincreasing deviation between A2 and zero decreasesthe probability of the data set being of a specificstatistical distribution.

If the AD test for normality fails, non-parametrictests must be used to check for equal means. In thispaper, the Kolmogorov-Smirnov (KS) [42] test is usedfor this purpose. The KS test quantifies the distancebetween two data sets, indicating if the data sets aredrawn from the same population.

2) Sensitivity analysis: Sensitivity analysis is im-portant since it provides a mapping of the behavior ofthe model and illustrates how the results are affectedby various parameters.

Factorial design is a more advanced form of sensi-tivity analysis used to investigate and understand theeffect of independent input variables on a dependentoutput variable [44]. One of the biggest advantages offactorial design compared to a conventional sensitivityanalysis, that only studies the effect of one variableat a time, is the ability to quantify both the influencethe variables have on the output variable and on theother investigated variables. However, this drasticallyincreases the number of combinations to investigate,which will lead to more simulation time, for example,a factorial design test with three factors that considertwo levels each leads to a total of 23 = 8 combinations.

IV. RELIABILITY TEST SYSTEM

A. Test network

The method is applied on the IEEE 33-bus systemseen in Fig. 5. The IEEE 33-bus system is chosen

since it is a commonly used network of a realisticsize that might fit well with a medium-sized microgrid.However, since the network is operated fully radiallywithout any backup connections, possible islandedoperation with the microgrid will be investigated.The total load in the distribution system is scaled toapproximately double the size of the traditional loadvalues in the IEEE 33-bus system [45] to have a totalload profile two to three times larger than the possibleproduction from the microgrid.

The distribution system includes different types ofloads, such as households, farms, industry, trade, andoffice buildings, to build a more dynamic customerprofile in the system and include the priority of theloads. The load profiles are generated based on the FA-SIT requirement specification [46] where the weatherdata is collected from a location in eastern Norway.The reliability data of the system can be seen in Tab.I.

Table I: Reliability data for the system

Section time [h] 1Outage time Line Transformer

[h] 4 8Failure rate Line Transformer[failure/year] 0.07 0.007

B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 B13 B14 B15 B16 B17 B18

B19 B20 B21 B22

B23 B24 B25

B26 B27 B28 B29 B30 B31 B32 B33

BM1 BM2

BM3

BM4

Line Bus Circuitbreaker Disconnector

Figure 5: The topology of the distribution system(blue) and the microgrid (green)

B. Microgrid

The microgrid is placed on bus 33 at the end of theradial arm in the distribution system The microgridcontains wind power, solar power, a battery, and someload. The specifications of the microgrid can be seenin Tab. II. The wind and solar power profiles are gen-erated based on weather data from the same locationin Norway.

The microgrid is operated in grid-connection modemost of the time, but if a line failure occurs in thedistribution system or the microgrid, the microgrid willshift to island operation.

8

Table II: Microgrid specifications

Component SpecificationBattery Max capacity: 1 MWh

Inverter capacity: 500 kWEfficiency: 0.95Min SOC: 0.1

Wind & Solar power Max power: ∼ 3.5 MWLoad Peak load ∼ 200 kWh

Battery strategy: The battery in the microgrid isassumed to follow the power market in the system,meaning that the battery will sell and store energybased on wanting to profit from the stored energy.However, since this paper focuses on the reliability ofelectricity supply, the market strategy will be simulatedby predicting the state of charge (SOC) levels of thebattery. The SOC level is assumed to follow a uniformdistribution between the minimum and the maximumSOC level of the battery. This can be assumed since noestimation has been done to evaluate a marked strategyfor the battery. This will result in a more realisticanalysis where the battery SOC level can vary duringfailures in the network.

C. Description of the different scenarios

The case study investigates three different scenarios,which are based on three different operating modesfor the microgrid, namely: 1) no support mode, 2)full support mode, and 3) limited support mode. Thispaper aims to investigate how a microgrid with DGand an energy storage system impacts the reliabilityof electricity supply in a distribution system. Thiswill be illustrated through the different scenarios forwhich the reliability of the electricity supply for boththe distribution system and the microgrid will beinvestigated and evaluated.

1) Scenario 1: No active participation, microgrid insurvival mode: The first scenario will include both thedistribution system and the microgrid, both radially op-erated. For failures in the network, the microgrid willdisconnect from the distribution system and operate inan island mode until the failure is removed.

