optimal load shedding tecniques(2009).pdf

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Published in IET Generation, Transmission & Distribution

Received on 21st January 2008

Revised on 6th October 2008

doi: 10.1049/iet-gtd.2008.0308

ISSN 1751-8687

Reliability assessment of restructured powersystems using optimal load sheddingtechnique

P. Wang1

Y. Ding2

L. Goel1

1Nanyang Technological University, Singapore2

Reliability Research Lab, University of Alberta, 4-9 Mechanical Engineering Building, Edmonton, Alberta, Canada, T6G 2G8

E-mail: [email protected]

Abstract: Power system restructuring and deregulation has changed the strategy of reliability management of a power

system. Load shedding, and generation and reserve re-dispatch methods used in the existing reliability evaluation

techniques have to be improved to incorporate these changes. An optimisation technique, incorporating those

changes, is proposed in this study to determine load curtailment and generation re-dispatch for each contingency

state in the reliability evaluation of restructured power systems with the Poolco market structure. The problem is

formulated using the optimal power flow (OPF) technique. The objective of the problem is to minimise the total

system cost, which includes generation, reserve and interruption costs, subject to market and network constraints.

A model for the contingency management of a Poolco power market is presented to include generation andreserve biddings, reliability considerations and transmission network constraints in reliability evaluation. Both

supply side reliability for a generation company (Genco) and demand-side reliability for a customer can be

calculated using the technique. The proposed technique can be used to evaluate both conventional and

restructured power systems, and can provide both economic and reliability information for the independent

system operator to manage system reliability, for Gencos to enhance their reliability, and for customer to select

suppliers. The modified IEEE-RTS with the Poolco market has been analysed to illustrate the techniques. The results

obtained using the proposed technique have been compared with those from the existing load shedding techniques.

Nomenclature0 normal state index (superscript)

j contingency state index (superscript)

i, k bus index (subscript)

g generation unit index (subscript)

c component index (subscript)

r reserve unit index (subscript)

s customer sector index (subscript)

G Genco index (subscript)

SG set of units in a Genco

N number of buses

NG number of Gencos

FGj set of the failed units in Genco G

NSi number of customer sectors

NC number of contingencies

Nc number of independent components

b number of failed components

CD Gj real power capacity not delivered (MW)

CS Gj real power capacity shortage (MW)

Pgi0 real power dispatched (MW)

Qgi0 reactive power dispatched (Mvar)

Pis0 real power load (MW)

Qis0 reactive power load (Mvar)

DPgij real power re-dispatched (MW)

DQgij reactive power re-dispatched (Mvar)

628 IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp. 628640

& The Institution of Engineering and Technology 2009 doi: 10.1049/iet-gtd.2008.0308

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Prij real power reserve utilised (MW)

Qrij reactive power reserve utilised (Mvar)

LCj total system real power load curtailment (MW)

Cj total system cost (k$/h)

TRj total reserve utilised (MW)

GCgij generation costs (k$/h)

RCrij reserve costs (k$/h)

CCisj customer interruption costs (k$/h)

CP ij, CP is

j real power load curtailment (MW)

CQij,

CQisj

reactive power load curtailment (Mvar)

jVijj/ui

j bus voltage (kV)

jVijmax voltage up limit (kV)

jVijmin voltage low limit (kV)

jSikj j magnitude of apparent power flow from bus ito

k (MVA)jSikj

max apparent power limit of line (MVA)

Pgimin minimum real power generation (MW)

Pgimax maximum real power generation (MW)

DPgilow ramp down limit (MW)

DPgiupp ramp up limit (MW)

Qgimin minimum reactive power generation (Mvar)

Qgimax maximum reactive power generation (Mvar)

Primin minimum real power reserve (MW)

Primax maximum real power reserve (MW)

Qrimin minimum reactive power reserve (Mvar)Qri

max maximum reactive power reserve (Mvar)

PLi0 bus real power load (MW)

QLi0 bus reactive power load (Mvar)

PLimax maximum real power load (MW)

QLimin minimum reactive power load (Mvar)

QLimax maximum reactive power load (Mvar)

jYikj j/dik

j bus admittance

CDFs customer damage function ($/kWh)

pj state probability

Dj state departure rate (departures/year)

dj state duration (h)

Ac component availability

Uc component unavailability

lc component failure rate ( failures/year)

mc component repair rate (repairs/year)

1 Introduction

One of the main objectives of power system restructuring is toprovide the customer more choices on their supply reliability

based on the associated costs [1, 2]. Reliability is not a freeproduct for customers in the restructured power industry.

The centralised reliability management in the verticallyintegrated power systems is moving forward to thedecentralised reliability management in the new environment.

