Load-shedding probabilities with hybrid renewable power generationand energy storage

Huan Xu, Ufuk Topcu, Steven H. Low, Christopher R. Clarke, and K. Mani Chandy

Abstract— The integration of renewable energy resources,such as solar and wind power, into the electric grid presentschallengs partly due to the intermittency in the power output.These difficulties can be alleviated by effectively utilizingenergy storage. We consider, as a case study, the integrationof renewable resources into the electric power generationportfolio of an island off the coast of Southern California,Santa Catalina Island, and investigate the feasibility of replacingdiesel generation entirely with solar photovoltaics (PV) andwind turbines, supplemented with energy storage. We use asimple storage model alongside a combination of renewablesand varying load-shedding characterizations to determine theappropriate area of PV cells, number of wind turbines, andenergy storage capacity needed to stay below a certain thresholdprobability for load-shedding over a pre-specified period of timeand long-term expected fraction of time at load-shedding.

I. INTRODUCTION

Santa Catalina Island is 26 miles off the coast of SouthernCalifornia, USA. It is 21 miles long and 76 square miles inarea with 54 miles of coastline. The island’s population inthe 2000 census was 3,696 with 90 percent living in the twolargest cities Avalon and Two Harbors [1]. Catalina is servedby three 12kV distribution circuits that are separate fromthe grid on the mainland of California with a peak demandin 2008 of 5.7MW [2]. Currently, electricity is generatedfrom a central diesel plant. It is desirable to reduce diesel-based generation for several reasons (both environmentaland economical): to reduce the emissions from diesel powerplants and to eliminate the cost of transportation of dieselfuel to the island. Furthermore, the diesel generators aremost efficient when running at full capacity and are thereforeinefficient for Santa Catalina Island, where the daily demandpeaks for a relatively short period of time. This paper aimsto asses the feasibility of replacing diesel generation withgeneration from intermittent renewable resources (e.g., solar,wave, and wind power generation).

Fluctuation in both the generation and demand as well asphysical distance between generation sites and users makethe integration of renewable sources into the electricity griddifficult [3], [4]. The potential benefits, however, can beutilized by supplementing the system with energy storage[5], [6]. Different techniques have been used in determin-ing the correct sizing for storage needed in the presenceof intermittent renewable sources as a function of storageefficiency, conventional fossil fuel generation and life cycle

H. Xu, U. Topcu, S. H. Low, and K. M. Chandy are with CaliforniaInstitute of Technology. E-mails addresses: mumu@caltech.edu,utopcu@cds.caltech.edu, slow@caltech.edu,mani@cs.caltech.edu. C. R. Clarke is with Southern CaliforniaEdison. E-mail address: Christopher.R.Clarke@sce.com.

LOADS!d(t)!

PV CELLS!s(t)!

WIND TURBINES!

w(t)!

DIESEL GENERATION!

f(t)!

STORAGE!b(t)!

r(t)!

Fig. 1. A schematic for a power network with renewable sources. Demandloads are met by base diesel generation, supplemented with PV cells, windturbines, and energy storage.

costs [7], [8], [9]. Reference [10] uses a probabilistic modelto predict the feasibility of increased renewable penetrationby investigating different types, sizes, and time scales ofstorage technologies. References [11] and [12] examine theintegration of wind power into isolated systems, similar toCatalina Island.

In this paper, we incorporate simple storage dynamics intoa load-shedding model to understand the effects of intermit-tency in generation and/or demand on the characteristics ofthe electricity network. We consider a scenario where theelectricity is supplied by a combination of solar photovoltaic(PV) cells, wind turbines, and base diesel generation. Withthese resources, we address the following design questions:Given information about typical demand and generationprofiles over certain periods of time, what are the appropriateamounts of each resource and energy storage capacity neededto ensure a certain level of reliable operation [13], e.g.,that the probability of load-shedding is smaller than a pre-specified threshold and the long-term expected fraction oftime spent at load-shedding is sufficiently small?

II. SET UP AND PROBLEM FORMULATION

A. Simple storage model

Consider the configuration in Figure 1. In this figure andhereafter, d(t) denotes the total amount of all loads over thetime period [t, t +1]. Additionally, s(t),w(t), and f (t) denotethe amount of energy generation by a unit area of PV cells,a wind turbine, and diesel (fossil fuel) based generators over[t, t +1]. The amount of energy storage at time t is denotedby b(t) and is considered to follow the first order difference

equation

b(t +1) = b(t)+g(t)!d(t), for t = 0, . . . ,T !1 (1)

with the initial amount b(0) of storage and the length T ofthe time horizon. Total generation g(t) is defined as

g(t) := !1s(t)+ !2w(t)+ f (t),

where !1 and !2 are the numbers of unit area of PV cells andwind turbines in the system. Additionally, the following non-negativity and capacity constraints on the amount of storageare imposed

0" b(t)" B, for t = 1, . . . ,T,

where B denotes the storage capacity.At time t + 1, we call the case in which the difference

between demand and total generation is strictly less than theavailable stored energy, i.e.,

b(t) <!g(t)+d(t), (2)

as load-shedding.

