Kleinberg - ISCAS 09

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Service Restoration of Power Distribution Systems

Incorporating Load Curtailment

Michael Kleinberg Karen Miu Hsiao-Dong Chiang

Electrical and Computer Engineering Department School of Electrical and Computer EngineeringDrexel University Cornell University

Philadelphia, PA U.S.A Ithaca, NY, U.S.A

Abstract Power distribution systems are nowintegrating Smart Grid technologies allowing for widespreadimplementation of demand response and direct load control.This paper will present a service restoration problemformulation and solution algorithm for power distributionsystems which incorporates load curtailment. A rankingbased heuristic search algorithm is proposed whichprioritizes the multi-objective service restoration problemand returns a new system configuration and load curtailmentscheme to restore power to out-of-service customersfollowing fault isolation. A numerical example is presentedto illustrate the mechanics of the proposed algorithm.

I. INTRODUCTIONIn modern power distribution systems, increasing

attention and investment is being focused on Smart Gridtechnologies allowing generation, network, and load tointeract in real time using existing and new communicationinfrastructures [1]. These technologies facilitate new controland automation applications. Among these applications isbroader implementation of demand response and direct loadcontrol via smart metering and intelligent appliances [2].

Power distribution systems serve as the direct link toelectric power for the majority of consumers. These systemsare operated as unbalanced networks where loading levelsmay vary across each phase. In addition, they are generallyoperated in a radial configuration for ease of protection andsystem design. Normally closed sectionalizing switches (ss)and normally open tie switches (ts) are available toreconfigure distribution networks.

Meanwhile, the availability of demand side managementis increasing. Direct load control allows utility operators toautomatically curtail customer power demands bycontrolling air conditioning and water heater temperature setpoints and other intelligent appliances. Customers

participating in load curtailment (LC) may be compensatedthrough discounted electricity rates while still beingguaranteed a base load condition. A similar concept isdemand response which allows customers to automaticallycontrol power consumption based on market signals.

One application which can benefit from the Smart Gridparadigm is service restoration. Service restoration istypically formulated to maximize out-of-service load

restored after fault isolation e.g. [3, 4]. This goal is achievedthrough network reconfiguration and determining the statusof each switch in the network. Work has extendedformulations to include network operating constraints andpriority customer considerations e.g. [5]. This paper willfurther extend the service restoration problem formulation toinclude load curtailment during outage conditions.

By curtailing the load of participating customers, systems

can increase capacity to restore priority and non-priorityload. The addition of LC significantly increases problemcomplexity. Yet, this approach has the human centricobjective of ensuring a maximum number of customers willreceive at least a minimum amount of service during anoutage. Specifically, this paper will present:

a service restoration problem formulation incorporatingload curtailment

selection indices used for ranking switches and loads a heuristic, ranking based, service restoration algorithma numerical example to illustrate the algorithm

II. PROBLEM FORMULATIONThe service restoration problem with priority customers

is formulated as a constrained non-differentiable multi-objective optimization problem. The objectives consideredare: (1) to maximize priority load restored, (2) to maximizetotal load restored, (3) to minimize number of switchingoperations, and (4) to minimize the amount of LC.

A feasible solution satisfies electrical constraintsexpressed as the three-phase power flow equations. Currentflows, feeder loading and voltage magnitudes must lie withinacceptable operating ranges. Also, switch voltagedifferentials, load curtailment limits and a radial networkstructure must be satisfied after reconfiguration. In summary:

,maxkpc

k

L kg G

k Nu U

R I (1)

,max

k

k

L kg G

k Nu U

RR

I

(2)

minops

n (3)

,

min

LC

LC k

k Nku U

I

(4)

This work was supported by the US Office of Naval Research under grantno. ONR #N0014-04-1-0404.


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subject to:

( , , ) 0k k

f V g u = (5)

max

p

k kI I k N (6)

2 2 max 2( )

f f f P Q S f F + (7)

min max, , ,

p

k k kV V V k N p a b c (8)

( ) max closej k l k

V V V V j g = (9)