The microgrid will not support the distribution sys-tem during the failure event and the microgrid mustbe self-sufficient during the outage time.

2) Scenario 2: Active participation of the microgrid,microgrid with full support: In the second scenario,active participation of the microgrid during failureswill be considered. During failures, the microgrid willreconnect after the sectioning and provide electricalpower support to the distribution system.

This is a more aggressive scenario where the out-come might result in the battery becoming emptyrelatively fast. In cases of no generation from the DGsin the microgrid, the loads in both the distributionsystem and the microgrid will be shedded.

3) Scenario 3: Active participation of the microgrid,microgrid with limited support: The third scenario also

considers active participation from the microgrid andthe resources in the microgrid. However, the batterywill rather serve as a backup source and will thereforenot participate in any market strategy. The batterywill therefore be fully charged when a failure occurs.Instead of giving full support, the microgrid will focuson storing enough power in the battery to ensure thatthe load in the microgrid can be supported at peak loadfor at least four hours. This will result in a prioritizingof the microgrid load.

V. RESULTS AND DISCUSSION

This section presents the reliability results for thethree described scenarios from both the distributionsystem and microgrid perspective. Finally, a factorialdesign study is presented for Scenario 3.

A. The distribution system perspective

Fig. 6 illustrates the distribution of ENS for the threedifferent scenarios in the distribution system. A boxplot of the same is illustrated in Fig. 7.

It can be observed from the box plot that thereis a slight decrease in ENS for Scenarios 2 and 3where the microgrid supports the distribution systemduring failures, compared to Scenario 1 where themicrogrid operates in island mode during failures. Thiscan be seen from the plots of the distributions, asthe distribution for Scenarios 2 and 3 are slightlyshifted to the left compared to Scenario 1. This resultindicates that load shedding with lower ENS happensmore frequently for Scenarios 2 and 3 compared toScenario 1. In comparing Scenarios 2 and 3, there is aslight improvement when applying Scenario 3. This ismostly due to the state of the battery, which operateson standby and will most likely be fully charged whenan outage appears in the system.

0

200

400

Fre

qu

ency

S1

0

200

400

Fre

qu

ency

S2

0 20 40 60 80 100 120 140

ENS [MWh]

0

200

400

Fre

qu

ency

S3

Figure 6: Frequency distribution of ENS in the distri-bution system for the three scenarios

9

S1 S2 S3

0

20

40

60

80

100

120

140

EN

S[M

Wh

]

Figure 7: Box plot of the frequency distribution of ENSin the distribution system

The result is more readable in Tab III where thediscussed reliability indices are presented for the threescenarios. Overall, the ENS decreases by 4.54% forScenario 2 and 5.69% for Scenario 3 compared toScenario 1. This effect could be increased by hav-ing multiple generation sources or microgrids spreadaround in the network. Since the microgrid is locatedat one spot in the microgrid, there will be several caseswhere the microgrid is unable to support the grid sincethe isolated part is not connected to the microgrid. Inaddition, since the microgrid prioritizes its own load,the distribution system will get less support.

Table III: Reliability indices for the distribution system

S1 S2 S3ENS 39.8014 37.9930 37.5350

SAIFI 5.3953 5.4143 5.4655SAIDI 9.9347 9.8339 9.7854CAIDI 1.8414 1.8163 1.8032

By analyzing the other reliability indices, somesmall differences between the different scenarios canbe observed. The failure duration for the system de-creases with support from the microgrid. The reasonfor this is that some load points will experience adecreased period of shedding. However, the failurefrequency will increase. In most cases, the microgridgeneration and battery are unable to preserve supplyduring the whole outage period, resulting in some loadpoints experiencing a new outage period when thegeneration is low or the battery is empty.

Statistical testing was performed on the resultsfrom the three different scenarios. First, the AD testwas used to investigate if the results are normallydistributed. As seen in Tab. IV, the normality testfailed, indicating that none of the results are nor-mally distributed. The test statistics, A2, are veryhigh, indicating with high certainty that the result isnot normally distributed. This can also be seen byinvestigating the distributions in Fig. 6.