B. Problem descriptionIn general, the load d and renewable generation s and

w feature intermittency and fluctuations [10]. For example,consider the daily d, s, and w profiles shown in Figure2, which are also used in the analysis in the subsequentsections. Each of the figures show d, s, and w at eachhour for 31 days in the month December 2009 obtained atgeographical locations with characteristics similar to those ofSanta Catalina Island (discussed at the end of this section).The load and solar graphs follow a profile with considerablefluctuations, while the wind profile is even less predictable.

In the face of stochasticity in both loads and generation,we are interested in quantifying the probability of havingload-shedding and the effects of energy storage on thisprobability in configurations with different levels of inter-mittent renewable and diesel-based generation. Note that,in general, the likelihood of load-shedding is a functionof the generation and demand profiles and, therefore, it isa function of !1, !2, and B (potentially along with otherfactors not modeled here). The total area of PV cells, numberof wind turbines, and capacity of storage are design-timedecisions to be made before investing in the deploymentof large renewable energy generation and accompanyingenergy storage. In the following, we consider the questionof determining appropriate values for !1, !2, and B that leadto an acceptable probability of load-shedding.

C. Description of data setIn this study, we use a data set including load (d), solar

generation (s), and wind generation (w) obtained in thefollowing way.

• Load profile: Only peak demand values are available forSanta Catalina Island1. Therefore, load profiles are gen-erated from a proxy distribution circuit statistically sim-ilar to that for the island, e.g., line distances, customer

1These peak values are obtained through personal communication withresearchers from Southern California Edison

0 5 10 15 200

0.5

1

1.5

2

2.5

3

Time of day (hours)

Load

(MW

)

0 5 10 15 200

0.5

1

1.5

2

2.5

3

Time of day (hours)

Per U

nit S

olar

Gen

erat

ion

(kW

)

0 5 10 15 200

200

400

600

800

1000

Time of day (hours)

Per T

urbi

ne W

ind

Gen

erat

ion

(kW

)

Fig. 2. Daily load profiles on Santa Catalina Island (top), solar generationprofiles in Long Beach, CA (middle), and wind generation profiles on anisland near Santa Barbara, CA (bottom), for the month of December.

count, and climate. The proxy circuit peak is scaled tomatch the peak data for each of the three distributioncircuits on the island. Hourly averages calculated fortimes where more than one data point was available,while data for missing hours are interpolated.

• Solar generation profile: Hourly data are obtainedusing the Solar Advisor Model (a detailed per-formance model for solar technologies - availableat https://www.nrel.gov/analysis/sam/) withLong Beach, California as a geographic location andeach PV cell unit measuring 35 m2.

• Wind generation profile: Hourly data isobtained from the National Renewable EnergyLaboratory (NREL) website available athttp://wind.nrel.gov/Web nrel/ for a stationlocated off an island near Santa Barbara, California.

III. PROBABILITY OF LOAD-SHEDDING

In this section, we investigate the effects of varyingamounts of renewable generation (i.e., !1 and !2) and storagecapacity (B) on the probability of load-shedding over the timehorizon t = 1, . . . ,T defined as

PLS := Prob[b(t)<!g(t)+d(t) for some t # {0,1, . . . ,T!1}].

A. Estimating the probability of load-shedding

In order to estimate the load-shedding probability, wequantize the allowable range [0,B] of energy storage using nequally spaced points "1 = B > "2 > · · · > "n!1 > "n = 0 and

!1= B"

t"!n+1"

!n= 0"

b(t) = !j"

mean of !j+g(t)-d(t)"

t+1"

empirical distribution of !j+g(t)-d(t)"

M1,j(t)"

M2,j(t)"

Mn+1,j(t)"

Mn,j(t)"

Fig. 3. Construction of matrix Mi, j(t) is based on the empirical distributionfor g(t)!d(t).

assigning "n+1 as the state corresponding to load-shedding.For i, j = 1, . . . ,n+1 and time t = 0, . . . ,T !1, define

Mi, j(t) := Prob[b(t +1) = "i | b(t) = " j]

as the probability of having a storage level of "i at timet +1 given that the storage level B is " j at time t. Using thedata shown in Figure 2, we empirically estimate Mi, j(t) fort = 0, . . . ,T !1 as follows. Define the intervals

I1 := ["1 = B,!),Ii := ["i,"i!1), for i = 2, . . . ,n,In+1 := (!!,"n = 0).