In vertically integrated power systems, similar reliabilitypolicy is usually applied for the same type of customers. Thesystem operator is concerned more with the system reliabilityrather than with the reliability of an individual customer.Generation re-dispatch and load shedding after a systemcontingency state in real-time operation are therefore usuallydetermined by system operators based on their experienceand judgement on system security. The main objective ofcontingency management in real-time system operation is tosolve the system voltage, line overloading, frequency andpower balance problems. The load shedding and generationre-dispatch techniques for conventional power systems have

been well-developed and applied in real-time operation.These techniques have also been applied in reliabilityevaluation. Most reliability evaluation techniques [35]perform DC and AC power flow analysis to determinenetwork violations for each contingency state. Proportionalor priority load shedding are commonly used to alleviatenetwork violations. The system and load point reliabilityindices are determined based on the load curtailments.Optimisation techniques have been used for load sheddingand generation re-scheduling in the reliability evaluation ofconventional power systems [68]. The objective of loadshedding and generation re-scheduling is to minimise thetotal load curtailment subject to system voltage and powerflow constraints. However, different customers have differentinterruption costs for the same amount of load curtailment.

The techniques [68] cannot guarantee the minimuminterruption costs although the curtailment is the minimum.Minimum total interruption cost is used to determineoptimal load shedding in distribution system reliabilityevaluation [9]. Generation costs have not been consideredduring the generation re-dispatch [69]. The changesintroduced by the restructuring have not been considered inthese techniques.

The most important changes in restructured power systems

are the introduction of electricity and ancillary service marketswhich allow customers to participate in reliabilitymanagement. Reserve is a product in the ancillary servicesmarket just like power is in the energy market. Customerscan purchase electricity from the energy market andreliability in the ancillary services market based on their priceand reliability expectations. Some customers may beprepared to pay more to receive higher reliability and othersmay be willing to pay less for lower reliability. Generationcompanies also have different offers for providing reserve andenergy. The independent system operator (ISO) willschedule the reserve considering customer reliability

requirements implicitly by incorporating the customerinterruption cost in the decision making. These changesmake the process of load shedding and generation

IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp. 628 640 629

doi: 10.1049/iet-gtd.2008.0308 & The Institution of Engineering and Technology 2009

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re-dispatch more complicated than that used in conventionalpower systems. The decision making for load shedding,reserve and generation re-dispatch in reliability evaluationshould incorporate these changes. The existing techniquesfor the reliability evaluation of restructured power systems[10, 11] utilise the AC power flow analysis. In these

methods, the proportional load shedding and generation re-dispatch were used in conjunction with a bilateral marketmodel. Generation cost, reserve costs and customer reliabilitychoices have not been considered in these techniques.

Goel et al. [12] proposed a framework to implementsupply- and demand-side contingency management in thereliability assessment of hybrid power markets. Thecontingency management model was formulated as a linearprogramming problem with the objective of minimising thecurtailment bidding costs from bilateral and spot marketcustomers, and the reserve bidding costs from each Genco.

All the bidding costs are assumed to be constant. Thereforethe cost models are not practical and accurate. Generationre-dispatch costs and transmission line contingencies werenot included in the objective function. The main focus of[13] is to propose a general methodology to evaluate bothreliabilities and prices for the pool and bilateral customersin a hybrid power market. An optimal power flow (OPF)technique was used to determine reliability indices andelectricity prices for each contingency state. The reservecost from the power pool was not included in the objectivefunction. The unreliability contributions from supply-side(Gencos) and the associated demand-side reliability(customers) were not clearly defined and classified. Goelet al. [14] focus on the impact of customer responses onthe system and nodal reliabilities. The customer responsesare modelled by the elasticity of the demand forelectricity in the normal state. In these references, theevaluation of the unreliability contribution from differentmarket participants including Gencos and transmissionnetwork were not investigated. Total reliability cost has notbeen investigated. The reliability-requirement-based loadcurtailments to customers in the load shedding process

were not calculated.

This paper investigates the cost-based load shedding and

generation re-dispatch techniques used in real-time marketand system operation for contingency management. A all-in-one technique for load shedding, generation re-dispatchand reserve dispatch in the reliability evaluation ofderegulated power systems with the Poolco marketstructure has been proposed. The objective of the problemis to minimise the total system cost including generationcosts, reserve costs and customer interruption costs.Reliability requirements are considered in an implicitmanner by incorporating the customer interruption cost inthe objective function. It does not mean a customer canarbitrarily select its reliability level, which is limited by

its interruption cost. More importantly reliability andeconomic indices for different market participants areintroduced and calculated to represent their reliabilities

and the associated responsibilities in the restructuredenvironment. Both supply- and demand-side reliabilityindices for each Genco and customer are calculated,respectively. Therefore the ISO and the power marketadministrator cannot only obtain the reliabilities of specificcustomers but also know the final obligation of a Genco to

its energy supply and the associated reliability.