The non-parametric test was then performed toinvestigate if the result originates from the samepopulation. A significance level of 5% was chosen,meaning that the null hypothesis, which is that theresult is taken from the same population, is rejectedif the p-value of the test is lower than 0.05. Tab.IV shows that Scenario 1 is significantly differentfrom Scenarios 2 and 3. However, Scenario 2 and 3are not significantly different, meaning they are fromthe same population. This result indicates that thereis a significant difference between receiving supportfrom the microgrid during unintentional outages in thedistribution system compared to no support. However,the actual network impact might vary.

Table IV: Statistical test of the three scenarios in thedistribution system

Scenario AD test Scenarios KS testA2 p-value

S1 22.61 S1 vs. S2 0.00S2 22.46 S1 vs. S3 0.00S3 24.73 S2 vs. S3 0.63

B. The microgrid perspective

The result as seen from the microgrid perspectiveis evident in Fig. 8. A box plot of the same resultis illustrated in Fig. 9. The differences between thescenarios are more clear when investigating them fromthe perspective of the microgrid. Here, Scenario 1will result in most shedding, since the microgrid hasto survive in island mode during the failure in thedistribution system without backup supply. Scenario 2is somewhat better since the microgrid will reconnectto the distribution system. For situations where themicrogrid is in the same sub-system as the mainfeeder of the distribution system, the microgrid willbe ensured supply. Scenario 3 is the case that willgive a very small ENS. The outliers in the box plotare a result of failures inside the microgrid and a highimpact low probability case. This case is a result of twofailures happening in close approximation, leading toa discharged battery before the last failure is removed.

The reliability indices for the microgrid can beseen in Tab. V. The indices indicate that there is aconsiderable decrease in the interruption duration forScenario 3 compared to the other. Scenario 3 also leadsto a lower interruption frequency. The ENS for Scenario3 has decreased by 71.55% compared to Scenario 1,whereas the ENS for Scenario 2 has decreased by16.87%.

10

0

1000

2000

Fre

qu

ency

S1

0

1000

2000

Fre

qu

ency

S2

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

ENS [MWh]

0

1000

2000

Fre

qu

ency

S3

Figure 8: Frequency distribution of ENS in the micro-grid for the three scenarios

S1 S2 S3

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

EN

S[M

Wh

]

Figure 9: Box plot of the frequency distribution of ENSin the microgrid

Table V: Reliability indices for the microgrid

S1 S2 S3ENS 0.2851 0.2370 0.0811

SAIFI 0.9609 1.0350 0.8616SAIDI 4.0188 3.254 1.024CAIDI 4.1825 3.1440 1.1884

The same statistical tests were performed on thereliability result for the microgrid. The result canbe seen in Tab. VI. None of the results passed thenormality test, which can be expected based on thedistribution of the result. The same non-parametric testwas performed with a significance level of 5%. Theoutcome shows that all the scenario results are signif-icantly different from each other. The result illustratesthe importance of backup supply for the microgrid,either through the distribution system or through themicrogrid sources. Since all the scenarios are signif-icantly different, Scenario 3 is the best scenario seenfrom the microgrid perspective when measuring ENS.

Table VI: Statistical test of the three scenarios in themicrogrid

Scenario AD test Scenarios KS testA2 p-value

S1 173.86 S1 vs. S2 0.00S2 174.77 S1 vs. S3 0.00S3 271.12 S2 vs. S3 0.00

C. Sensitivity analysis

A factorial design study was performed on Scenario3, which was chosen based on the results obtainedfrom the reliability assessment. Since there was no sig-nificant difference between the results from Scenario2 and 3 for the distribution system and Scenario 3gave the best overall result for the microgrid, this isthe most appropriate case to investigate further. Theinvestigated parameters and parameter values can beseen in Tab. VII. Any lower limit value for the batterycapacity was not chosen since the battery in Scenario3 is dimensioned for the peak load in the microgrid.If the battery capacity value is decreased, Scenario 3as it is illustrated in this study is not applicable.

Table VII: Parameter values used in the factorial designstudy

NameBattery 1 Bat1[MWh] 2 Bat2

Line Transformer Name2 4 Out1

Outage time 4 8 Out2[h] 6 10 Out3

Line Transformer Name0.05 0.005 Fail1

Failure rate 0.07 0.007 Fail2[failures/year] 0.09 0.009 Fail3

The results from the factorial design study areillustrated in Fig. 10 for the distribution system and inFig.11 for the microgrid. The plots indicate both theeffect each parameter has on the ENS in the systemand how the parameters affect each other.