, max

, ,, , ,

p p

LC k LC k LC I I k N p a b c = (10)

k Rg G (11)

where:

kg : post-fault network configuration,

close

kg set ofts closed

RG : set of possible radial configurations

ku : load curtailment scheme

U : set of possible load curtailments

pcN : set of restored priority customers

RN : set of restored buses

LCN : set of buses participating in load curtailment

,L kI : total load current at bus k

opsn : number of switch operations to arrive at

kg

,LC k I : total load curtailment current at bus k

( , ) :,kk

f V g u three-phase power flow equations

V : node voltage vectorp

kV : voltage at bus k, phasep

p

kI : current flow entering bus k, phasep

f f f S P jQ= + : total apparent power entering feederf

max

f

S : max capacity of feederfor its supplying transformer,

whichever is least

,

p

L C k I : load to be curtailed by customers at bus k, phasep

, max

,

p

L C k I : maximum available load curtailment of customers

at bus k, phase p

The service restoration problem is NP-complete with

discrete and continuous variables related to switch status and

network voltage and current flows, respectively. The

addition of LC increases the complexity of the problem. An

upper bound on the search space can be calculated as the set

of all possible post-fault network configurations along with

all possible LC schemes. Letting ntbe the number of ts, ns

be the number of ss, and nLC be the number of candidatebuses for LC with two loading levels each, the upper bound

on dimension of the search space is:

( )1

21

LCt sn

k

tn n n

k k=

(12)

For example, an actual 416 bus system is presented in[5]. The system contains 53 ss, 30 ts, and 208 loads. If allloads are candidates for curtailment, the search spaceincreases by a factor of 2

208. Thus, to facilitate computation

of a restoration plan, a ranking based heuristic search methodis proposed.

The solution algorithm solves the multi-objectiveoptimization problem by prioritizing the objectives in theorder presented. A pareto optimal solution is then foundwhich favors the objectives in this order. That is, noimprovement in a lower priority objective is accepted if itdegrades the optimality of higher priority objective. Thismethod is chosen over other multi-objective optimization

solutions such as a weighted sum objective function orevolutionary algorithm [6].

Total load restored is given priority over LC to improvereliability indices such as system average interruptionduration index (SAIDI). The minimization of switchingoperations is prioritized based on the fact that networkswitches are inherently more expensive than LC hardwareand that repeated operation reduces lifetime and increasesrequired maintenance. To assist the heuristic search,analytically-based search indices are proposed.

III. SWITCH SELECTION INDICIESIn this work, four switch indices will be utilized. A new

index,avail

LCI , is proposed which quantifies capacity releaseavailable through LC. The three remaining indices areadapted from [5]. The indices allow the restoration algorithmto systematically distinguish between network switchesbased on graph theoretical arguments and analyticallydetermined criteria. In summary, the indices are:

availLC

I : the LC available to each ts

MI : the spare capacity of each ts pathZ : the electrical distance of each ts to other buses ssI : the amount of load each ss can transfer to a ts

The impact of LC will be to increase the spare capacity ofeach ts. As such, the indexIMis reviewed.

Each feeder has a maximum amount of current it cansustain before a branch becomes overloaded or a protectiondevice operates. For each ts, the spare capacity is as follows:

( )max,

minp

k kk C pu

MI I I

= (13)

where Cu is the set of upstream buses on the path from the

primary side of each ts to the substation. In addition to IM,

the critical branch at which this value is found is stored as bts.

The LC available for a ts,avail

LCI , is defined as the maximum

amount of load which may be curtailed from upstream busesto increaseIM. It is computed as the minimum over all phases

of the total load which may be curtailed from buses locatedon the path downstream from branch, bts, to theprimary/substation side of each ts:

, , max

,

LC

p avail p

LC LC k

btsk C N

I I

= (16)

,min avail p avail

LC LC p

I I= (17)


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fault

ss1 ts2

ss2 ss3 ss4

ss5ts1ILC,1

ILC,2 ILC,3

ILC,4 ILC,5

LegendSectionalizing Switch

Tie Switch

ts3

Substation

Load

Controllable Load

Transformer

2ts

b

1tsb

Figure 1: Sample 22 Bus Distribution System with Available LC

wheretsb

C is the set of all buses downstream ofbts to the ts.

In this work, each candidate for load curtailment has only

one level of load control,max

, ,

LC k LC k I I k = .

For illustration, a single phase sample distribution systemis shown in Figure 1. The system consists of 22 buseslocated along a main feeder with three laterals. A faultoccurs at the denoted location and is assumed to be isolated.The out-of-service area is located downstream of the fault.Assuming critical branches for each candidate ts as shown,

for ts1:avail

LCI =ILC,4 +ILC,5 and for ts2 :

avail

LCI =ILC,2 +ILC,3.