Section 2 classifies the existing techniques for loadshedding and generation re-dispatch used in systemreliability evaluation. Customer responses to theircurtailments as customer damage functions are discussed inSection 3. The model for reliability management ofrestructured power systems with the Poolco marketstructure is presented in Section 4. Section 5 formulatesload shedding and generation re-dispatch problem for acontingency based on the minimum total system cost. Theproblem has been solved using the sequential quadratic

programming (SQP). Both transmission and generationfailures have been considered in contingency enumeration.

The reliability indices for Gencos and customers arepresented in Section 6. The modified IEEE RTS systemhas been analysed using the proposed techniques and theresults are presented in Section 7.

2 Existing load shedding andgeneration re-dispatch techniques

Different load shedding and generation re-dispatchtechniques have been implemented for reliabilitymanagement of conventional and restructured powersystems [311]. Some techniques [39] have beendeveloped for security analysis in system operation. Theload shedding techniques used in reliability evaluation [311] can be classified into two categories of AC power flow-based methods [35, 911] and OPF-based approaches[68]. In the first category, AC power flow for acontingency state is calculated to determine the network

violations. The generation is re-dispatched to releasenetwork violations. If the network violations cannot bealleviated by the generation re-dispatch, loads have to be

curtailed. The proportional and priority load sheddingschemes are usually used in these techniques. The objectiveof these techniques is to alleviate power flow and voltageproblems. The proportional load shedding scheme cuts theload based on the percentage of load at each bus. Thistechnique ignores the different specific reliabilityrequirements of customers. The priority load sheddingmethod considers the importance of different types ofcustomers. However, it cannot consider the differentreliability requirements for the same type of customers.Both the techniques usually result in load being over-curtailed. In order to solve the problem of

over-curtailments, the objective of the OPF-based methods[68] is to minimise the total load curtailment byconsidering network security constraints. For contingencyj,



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the problem is usually formulated as

MinLCj XNi1

CPji (1)

subject to the following constraints:

Power balance constraints

XNGG1

Xg[G

P0gi DPj

gi

P0Li CP

ji

XNk1

Vji

Vjk Yjik cos u ji u jk djik (2)

XNG

G1X

g[G

Q0gi DQjgi Q0Li CQji

XNk1

Vji

Vjk Yjik sin uji ujk djik (3)Generator limits

Pmingi P0

gi DPj

gi

Pmaxgi (4)

Qmingi Q0

gi DQjgi

Qmaxgi (5)

Load curtailment limits

0 CPji PmaxLi (6)

QminLi CQji Q

maxLi (7)

Voltage limits

Vi min

Vji Vi

max(8)

Line flow constraints

Sjik

Sik max (9)The existing OPF-based methods can only guaranteeminimum load curtailment. However, the total system costunder minimum curtailment may not be the minimum.

The existing reliability techniques have to be improved inthe new environment to incorporate customer choices,reserve and generation offers by Gencos. The proposed

technique considers customer choices on reliability with theobjective of minimising the total cost of a restructuredpower system with the Poolco model.

3 Customer responses tocurtailment

In a Poolco market, customers can purchase energy andreliability in term of reserve from the centralised power pool.

The reliability of a customer depends on the cost he or shewants to pay or has paid. Customers usually pay nodal priceswhich are determined by demands and supply bids at systembuses after considering network constraints. When a powersystem transfers from the normal operating state to acontingency state because of random failures, nodal pricesmay vary with changes in system operating conditions. Ifsystem generation is inadequate or network is congestedduring a contingency state, system operator may commitreserve or cut demand based on the difference betweencustomer reliability considerations and reserve prices.

The Federal Energy Regulatory Commission (FERC)

requires public utility load serving entities to establishdemand response mechanisms in which they will identifythe price at which load would be curtailed [15] for acontingency state. In other words, this price reflects thecustomer willingness to pay to avoid a service interruptionfor a contingency state. It is difficult to directly determinethe price for a customer to pay for its reliability. Customerresponse to price increment associated with the interruptioncan be indirectly measured by the customer interruption cost.

Customer interruption cost is a function of customers andthe associated interruption characteristics [16]. Customer

characteristics include the diversity of customers, the natureof customer activities, size of the operation etc. Interruptioncharacteristics include duration, frequency and time ofoccurrence of interruptions etc. The models used in thispaper are the sector customer damage functions, whichreflect the diversities of customer sectors. A standardindustrial classification (SIC) [16] has been used to dividecustomers into large user, industrial, commercial,agricultural, residential, government and institutions andoffice and buildings categories. Postal surveys have beenconducted by the University of Saskatchewan and Canadianelectric power utilities to estimate customer interruptioncosts. The survey data have been analysed to give sector

customer damage functions (CDFs), which are a function ofthe interruption durations. The CDFs normalised by theannual consumption of electricity ($/kWh) for the fiverepresentative customer sectors are shown in Fig. 1.