When investigating the result for the distributionsystem (Fig. 10), an increased battery capacity doesnot contribute to any considerable improvement ofthe ENS. The effect is, as expected, mirrored by theincrease of the battery capacity. This illustrates thatthe saturation level is not met. In addition, since thesystem is relatively large, so multiple scenarios willlead to situations where the microgrid is unable tosupport all the isolated load points in the distributionsystem, the total support will be seen as small.

However, the change of ENS is larger when inves-tigating the failure rate and the outage time. This isespecially for situations with either a high failure rateor outage time compared to the other parameter. Over-all, the failure rate affects the ENS in the distributionsystem the most.

Investigating the result from the microgrid perspec-tive (Fig. 11), the battery capacity holds a more im-

11

portant role compared to the distribution system. Thiseffect is considered when comparing the battery ca-pacity with the higher outage time of the components.This is a result of the microgrid being able to supplythe microgrid load for more hours during specialhigh impact low probability events, as discussed. Theoutage time is also important for the reliability ofthe microgrid. This effect illustrates the importance ofbeing able to restore supply faster.

Bat

1

Bat

2

Battery capacity

25

50

75Mea

nlo

adsh

ed[M

Wh

]

Ou

t 1

Ou

t 2

Ou

t 3

Outage time

Fail1Fail2Fail3

Fai

l 1

Fai

l 2

Fai

l 3

Failure rate

Main effects

25

50

75

Out1

Out2

Out3

25

50

75

Bat1

Bat2

Battery

capacity

Ou

tage

time

Failu

rerate

Figure 10: Interaction plot for the distribution system

Bat

1

Bat

2

Battery capacity

0.0

0.5

Mea

nlo

adsh

ed[M

Wh

]

Ou

t 1

Ou

t 2

Ou

t 3

Outage time

Fail1Fail2Fail3

Fail

1

Fail

2

Fail

3

Failure rate

Main effects

0.0

0.5

Out1

Out2

Out3

0.0

0.5

Bat1

Bat2

Battery

capacity

Ou

tage

time

Failu

rerate

Figure 11: Interaction plot for the microgrid

VI. CONCLUDING REMARKS

The reliability of electricity supply has been inves-tigated from both the distribution system’s and themicrogrid’s point of view. The impact of the microgrid

on the reliability in the distribution system based onthe reliability indices is not considerably large in thiscase. This result is dependent on multiple factors suchas the availability of wind and solar power in themicrogrid, the load in the system at the time of thefault, where the fault occurred, the location of themicrogrid, and the dimensioning of the load in thedistribution system compared to the generation in themicrogrid. Based on the statistical tests, there is asignificant difference between having support from themicrogrid compared to no support. The results indicatethat this may have a significant impact on the reliabilityof the distribution system. However, this is dependenton to which criteria the DSO gives priority to. Thiswill most likely be either measured in ENS or basedon cost benefit analysis.

When investigating the different scenarios for boththe distribution system and the microgrid, Scenario3 is the best cross-over strategy for both systems.This is related to the statistical test illustrating thatScenario 3 does not differ significantly from the otherbest cases for the two systems. However, due to thepossibility of additional load shedding in the microgridfor Scenario 3 related to a low probability case, themicrogrid could take some precautions like having anadditional backup to give sufficient power. In addition,the distribution system and the microgrid should havesome agreement related to such support, which alsocould include agreements related to situations withhigh impact low probability for the microgrid.

In the end, the outage time, the sectioning time, andthe probability of failure were selected based on meanvalues from Norwegian failure statistics. However,if the parameters followed probability distributionsinstead, the results would have been different. Thisis considered an area for future work. Parts of theeffect are investigated and presented in the factorialdesign study, but it is not fully accounted for. Anadditional important finding is that, in such a network,different scenarios can occur where the microgrid isunable to provide support, for example, if the isolatedpart of the network does not include the microgrid.Since all of these scenarios are possible outcomes, theywill be added to the failure statistic and increase thesimilarity of the scenarios. Therefore, it could be ofvalue to investigate the same idea with multiple micro-grids or other sources scattered throughout the systemand therefore, obtain more possible sub-systems withgeneration units.