IV. SERVICE RESTORATION ALGORITHMLoad curtailment should be considered in cases where

full restoration of out-of-service loads, Iout, is not attainableand/or to possibly reduce the number of nops required, e.g.from [5]. Therefore, the service restoration algorithm in [5]

is run first and nops stored. If multiple out-of service areasexist the following algorithm works sequentially on each outof service area in decreasing order of priority load served.

Step 1: Build candidate tie and sectionalizing switch list

Step 2: Build candidate load curtailment list

Step 3: Select and operate one candidate ts and implement LCto attempt full restoration of out-of-service area.

Step 4: Select and operate candidate switch pairs orderedby transferable load,ISS, and implementLC.

Step 5: Determine which non-priority and then priorityloads not to restore by opening ss and implementingLC.

Step 6: Return new system configuration and loadcurtailment scheme

The details of each step are now presented.

Step 1: Candidate Switch Lists

Candidate ts and ss are identified as:

ts from energized feeders that can connect directlyinto the out-of-service area

ss located in the out-of service areaThe spare capacity,IM, associated with each ts is determined.

Step 2: Candidate LC List

Candidates for LC are participating customers on thepath upstream from each candidate ts to the correspondingcritical branch. Participating customers in the out-of-serviceareas are not considered. With each candidate ts is stored

avail

LCI and a LC options vector

,

opt

LC ts

LCnI where LCn is the

total number of participating customers in the network.

,

opt

LC tsI contains the values of each candidate LC available to a

given ts in its corresponding row, 0 if the LC customer is nota candidate. If LC candidates are associated with multiple ts,

avail

LCI and

,

opt

LC tsI are updated as LC is enacted for each ts.

For each ts, different combinations of customers can becurtailed to meet LC requirements. When required, theminimum amount of these LC options which is greater thanthe requirement, e.g. capacity release needed, is selected:

,

,

min

. .

T opt

LC LC ts

T opt

LC ts

cI c I

s t c I required

=

(18)

where LCn

c is a vector composed of 1s and 0s used to

compute a partial sum of,

opt

LC tsI .

Step 3: Select One ts

This step attempts to operate one ts and implementavailable LCto restore the entire out-of-service area. If thisis not possible, the candidate ts with the largestIM is chosenand the algorithm proceeds to Step 4. Specifically, an initialts is selected by the following steps:

S3.1: Create candidate ts list withavail

M LC outI I I + . Order

ts by increasingLC M out

I I I , if empty go to S3.5.

S3.2: Select the next ts on the list, set as ts1. If none, goto S3.5. Operate ts1,run power flow and check

constraints.

If all constraints are satisfied, go to Step 6. If a voltage violation occurs and other ts exist,

order the ts byZpath and select ts according to [5].

If an overload violation exists, record the amountas Ovrld. This represents the minimum amount

of LC required for one ts operation.

S3.3: Determine and implementLC

I Ovrld according

to (18) and continue, else go to S3.5

S3.4: Run power flow, check constraints, and recordLC

I .

If all constraints are satisfied, go to Step 6. Else undo ts1and go to S3.2

S3.5: Select ts with largestIMand set as ts1. Run power

flow, store Ovrldand go to Step 4.

Step 4: Select Candidate Switch Pairs

In this step, switch pairs and load curtailment aresequentially selected to alleviate the overload of ts1. For agiven ts, candidate ss are those lying in the out-of-servicearea and in the path from the secondary side of the ts towardsthe substation. Switches are not paired together if theiroperation would create a constraint violation.


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The algorithm first attempts to select one additional

switch pair (tsA, ssA) withA

AM

ts

ssI I and

Ass

I Ovrld to

alleviate the overload of ts1 similar to [5]. If this cannot beachieved and only one additional switch pair is to be utilized,then load curtailment is necessary (but not sufficient):

Determine (minimal) LC for ts1 and the operation of(tsA, ssA); i.e.

1

A

ts

LC ssI Ovrld I .

Determine (minimal) LC for tsA to allow for a new (tsA,ssnew) pair with

newssI Orlvd ; i.e. A A

new

ts ts

LC M ssI I I .

Determine combination of LC on tsA and ts1 wouldallow for a new (tsA, ssnew) pair;

A A

new

ts ts

LC M ssI I I

and 1new

ts

LC ssI Ovrld I .

Select the option with minimal LC from above andcheck constraints.

If the entire out-of-service area cannot be served, anadditional switch pair is needed. Select the (ts, ss) includingLC yielding the largest Iss transferred away from ts1 andrepeat the process. If no additional pairs are available,proceed to Step 5.