4 Model for contingencymanagement

Load shedding, generation re-dispatch and reserve dispatchare closely related to power market structures. The powermarkets in the world can be classified into bilateral markets,

Poolco markets and Hybrid markets. The method forcontingency management varies from one market to another.In the Poolco model used in this paper, Gencos bid to sell



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electricity and reserve for the maximum profit and customersbid to buy electricity and reserve based on their electricityand reliability considerations. Generating units whichprovide reserve can be classified into two categories [17] of

Tier 1 and Tier 2 resources. Tier 1 resources include thegenerating units are economically dispatched in the normaloperation and can ramp up or down from the normal outputto a new value during a contingency state; Tier 2 resourcesinclude the generating units are served only as spinningreserve providers, which supply additional reserve notavailable from Tier 1. These units usually operate at pointsthat deviate from economical dispatch in the normal state.

All those biddings have to be included in reliabilityevaluation with considering the various system constraints. Ageneral model for contingency management with the Poolcomarket is proposed as shown in Fig. 2. Market settlements,cost data and network constraints are included in this model.Generation re-dispatching (Tier 1) and reserve dispatching(Tier 2) under a given load condition are determined basedon the biddings from Gencos and customers through themarket clearing process. Quadratic equations are used in thispaper to represent generation dispatching and reservedispatching costs. The customer damage functions are used

to evaluate customer interruption costs for different customersectors [16]. When a system outage or a contingency occurs,load curtailment, generation re-dispatched and reservedispatched will be determined based on the proposed modelusing the optimisation technique.

5 Problem formulation

Reliability evaluation of a restructured power system is theprocess of predicting reliability of customers, Gencos andtransmission network for a given system operating conditionand market settlement. The system capacity inadequacy has

to be determined when a power system transfers from thenormal operating state to a contingency state because ofgeneration and/or transmission outages. If there is

generation inadequacy for a contingency state, generationand reserve has to be re-dispatched and load has to becurtailed to maintain system operation. The problem is acomplicated multi-objective optimisation which includesminimise the system operation cost and to maximisecustomer reliabilities. The proper solution for multi-objectiveoptimisation is Pareto optimal. In this paper, customerreliability requirements are implicitly represented usingcustomer interruption costs. Therefore the multi-objectiveproblem becomes the single-objective problem. The objective

is to minimise the total system cost after consideringnetwork and market constraints. The generation re-dispatched, reserve dispatched and load curtailed under theminimum total system cost are used to calculate thereliability indices of customers, Gencos and transmissionnetwork. For contingency state j, the objective function is

Min Cj XNi1

PNGG1

Pg[SG

GC

jgi P

0gi DP

jgi

Pr[SGRC

jri P

jri

PNSi

s1CCjis CP

jis

8>>>>>>>:

9>>>>=>>>>;

(10)

CCjis CPjis

CPjis CDFs d

j

(11)

subject to the following constraints for contingency state j:

Power balance constraints

XNGG1

Xg[SG

P0gi DPj

gi

Xr[SG

Pjri

0

@

1

AXNSis1

Pois CPjis

XNk1

Vji

Vjk Yjik cos uji ujk djik (12)

Figure 1 Customer damage functions for different sectors

Figure 2 Model for contingency management



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XNGG1

Xg[SG

Q0gi DQjgi

Xr[SG

Qjri

0@

1AXNSi

s1

Qois CQjis

XN

k1

Vji Vjk Yjik sin uji ujk djik (13)

Equations (12) and (13) show that the total system supply hasto be balanced with the total system load and transmissionloss after the generation re-dispatch, reserve dispatch andload curtailment.

Generation constraints

Pmingi DPj

gi P0

gi Pmax

gi (14)

DPlowgi DPj

gi DPupp

gi (15)

Qmingi DQjgi Q

0gi Q

maxgi (16)

Reserve constraints

Pminri Pjri P

maxri (17)

Qminri Qjri Q

maxri (18)

DPgij in the above equations is Tier 1 reserve and can ramp up

or down when required in the contingency state, which canbe positive and negative. P

jri in the above equations is Tier

2 reserve dispatched which is positive. Both Tier 1 and

Tier 2 reserve in a contingency state j must be within itslower and upper limits.