Then, for t = 0, . . . ,T ! 1, j = 1, . . . ,n, and i = 1, . . . ,n + 1,an empirical estimate of Mi, j(t) is the fraction of points inthe set of data points " j +g(t)!d(t) that fall in Ii with theconvention Mn+1,n+1(t) = 1 and Mi,n+1(t) = 0 for i = 1, . . . ,n.Note that "n+1 is an absorbing state of the Markov chaingoverned by the matrix M(t) [14].

Define

M := M(T !1)M(T !2) . . .M(0).

Then, the probability

Prob[b(t) <!g(t)+d(t)for some t # {0, . . . ,T !1} | b(0) = " j]

of transitioning to the load-shedding state "n+1 at some timet = 1, . . . ,T from b(0) = " j is estimated by Mn+1, j.

B. ResultsIn this section, we discuss the results obtained for T = 24

hours and storage capacity B = 20 MWh using the hourlyload, solar generation, and wind generation profiles in Figure2. A quantization increment of " j!1 ! " j = 100 kWh fori = 2, . . . ,n over [0,B] is used.

Figure 4 shows the load-shedding probabilities versusinitial storage levels for different levels of solar generationwith !2 = 1 (i.e., a single wind turbine), and without any base

0 2 4 6 8 10 12 14 16 18 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Initial Storage Level (MWh)

Prob

abili

ty o

f Loa

d−Sh

eddi

ng

! "1 = 2x103

! "1 = 3x103! "1 = 7x103

! "1 = 1x106

Fig. 4. Probability of load-shedding versus amount of initial energy storagewith B = 20 MWh, T = 24 hours, and !2 = 1 (i.e., a single wind turbine),for different levels of solar generation.

diesel generation (i.e., w(t) = 0 and f (t) = 0 for t = [0,T ]).Note the transition from high PLS at low levels of initialenergy storage to low PLS at higher levels of initial energystorage. There are at least two trends in Figure 4 that are ofinterest.

• As the level of solar generation increases, (i.e., !1increases), the amount of initial storage at which thetransition from high PLS to low PLS decreases andapproaches a constant value which can be attributed tothe mismatch between the generation and demand at thebeginning of the day due to lack of any solar generation.

• As !1 increases, the transition from high PLS to low PLSsharpens. This can be attributed to the higher variabilityin wind generation compared to that in solar generation(see Figure 2).

Figure 5 shows the load-shedding probabilities versusinitial amount of storage with a fixed level of solar generation!1 = 2 ·103 for different number !2 of wind turbines. Similarto the trend with increasing !1 in Figure 4, the level of initialstorage at which the transition from high PLS to low PLSoccurs decreases with increasing !2. Unlike the trends forincreasing !1, PLS can be made less than one even for zeroinitial storage with sufficiently large !2 (e.g., !2 = 5 in Figure5).

In the results shown in Figures 4 and 5, the base dieselgeneration is considered to be zero. We now include basegeneration and investigate the probability of load-sheddingas a function of the level of penetration of renewable energyresources. Assuming that the base generation PB is constantover time, we define the penetration ratio # as

# :=PR

PR +PB,

where PR is the largest value of renewable generation over apre-specified time window of interest. For ease of presenta-tion, we consider that solar generation is the only renewableresource, and take PR as !1 times the largest value of solargeneration in the data set shown in Figure 2 (middle). Then,# is a function of !1 and PB. Figure 5 shows PLS versus initial

0 2 4 6 8 10 12 14 16 18 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Initial Storage Level (MWh)

Prob

abili

ty o

f Loa

d−Sh

eddi

ng ! "2 = 0

! "2= 5

! "2= 2

! "2= 1

Fig. 5. Probability of load-shedding versus amount of initial energy storagewith B = 20 MWh, T = 24 hours, and fixed !1 = 2 ·103 for different valuesof !2 (i.e., number of wind turbines).

0 2 4 6 8 10 12 14 16 18 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Initial Storage Level (MWh)

Prob

abili

ty o

f Loa

d−Sh

eddi

ng

! " = .13! " = .15 ! " = 0.2

! " = .65

! " = .75

! " = 0.8

! " = .95

! " = .97

! " = 1.0

Fig. 6. The probability of load-shedding versus initial amount of storagewith B = 20 MWh and !2 = 0 for different penetration levels # . !1 = 102

(solid curves), !1 = 103 (dashed curves), !1 = 104 (solid curves with squaremarkers).

amount of storage with !2 = 0 for different penetration levels# .