VII. ACKNOWLEDGMENT

This work is funded by CINELDI - Centre for intel-ligent electricity distribution, an 8-year Research Cen-tre under the FME-scheme (Centre for Environment-friendly Energy Research, 257626/E20). The authorsgratefully acknowledge the financial support from theResearch Council of Norway and the CINELDI part-

12

ners. The authors will also like to thank Gerd Kjølle forher bright insight and the knowledgeable discussions.

REFERENCES

[1] National Academies of Sciences, Engineering, and Medicineand others, The power of change: Innovation for developmentand deployment of increasingly clean electric powertechnologies. National Academies Press, 2016. [Online].Available: https://doi.org/10.17226/21712

[2] I.-I. E. Agency, “Status of power system transformation2019.” [Online]. Available: https://www.iea.org/reports/status-of-power-system-transformation-2019

[3] “Eei future of us electric distribution.”[4] A. Ipakchi and F. Albuyeh, “Grid of the future,” IEEE Power

and Energy Magazine, vol. 7, pp. 52–62, 3 2009. [Online].Available: 10.1109/MPE.2008.931384

[5] G. Prettico and M. G. Flammini, “Distribution systemoperators observatory 2018 overview of the electricitydistribution system in europe,” 2019. [Online]. Available:http://doi.org/10.2760/104777

[6] I. B. Sperstad, M. Z. Degefa, and G. Kjølle, “The impactof flexible resources in distribution systems on the securityof electricity supply: A literature review,” Electric PowerSystems Research, vol. 188, p. 106532, 2020. [Online].Available: https://doi.org/10.1016/j.epsr.2020.106532

[7] B. Mohandes, M. S. El Moursi, N. Hatziargyriou, andS. El Khatib, “A review of power system flexibility withhigh penetration of renewables,” IEEE Transactions on PowerSystems, vol. 34, no. 4, pp. 3140–3155, 2019. [Online].Available: http://doi.org/10.1109/TPWRS.2019.2897727

[8] A. Escalera, B. Hayes, and M. Prodanovic, “A surveyof reliability assessment techniques for modern distributionnetworks,” Renewable and Sustainable Energy Reviews,vol. 91, pp. 344–357, 2018. [Online]. Available: https://doi.org/10.1016/j.rser.2018.02.031

[9] C. L. T. Borges, “An overview of reliability models andmethods for distribution systems with renewable energydistributed generation,” Renewable and sustainable energyreviews, vol. 16, no. 6, pp. 4008–4015, 2012. [Online].Available: https://doi.org/10.1016/j.rser.2012.03.055

[10] G. Celli, E. Ghiani, F. Pilo, and G. G. Soma, “Reliabilityassessment in smart distribution networks,” Electric PowerSystems Research, vol. 104, pp. 164–175, 2013.

[11] F. Mohamad, J. Teh, C.-M. Lai, and L.-R. Chen, “Developmentof energy storage systems for power network reliability: Areview,” Energies, vol. 11, no. 9, p. 2278, 2018. [Online].Available: https://doi.org/10.3390/en11092278

[12] A. Kwasinski, V. Krishnamurthy, J. Song, and R. Sharma,“Availability evaluation of micro-grids for resistant powersupply during natural disasters,” IEEE Transactions on SmartGrid, vol. 3, no. 4, pp. 2007–2018, 2012. [Online]. Available:http://doi.org/10.1109/TSG.2012.2197832

[13] S. M. Nosratabadi, R.-A. Hooshmand, and E. Gholipour,“A comprehensive review on microgrid and virtual powerplant concepts employed for distributed energy resourcesscheduling in power systems,” Renewable and SustainableEnergy Reviews, vol. 67, pp. 341–363, 2017. [Online].Available: https://doi.org/10.1016/j.rser.2016.09.025

[14] Microgrid Knowledge Editorial Team, “Microgrid cybersecu-rity: Protecting and building the grid of the future,” S&CElectric Company, pp. 1–12, 2019.