Load curtailment limits

0 CPjis CP

maxis (19)

Voltage limits

Vi min Vji Vi max (20)

Line flow constraints

Sjik

Sik max (21)Equations (10) and (11) can be represented as a generaloptimisation problem which includes the general objectivefunction

Minfj(xj) (22)

where xj DPj11, . . . , DPj

gi, . . . , Pj11, . . . P

jri, . . . ,

CP

j11, . . . , CP

jis, . . . subject to the general equality

constraints (12) and (13)

gj(xj) 0 (23)

General inequality constraints of (1421)

hj(xj) 0 (24)

The SQP algorithm [18], which combines the Newtonmethod with the quadratic programming (QP), has proved

to be a highly efficient method for solving the non-linearprogramming problem. Using the SQP algorithm, theoriginal non-linear programming problem is reduced to aQP problem

Min1

2dk T

Hkdk rf xjk

Tdk (25)

subject to

gj xjk

rgj x

jk

Tdk 0 (26)

hj xjk

rhj xjk

Tdk 0 (27)

where k is the iteration number and Hk is the a positivedefinite matrix approximating the Hessian matrix of theLagrange function Lj. Hk will be updated so as to convergeto the real Hessian matrix of Lj.

The Lagrange function Lj of (2224) is

Lj fj(xj) rgj(xj) hhj(xj) (28)

where rand hthe Lagrange multiplier vectors forgj(xj) andhj(xj), respectively.

The new search direction dk can be obtained by solving theQP problem. The new pointx

jk1 can be reset as follows until

it converges

xjk1 x

jk ak dk (29)

where ak is the optimal step length parameter along thesearch direction dk.

After solving the problem, the load curtailment, generation

re-dispatched and reserve dispatched can be determined foreach contingency state. These data are used to calculate thereliability indices in the following section. The randomfailures up to the second order are considered in this paper.

Therefore the optimisation technique can converge to aproper solution in most contingency states. However, itshould be noticed that the optimisation technique cannotconverge for some contingencies with the severe networkisolations. For those states that result in the violations ofphysical circuit laws related to generation limits andnetwork constraints, the corresponding loads andgenerators have to be cut and shut down, respectively, to

guarantee the system stable operation. In such special cases,load curtailments will be determined according to thephysical laws. The procedure of the proposed technique



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includes three main steps: to check the network isolations andto find the network islands, to determine load curtailmentsand generation and reserve re-dispatches for each islandusing the optimisation technique if it is possible, otherwiseto determine load curtailments using the circuit laws.

6 Reliability indices

Contingency enumeration and state selection techniques areused to determine contingency states and the associated stateparameters. Considering a power system with Nc independentcomponents, the state probability pj, departure rate Dj

and failure duration d j for contingency state j with exactly bfailed components can be determined using the followingequations

pj Yb

c1

Uc YNc

cb1

Ac (30)

Dj Xbc1

mcXNc

cb1

lc (31)

d j 8760

Dj(32)

The definition of these parameters can be found in [19].

Equation 30 illustrates that the probability of the state jequals to the multiplication of unavailabilities of failed

components multiplies the multiplication of availabilities ofworking components. The reliability indices of customers,Gencos and transmission network will be determined inthe following section using the proposed technique.

6.1 Customer reliability indices

Customer choices on price and reliability make non-uniformreliability possible in restructured power systems, wherecustomer satisfaction is the main concern of reliabilityevaluation. Customer reliability indices can be calculatedfrom the load curtailment for each contingency state

obtained from the load shedding technique.

The total customer interruption cost for state j(TCOSTj)and the total customer load curtailment for state j (LCj) canbe calculated as

TCOSTj XNi1

XNSis1

CCjis CPjis

(33)

LCj XNi1

XNSis1

CPjis

!(34)

The expected energy not supplied (EENSi) and expectedcustomer interruption cost (ECOSTi) for customer at bus i

can be calculated using the following equations

EENSi XNCj1

8760 pj XNSis1

CPjis

!(35)

ECOSTi XNCj1

8760 pj XNS

i

s1

(CCjis CPjis

!(36)

6.2 Genco and transmission networkreliability indices

In a restructured power system the ISO and power marketadministrator usually want to know the final obligation of aGenco to its energy supply and the associated reliability. Inthis paper, the expected energy not supplied (EENS) andthe expected interruption cost (ECOST) for each Gencoand the transmission network are introduced and calculated

using the proposed technique. In order to calculate thesetwo indices, the real power capacity not delivered from eachGenco for each state has to be determined in the loadshedding process.

The capacity not delivered CDjG for Genco G is thedifference between the generation dispatched in the normalstate and one in the state j. This index can be calculated as

CDjG

XN

i1 Xg[SGP0gi

Xg[SGgFjG

Pmaxgi

0

BBB@

1

CCCA(37)

If CDjG is less than or equal to zero, Genco G has enoughcapacity to meet its supply obligation. If CD

jG is greater

than zero Genco G has not enough capacity to meet itssupply obligation. The capacity shortage CS

jG can be

calculated as

CSjG max(CDjG, 0) (38)

In this case, the reserve from Gencos will be dispatched or/and the loads will be curtailed based on the costs to balance

the generation and load for system stable operation.