IV. LONG-TERM DURATION OF LOAD-SHEDDING ANDENERGY NOT-SERVED

The simple load-shedding characterization discussed inSection III does not distinguish between levels of load-shedding by using a single lumped state "n+1 correspondingto the condition b(t) < !g(t) + d(t). In this section, wemodify the load-shedding characterization to allow multiplelevels of deficit d(t) ! g(t) ! b(t). This characterizationallows us to estimate the (long-term) expected time spentat a load-shedding state and expected amount of energy not-served. To this end, let "1 = B > "2 > · · · > "n!1 > "n = 0 beequally spaced points over [0,B] as before and "n+1, . . . ,"n+mbe such that "n = 0 > "n+1 > · · · > "n+m. Consider that theamount of energy storage evolves by following the difference

B!

t!

b3(t)!

0!

t+1!

b2(t)!

b1(t)!

b1(t)+g(t)-d(t)!

b1(t+1)!

b3(t)+g(t)!

b2(t+1)=b2(t)+g(t)-d(t)!

Fig. 7. Transitions governed by (3) for three different values of storage attime t.

equations

b(t +1) = b(t)+g(t)!d(t) if b(t)$ 0b(t +1) = b(t)+g(t) if b(t) < 0 (3)

with saturation at "n+m and B. Figure 7 depicts the transitionsfor three different values of “storage” at time t.

Note that, in (3), a negative value for b(t) is used toquantify the severity of load-shedding and "n+1, . . . ,"n+m < 0correspond to different levels of load-shedding. With poten-tial abuse of terminology, we do not distinguish betweenb(t) $ 0 and b(t) < 0 and refer to both as the amount ofstorage. The transitions from storage levels " j $ 0 and " j < 0are different. If the storage level b(t) = " j < 0, then generatedenergy is used to reduce the severity of load-shedding.

We again empirically estimate the probability Mi, j(t) ofhaving b(t +1) = "i given that b(t) = " j for t = 0, . . . ,T !1.To this end, define

I1 := ["1 = B,!),Ii := ["i,"i!1), i = 2, . . . ,n+m!1,In+m := (!!,"n+m!1).

Then, an empirical estimate of Mi, j(t) is the fraction of pointsin the set of points

• " j + g(t)! d(t) that fall in Ii for i = 1, . . . ,n + m andj = 1, . . . ,n

• " j +g(t) that fall in Ii for i = 1, . . . ,n+m and j = n+1, . . . ,n+m.

DefineM := M(T !1)M(T !2) . . .M(0).

In the case where M has a unique eigenvalue at 1 withM$ = $, $i $ 0 for i = 1, . . . ,n+m and $1 + · · ·+$n+m = 1,$ is called the stationary distribution for the Markov chaininduced by M and the i!th entry $i of $ can be interpreted asthe long-term expected fraction of time b(t) spends at state "i(regardless of the initial conditions) at times t = 0,T,2T, . . .

0.150.15

0.15

0.2

0.20.2

0.2

0.3

0.30.3

0.30.5

0.50.5

0.5

0.7

0.7

0.9

!1

! 2

0.150.15

0.15

0.2

0.20.2

0.2

0.3

0.30.3

0.30.5

0.50.5

0.5

0.7

0.7

0.9

0.050.05

0.05

0.1

0.1

0.10.1

0.18

0.18

0.18

0.18

0.3

0.3

0.3

0.3

0.5

0.5

0.5

0.50.65

0.65

0.8

500 1000 1500 2000 2500 3000 3500 40000

1

2

3

4

5

Fig. 8. The long-term expected fraction of time spent at a load-sheddingstate as a function of solar !1 and !2 for a storage capacities of B = 3 MWh(solid) and B = 10 MWh (dash-dot).

0.1

0.10.1

0.15

0.150.15

0.15

0.2

0.20.2

0.20.3

0.3 0.3 0.30.4

0.4 0.4

!1

! 2

0.1

0.10.1

0.15

0.150.15

0.15

0.2

0.20.2

0.20.3

0.3 0.3 0.30.4

0.4 0.4

0.020.02

0.020.03

0.03

0.03

0.05

0.05

0.050.05

0.1

0.1

0.1

0.1

0.2

0.2

0.20.2

0.35 0.35 0.35

500 1000 1500 2000 2500 3000 3500 40000

1

2

3

4

5

Fig. 9. The long-term expected energy not-served due to load-sheddingnormalized by the expected (normalized) total demand as a function of !1and !2, for B = 3 MWh (solid) and B = 10 MWh (dash-dot).