[15] P. M. Costa and M. A. Matos, “Assessing the contribution ofmicrogrids to the reliability of distribution networks,” ElectricPower Systems Research, vol. 79, no. 2, pp. 382–389, 2009.[Online]. Available: https://doi.org/10.1016/j.epsr.2008.07.009

[16] S. Conti, R. Nicolosi, and S. Rizzo, “Generalized systematicapproach to assess distribution system reliability withrenewable distributed generators and microgrids,” IEEETransactions on Power Delivery, vol. 27, no. 1, pp. 261–270,2011. [Online]. Available: http://doi.org/10.1109/TPWRD.2011.2172641

[17] S. Conti and S. A. Rizzo, “Monte carlo simulation by usinga systematic approach to assess distribution system reliabilityconsidering intentional islanding,” IEEE Transactions onPower Delivery, vol. 30, no. 1, pp. 64–73, 2014. [Online].Available: http://doi.org/10.1109/TPWRD.2014.2329535

[18] P. M. de Quevedo, J. Contreras, A. Mazza, G. Chicco,and R. Porumb, “Reliability assessment of microgrids withlocal and mobile generation, time-dependent profiles, andintraday reconfiguration,” IEEE Transactions on IndustryApplications, vol. 54, no. 1, pp. 61–72, 2017. [Online].Available: http://doi.org/10.1109/TIA.2017.2752685

[19] A. Syrri, E. M. Cesena, and P. Mancarella, “Contributionof microgrids to distribution network reliability,” in 2015IEEE Eindhoven PowerTech. IEEE, 2015, pp. 1–6. [Online].Available: http://doi.org/10.1109/PTC.2015.7232486

[20] H. Farzin, M. Fotuhi-Firuzabad, and M. Moeini-Aghtaie, “Roleof outage management strategy in reliability performance ofmulti-microgrid distribution systems,” IEEE Transactions onPower Systems, vol. 33, no. 3, pp. 2359–2369, 2017. [Online].Available: http://doi.org/10.1109/TPWRS.2017.2746180

[21] M. Barani, J. Aghaei, M. A. Akbari, T. Niknam, H. Farahmand,and M. Korpas, “Optimal partitioning of smart distributionsystems into supply-sufficient microgrids,” IEEE Transactionson Smart Grid, vol. 10, no. 3, pp. 2523–2533, 2018. [Online].Available: http://doi.org/10.1109/TSG.2018.2803215

[22] N. Nikmehr and S. N. Ravadanegh, “Reliability evaluationof multi-microgrids considering optimal operation of smallscale energy zones under load-generation uncertainties,”International Journal of Electrical Power & Energy Systems,vol. 78, pp. 80–87, 2016. [Online]. Available: https://doi.org/10.1016/j.ijepes.2015.11.094

[23] THEMA Consulting Group, “Descriptive studyof local energy communities,” 2018. [Online].Available: https://thema.no/wp-content/uploads/THEMA-Reort-2018-20-Local-Energy-Communities-Report-Final.pdf

[24] L. Hancher and B. Winters, “The eu winter packagebriefing paper,” 2017. [Online]. Available: https://fsr.eui.eu/wp-content/uploads/The-EU-Winter-Package.pdf

[25] D. T. Ton and M. A. Smith, “The us departmentof energy’s microgrid initiative,” The Electricity Journal,vol. 25, no. 8, pp. 84–94, 2012. [Online]. Available:https://doi.org/10.1016/j.tej.2012.09.013

[26] S. F. Myhre, O. B. Fosso, P. E. Heegaard, and O. Gjerde, “Atool for reliability assessment of smart and active distributionsystems – relsad,” 2021.

[27] E. Commission, “Directive 2005/89/ec of the europeanparliament and of the council of 18 january2006 concerning measures to safeguard security ofelectricity supply and infrastructure investment,” 2006.[Online]. Available: https://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:32005L0089&from=EN

[28] “Driften av kraftsystemet 2018,” https://publikasjoner.nve.no/rapport/2019/rapport2019 27.pdf, accessed: 2021-05-30.

[29] E. R. Larsen, S. Osorio, and A. van Ackere, “A framework toevaluate security of supply in the electricity sector,” Renewableand Sustainable Energy Reviews, vol. 79, pp. 646–655, 2017.[Online]. Available: https://doi.org/10.1016/j.rser.2017.05.085

[30] “Forskrift om leveringskvalitet i kraftsystemet,” https://lovdata.no/dokument/SF/forskrift/2004-11-30-1557, accessed: 2021-05-30.