If there is load curtailment, the energy not supplied ENS jGcaused by Genco Gfor state jcan be calculated based on thetotal reserve dispatched TRj and the capacity not deliveredCD

jG using (39) and (40)

ENSjG dj

CSjGPNGG1 CS

jG

XNGG1

CDjGTRj

!(39)

TRj XN

i1XNR

ji

r1

Pjri (40)

wherePNG

G1 CDjGTR

j

is the total load curtailment



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caused by the generation system and CS jG=PNG

G1 CSjG is the

percentage of load curtailment contribution from Genco G.

The interruption cost COSTjG (k$) caused by Genco Gforstate j can be calculated as

COSTjG dj

CSjGPNGG1 CS

jG

PNGG1 CD

jGTR

j

LCj

TCOSTj (41)

wherePNG

G1 CDjGTR

j

=LCj is the percentagecontribution of the generation system to the total customerload curtailment.

Considering all the system states, expected customerenergy not supplied EENS (MWh/year) and interruptioncost ECOST (k$/year) for Genco G can be defined andevaluated based on the state departure rate Dj and stateprobabilitypj for state j using

EENSG XNCj1

Dj pj ENSjG

(42)

ECOSTG XNCj1

Dj pj COSTjG

(43)

In (42) and (43), multiplying Djwith pj we can obtain thefrequency (occ/year) of encountering the state j using thefrequency and duration technique [19]. Multiplying thefrequency of state j with the ENS

jG (MWh) and COST

jG

(k$), we can obtain the EENS and ECOST caused byGenco G for state j. Considering all the system states, wecan obtain the EENS (MWh/year) and ECOST (k$/year)caused by Genco G. The EENS and ECOST of thetransmission network can be calculated from the EENSand ECOST of the whole system minus the EENS andECOST of the generation system using the following

equations, respectively

EENST EENSS XNGG1

EENSG (44)

ECOSTT ECOSTS XNGG1

ECOSTG (45)

Considering all the states, the system indices EENSS and

ECOSTS can be calculated based on the total loadcurtailment LCj, total interruption cost TCOSTj and stateprobability pj for state j using the following two equations,

respectively

EENSS XNCj1

8760 pj LCj

(46)

ECOSTS XNCj1

8760 pj TCOSTj

(47)

The definitions and theoretical justifications of the reliabilityindices in this section can be found in [19].

7 The IEEE RTS studies

The proposed technique was used to analyse IEEE-RTS[20] shown in Fig. 3. The system is modified as arestructured power system with Poolco market. It isassumed that all generators at bus i belong to a single

Genco i. The system has 10 generation companies and 17bulk load point customers. The bidding curve for energy orreserve utilisation from each Genco is assumed to be aquadratic equation, which is illustrated in Table 4. Thecustomer damage functions for different customer sectorsare shown in Fig. 1. Parameter values for customer damagefunctions of different sectors and customer sectorallocations at load buses have also been supplemented in

Tables 5 and 6, respectively. The random failures up to thesecond order, which include the failure of a generating unitor a transmission line, and the failure of two units or two

Figure 3 IEEE RTS



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lines or the combination of one unit and one line, areconsidered in the illustrative example. For each system stateconsidered, load shedding and generation re-dispatch areevaluated by using the proposed OPF technique andcorresponding reliability parameters are also calculated. Theannual constant peak load is used in the analysis. After

considering all the contingency states, the annual reliabilityindexes can be calculated.

A Pentium IV 2.4 GHz personal computer is used in thesimulation studies. The computing time considering all thesystem states is about 2 h. The technique is proposed foroff line calculation. Therefore the computing time was notoptimised when we wrote the program. The computingtime can be significantly reduced to minutes when high-speed computer is used and the program is optimised.

The results obtained from the proposed technique are

compared with those from the existing methods in the casestudies. The conventional AC power flow is used tocalculate the reliability indices in the first two cases. Theproportional and priority load shedding techniques areapplied in Cases 1 and 2, respectively. In Case 2 it isassumed that load points 1 and 2 have higher servicepriority than other load points. The OPF-based reliabilitytechnique with the minimum total system load curtailmentis used in Case 3. In Case 4, the proposed OPF-basedreliability technique with optimal generation re-dispatchingand load shedding considering customer interruption costis used to calculate the load point reliability indices. InCase 4, it is supposed that the two 20 MW units at bus 1and two 20 MW units at bus 2 are served as reserveproviders (Tier 2 resource). In the normal state, these unitsare not economically dispatched by the system operator.Other generating units are economically dispatched in thenormal state (Tier 1 resource).