Step 5: Determine Loads Not to be Restored

If the entire out-of-service area cannot be restoredthrough Steps 3 and 4 without constraint violations, thennon-priority and, if necessary, priority loads must bedropped. This is accomplished by opening ss according tothe method described in [5].

Step 6: Return Feasible Control Scheme

To implement the service restoration plan, the loadcurtailment scheme should be executed before any switching

operations are performed. Then, ss identified in Step 4 andStep 5 are opened.Next,all (tsA, ssA) pairs are operated andts1is closed.

V. NUMERICAL ILLUSTRATIONThe system shown in Figure 1 is used to demonstrate the

service restoration algorithm. An initial power flow is run to

calculate the post-outage system states. The candidate

switches are identified as ts1, ts2, ss2, ss3 and ss4. The spare

capacity, critical branch, and available LC of each tie switch

are calculated., 1

opt

LCtsI and

, 2

opt

LCtsI for ts1 and ts2 are:

[ ] [ ]T T

, ,4 ,510 0 0 0 0 0 1 7

opt

LC ts LC LC I I I = =

[ ] [ ]T T

, ,2 ,320 0 0 0 1 3 0 0

opt

LC ts LC LC I I I = =

Here, 80 different settings of load curtailment and radialnetwork reconfiguration are possible: 8 settings correspondto nops = 1, operating one ts; 48 correspond to nops = 3,operating a tie switch and a switch pair; 24 settingscorrespond to nops = 2, operating a tie switch and asectionalizing switch, where not all load can be restored.

Table 1. Numerical values for switch indices corresponding to Figure 1.

ss2 ss3 ss4

IMavail

LCI Iss2 Iss3 Iss4

ts1 8 8 17 10 5

ts2 7 4 0 7 12

For this example, Iout = 17 and additional index valuesmay be seen in Table 1. By utilizing the switch selectionindices, the proposed analytically-based heuristic can quicklyidentify potentially infeasible options and significantlyreduce the effective search space. With respect to the

numerical example, in Step 3, utilizingavail

M LC I I+ the

algorithm quickly avoids all 8 settings corresponding to

nops = 1. In Step 4, the use of ,M ssI I andavail

LCI eliminates 36

settings (or ) of the possible nops = 3 settings and orders theremaining 12 options with respect to the minimal LC. Then,control options are selected sequentially in increasing orderof LC if needed. For this case, the algorithm determines thatthe entire out-of-service area can be restored by operating ts1,

(ts2, ss3) andILC,3.

VI. CONCLUSIONSThe problem of service restoration in power distribution

systems including load curtailment has been addressed. Withthe inclusion of LC, it is expected that previous servicerestoration options can be improved with respect to either thetotal load restored and/or the number of required switchoperations. A multi-objective problem formulation has beenpresented together with switch selection indices based ongraph theory and power flow analysis. The proposed methodrelies on ranking the switch indices to eliminate infeasiblesolutions while identifying sets of non-inferior optimalsolutions. A sample implementation was presented todemonstrate the mechanics of the proposed method. It isanticipated that the work can both justify the implementationof Smart Grid technologies in distribution systems andbenefit from the same technological advancements.

REFERENCES

[1] Garrity, T.F.; Getting SmartIEEE Power and Energy MagazineVol. 6, Issue 2, March-April 2008, pp. 38-45

[2] Heffner, G.C. Goldman, C.A., Moezzi, M.M., InnovativeApproaches to Verifying Demand Response of Water Heater LoadLontrol, IEEE Trans. Power Delivery.,Vol.21, Issue 1, pp. 388-397 Jan. 2006

[3] Morelato, A.L., Monticelli, A., Heuristic Search Approach ToDistribution System Restoration,IEEE Trans. on Power Delivery,Vol. 4, No. 4, Oct. 1989, pp. 2235-2241.

[4] Shirmohammadi, D., Service Restoration in Distribution Networksvia Network Reconfiguration,IEEE Trans. on Power Delivery, Vol.7, No. 2, April 1994, pp. 952-958.

[5] Miu, K. N., Chiang, H.-D., Yuan, B.Bentao, Darling, G., Fastservice restoration for large-scale distribution systems with prioritycustomers and constraints,IEEE Trans. Power Syst., Vol. 13, pp.789-795, Aug. 1998.

[6] Bui, Lam Thu; Alam, Sameer,Multi-Objective Optimization inComputational Intelligence: Theory and Practice, InformationScience Reference, 2008

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Kleinberg - ISCAS 09