[31] R. Billinton and R. N. Allan, Reliability Evaluation of PowerSystems. New York: Springer Science+Business Media, LLC,1996, no. 2nd ed.

[32] G. Kjolle and K. Sand, “Relrad-an analytical approach fordistribution system reliability assessment,” IEEE Transactionson Power Delivery, vol. 7, no. 2, pp. 809–814, 1992. [Online].Available: http://doi.org/10.1109/61.127084

[33] S. Wang, Z. Li, L. Wu, M. Shahidehpour, and Z. Li,“New metrics for assessing the reliability and economicsof microgrids in distribution system,” IEEE transactions onpower systems, vol. 28, no. 3, pp. 2852–2861, 2013. [Online].Available: http://doi.org/10.1109/TPWRS.2013.2249539

[34] D. Shirmohammadi, H. W. Hong, A. Semlyen, and G. Luo, “Acompensation-based power flow method for weakly mesheddistribution and transmission networks,” IEEE Transactionson power systems, vol. 3, no. 2, pp. 753–762, 1988. [Online].Available: http://doi.org/10.1109/59.192932

[35] M. Haque, “Load flow solution of distribution systems withvoltage dependent load models,” Electric Power SystemsResearch, vol. 36, no. 3, pp. 151–156, 1996. [Online].Available: https://doi.org/10.1016/0378-7796(95)01025-4

13

[36] O. B. Fosso, “Pydsal-python distribution system analysislibrary,” in 2020 IEEE International Conference onPower Systems Technology (POWERCON). IEEE, 2020,pp. 1–6. [Online]. Available: http://doi.org/10.1109/POWERCON48463.2020.9230554

[37] “Kile – kvalitetsjusterte inntektsrammer ved ikke-levert en-ergi,” https://www.nve.no/reguleringsmyndigheten/okonomisk-regulering-av-nettselskap/om-den-okonomiske-reguleringen/kile-kvalitetsjusterte-inntektsrammer-ved-ikke-levert-energi/, accessed: 2021-05-30.

[38] A. Chowdhury and D. O. Koval, “Application of customerinterruption costs in transmission network reliability planning,”IEEE Transactions on Industry Applications, vol. 37,no. 6, pp. 1590–1596, 2001. [Online]. Available: http://doi.org/10.1109/28.968165

[39] M. J. Sullivan, B. N. Suddeth, T. Vardell, and A. Vojdani,“Interruption costs, customer satisfaction and expectations forservice reliability,” IEEE Transactions on Power Systems,vol. 11, no. 2, pp. 989–995, 1996. [Online]. Available:http://doi.org/10.1109/59.496185

[40] T. Langset, F. Trengereid, K. Samdal, and J. Heggset, “Qualitydependent revenue caps-a model for quality of supplyregulation,” in 16th International Conference and Exhibitionon Electricity Distribution, 2001. Part 1: Contributions.CIRED. (IEE Conf. Publ No. 482), vol. 6, 2001, p. 5 pp. vol.6.[Online]. Available: https://doi.org/10.1049/cp:20010906

[41] S. G. W. and W. G. Cochran, Statistical Methods. Iowa StateUniversity Press, 1989, no. 8th ed.

[42] I. M. Chakravarti, R. G. Laha, and J. Roy, Handbook of meth-ods of Applied Statistics Volume I: Techniques of computation,descriptive methods, and statistical inference. John Wiley andSons, 1967.

[43] M. A. Stephens, “Edf statistics for goodness of fit and somecomparisons,” Journal of the American statistical Association,vol. 69, no. 347, pp. 730–737, 1974.

[44] R. E. Walpoe, R. H. Myers, S. L. Myers, and K. Ye, Probability& Statistics for Engineers & Scientists. Pearson Education,Inc, 2011, no. 9th ed.

[45] M. E. Baran and F. F. Wu, “Network reconfiguration indistribution systems for loss reduction and load balancing,”IEEE Power Engineering Review, vol. 9, no. 4, pp. 101–102,1989. [Online]. Available: http://doi.org/10.1109/61.25627

[46] A. O. Eggen, “Fasit kravspesifikasjon,” Sintef Energy AS.,Tech. Rep., 2016.