Table 1 shows the expected energy not supplied (MWh/year) of each bulk load point for the four cases. Theexpected interruption costs (k$/year) of each bulk loadpoint for the four Cases are shown in Table 2. Tables 1and 2 also show the total system EENS and ECOST forthe four cases.

It can be seen from Table 1 that the priority load shedding(Case2) has the largest total system EENS (39 687 MWh/

year) followed by Case 1 (39 128 MWh/year) withproportional load shedding and Case 4 (30 625 MWh/

year) with minimum total system cost. The total systemEENS calculated using the minimum load curtailmenttechnique (Case 3) is the minimum (30 417 MWh/year).

The difference between Cases 3 and 4 is only 0.68%. Thedifference between Case 3 and other two cases is over 28%.

This means that the proposed technique also has the lowerEENS.

It can be seen from Table 2 that the total system ECOSTs(k$/year) for Cases 4, 3, 1 and 2 are 97 101, 747 221,

367 290 and 364 650, respectively. The difference betweenCase 4 and other cases is over 73%.

The total system cost including generation costs, reserve costsand customer interruption costs (k$/year) for the four cases are875 890, 873 190, 1239 000 and 588 930, respectively.

Tables 1 and 2 also show that optimal load shedding andgeneration re-dispatching in reliability management usingthe proposed technique (Case 4) guarantee both the lowertotal system EENS and the minimum total system ECOST.

Although the system EENS for Case 3 is the lowest amongthe four cases, the corresponding total system ECOST is thehighest because the minimum load curtailment ignores thecustomer willingness to pay for its reliability. It can be seenfrom the results that the customer response to its reliabilityis a very important factor in reliability evaluation ofrestructured power system with Poolco market and thereforeshould be considered in the evaluation.

The results in Tables 1 and 2 are calculated based on2850 MW system load. However, the proposed technique

can be easily implemented to calculate reliability indices fordifferent operating conditions and assumptions. The resultsfor the high load level of 2992.5 MW has been calculated

Table 1 The EENS for four cases

Bus Case 1 Case 2 Case 3 Case 4

1 1482.7 0 0 327.16

2 1331.7 0 0 1047.3

3 2471.2 2701.1 6974 7329.7

4 1015.9 1110.3 3392.6 2227.1

5 974.78 1065.3 0.021 0.021

6 1867.3 2040.6 11576 5642.4

7 1716.1 1875.6 0 0

8 2347.7 2565.8 0 1269.9

9 2402.6 2625.8 191.7 3641.9

10 2677.1 2925.9 2634.3 3876.3

13 3638.2 3976.2 0 413.42

14 2663.5 2911.3 3760.4 1637.6

15 4352.2 4756.5 0 854.16

16 1372.9 1500.4 0 414.27

18 4571.7 4996.5 1888.2 1426.9

19 2484.9 2715.8 0 102.5

20 1757.3 1920.6 0 414.27

Total 39 128 39 687 30 417 30 625



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and compared with those for 2850 MW using the proposedtechnique (Case 4). In this case, the system EENS andECOST for 2992.5 MW are 116 530 MWh/year and322 940 k$/year, respectively, which increase 280.5 and232.6% comparing with those for 2850 MW. The totalexpected system cost is 861 890 k$/year, which increases46.35% comparing with those for 2850 MW. That meanssystem reliability and economical indices are very sensitiveto the system demand.

The proposed technique can also provide the reliability of

each Genco and the overall transmission system. Table 3shows the EENS (MWh/year) and the associated ECOST(k$/year) for each generation company. The EENS andECOST caused by the Gencos at buses 1, 2, 7 and 22 arezeros because these Gencos have more generating units andcapacity to meet the obligation. The EENS caused by theGencos at buses 18 and 21 are 12 133 and 11 988,respectively, which are 39.62 and 39.15% of the EENScaused by the generation system, respectively. The ECOSTcaused by the Gencos at buses 18 and 21 are 38 840 and38 539, respectively, which are 40 and 39.7% of theECOST caused by the generation system, respectively. Inthose companies, the high EENS and ECOST values isdue to the less generating units each with high capacity.

Therefore those Gencos should be responsible for customer

interruptions and should purchase additional reserve toincrease their reliabilities. It can be concluded that theGencos with less units each with high install capacity mayhave large contribution to system unreliability. The EENScaused by Gencos is 30 623.72 MWh/year, which isslightly less than the 30 625 MWh/year considering both

Table 2 The ECOST for four cases

BUS Case 1 Case 2 Case 3 Case 4

1 15 541 0 0 588.31

2 17 295 0 0 1883.2

3 9324.3 10 215 164 120 34 176

4 22 675 24 840 111 510 6575.5

5 23 629 25 884 0.1721 0.1721

6 30 547 31 725 284 840 24 349

7 13 421 14 703 0 0

8 19 614 21 486 0 2049.7

9 48 117 52 710 5258.3 5849.8

10 8606.1 9427.7 45 558 6315

13 34 319 37 595 0 743.44

14 6852.9 7506.8 94 060 6020.3

15 40 229 44 069 0 1536

16 16 777 18 378 0 744.97

18 26 854 29 418 41 876 5340.6

19 16 718 18 314 0 184.32

20 16 777 18 378 0 744.97

Total 36 7290 364 650 747 221 97 101

Table 3 The EENS and ECOST of Gencos for case 4

Bus EENSG ECOSTG

1 0 0

2 0 0

7 0 0

13 53.56 75.367

15 209.09 271

16 206.97 268.26

18 12 133 38 840

21 11 988 38 539

22 0 0

23 6033.1 19105Total 30623.72 97098.63



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generation and transmission. Therefore the EENS caused bythe transmission network is very small because of the stronggrid connection.

8 Conclusion

An optimisation problem to determine load shedding,generation re-dispatching and reserve utilisation forreliability evaluation of restructured power systems with thePoolco market model has been formulated in this paper.

The objective of the problem is to minimise the totalsystem cost including generation costs, reserve utilisationcost and customer interruption costs. Customer reliabilityrequirements in terms of interruption costs are implicitlyincorporated in the decision process when a systemdeficiency occurs. The OPFtechnique based on SQP hasbeen used to solve the problem. Reliability indices forGencos, bulk load point customers and overall system have

been introduced for the new market environment. Differentexisting techniques have been compared with the proposedtechnique. The proposed technique can be used to evaluatethe reliability of both conventional and restructured powersystems. The modified IEEE RTS has been analysed toillustrate the technique. The results show that the proposedtechnique can provide useful information for reliabilitymanagement of restructured power systems. The evaluatedreliability indices can be used in power markets todetermine the penalty payments of each Genco.

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Bibliography on the application of probability methods in

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10 Appendix

Table 4 Bidding parameters for energy and reserve utilisation of RTS

generating units

Unit no. Bus no. Installed capacity (MW) Bidding functioncoefficients

a ($/MWh) b ($/MW2 h)

11 1 20 103.6 0.4

12 1 20 103.6 0.4

13 1 76 15.30 0.5

14 1 76 15.30 0.5

21 2 20 103.6 0.4

22 2 20 103.6 0.4

23 2 76 15.30 0.5

24 2 76 15.30 0.5

31 3 100 52.8 0.3

32 3 100 52.8 0.3

33 3 100 52.8 0.3

131 13 197 50.6 0.4

132 13 197 50.6 0.4

133 13 197 50.6 0.4

151 15 12 63.3 0.7

152 15 12 63.3 0.7

153 15 12 63.3 0.7

154 15 12 63.3 0.7

155 15 12 63.3 0.7

156 15 155 12.44 0.3

161 16 155 12.44 0.3

181 18 400 6.3 0.1

211 21 400 6.3 0.1

221 22 50 0.5 0.1

222 22 50 0.5 0.1

223 22 50 0.5 0.1

224 22 50 0.5 0.1

225 22 50 0.5 0.1

226 22 50 0.5 0.1

231 23 155 12.44 0.1

232 23 155 12.44 0.1233 23 350 12.10 0.1



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Table 6 Customer sector allocations at load buses for the RTS

Bus Peak load (MW) Large users (MW) Agriculture (MW) Residential (MW) Industrial (MW) Commercial (MW)

1 108 23.83 0 34.03 36.94 13.20

2 97 32.26 0 50.05 0 14.69

3 180 80.0 6.33 52.50 33.25 7.92

4 74 20.22 0 34.52 0 19.26

5 71 0 0 22.83 28.10 20.07

6 136 38.1 8.38 49.70 29.34 10.48

7 125 25 18.24 38.44 31.92 11.4

8 171 87.67 0 55 11.67 16.66

9 175 91.13 19.54 23.46 0 40.87

10 195 116.92 8.77 41.54 20.46 7.31

13 265 74.84 17.91 82.15 60.95 29.15

14 194 84.55 0 62.89 40.74 5.82

15 317 228.0 0 54.84 0 34.16

16 100 59.85 0 25.90 0 14.25

18 333 207 0 63.27 39.96 22.77

19 181 111.2 0 55.60 0 14.20

20 128 59.85 0 53.9 0 14.25

Table 5 Parameter values for customer damage functions of different sectors

Failure duration (min) Interruption cost ($/kWh)

Large users Industrial Commercial Agricultural Residential

1 0.073 0.46 0.129 0.027 0.0004

20 0.111 1.332 1.014 0.155 0.044

60 0.163 2.99 2.951 0.295 0.243

240 0.291 8.899 10.992 1.027 2.235

480 0.604 18.156 28.02 2.134 6.778



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