[14], [15]. Let $ be the stationary distribution for M (if itexists) and define

p(t) := M(t!1) . . .M(0)$, t = 0, . . . ,T !1.

Then, the long-term expected fraction of time spent at a load-shedding state can be estimated as

1T

T!1

"t=0

n+m

"i=n+1

pi(t).

The long-term expected (normalized) energy not-served overthe period [0,T ] due to load-shedding can be estimated as

1T

T!1

"t=0

n+m

"i=n+1

|"i|pi(t).

Figure 8 shows the long-term expected fraction of timespent at a load-shedding state with two different storagecapacities B = 3MWh (solid, black curves) and B = 10MWh

Fig. 10. The long-term expected fraction of time spent at a load-sheddingstate as a function of !1 (with !2 = 0) for varying levels # of penetration ofrenewable generation for two different storage capacities B = 3 MWh (solid)and B = 10 MWh (dash-dot). The curves for # = 0.5 are indistinguishable.

(dash-dot, magenta curves) for varying values of solar (i.e.,!1) and wind (i.e., !2) generation. Figure 9 shows the ratioof the long-term expected normalized energy not-served dueto load-shedding over a day to the expected total demand(load), i.e., mean of 1

T "T!1t=0 d(t), as a function of !1 and !2

for B = 3 MWh (solid) and B = 10 MWh (dash-dot). Bothquantities decrease with increasing levels of generation withfixed storage capacity and with increasing storage capacityfor fixed !1 and !2. Figure 10 shows the effect of penetration# of renewable generation (as a function of the level ofsolar generation – with no wind generation) on the long-termexpected fraction of time spent at a load-shedding state. Notethree trends in the expected fraction of time at load-shedding:

• it increases with increasing penetration;• for fixed # , it is smaller with B = 10 MWh compared

to that with B = 3 MWh; and• the down-shifting effect reduces with the decrease in

the penetration level.This final observation suggests that the amount of storagecapacity will affect the tolerable level of penetration ofrenewable generation.

V. EFFECTS OF STORAGE EFFICIENCY

Let us re-write the evolution of the amount of storage as

b(t +1) = b(t)+ e(t), for t = 0, . . . ,T !1, (4)

where e(t) denotes the amount of energy flow into the storageover [t, t + 1] if e(t) $ 0 and the amount of energy flowout from the storage if e(t) " 0. Let 0 < %i " 1 denote theefficiency of energy in-flow and 0 < %o " 1 the efficiencyof energy out-flow. In order to investigate the effects of theefficiency of charging and discharging, let us modify (4) as

b(t +1) = b(t)+min!

%ie(t),1%o

e(t)"

. (5)

Define the overall efficiency of storage as % := %i%o. Figure 11shows the probability of load-shedding over 24 hours versus

2 4 6 8 10 12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Initial Storage Level (MWh)

Prob

abili

ty o

f Loa

d−Sh

eddi

ng

! = .7! = .8! = .9! = 1

Fig. 11. Probability of load-shedding versus amount of initial storage withB = 20MWh, !1 = 104, !2 = 1, and T = 24 hours for varying values of %.

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

!2

Frac

tion

of T

ime

of L

oad−

Shed

ding

" = 0.7" = 0.8" = 0.9" = 1

Fig. 12. The effect of varying efficiency % of storage as a function of !2on the long-term expected fraction of time spent at a load-shedding statefor fixed values of !1 = 104 and B = 3 MWh.

amount of initial storage with B = 20MWh, !1 = 104, !2 = 1,and T = 24 hours for varying values of % where %i = %o =

%%

(obtained by repeating the analysis of section III.A with thedynamics in (5)). Note that the amount of initial storageat which the transition in the probability of load-sheddingfrom high to low values happens increases with decreasingefficiency.

Figure 12 shows the effect of variations in storage effi-ciency on the long-term expected fraction of time spent ata load-shedding state as a function of !2 for fixed !1 = 104

and B = 3 MWh. A similar trend is observed for increasing!1 with fixed !2. Note that, for fixed !1, !2, and B, a decreasein the storage efficiency leads to an increase in the expectedfraction of time at load-shedding.

VI. DISCUSSION

We now discuss how the results presented in the previoussection can be translated into design choices. To this end,consider the four configurations corresponding to the crossesin Figure 13 for fixed storage capacity B = 20 MWh. TableI shows the area of PV cells and number of wind turbines

0.001

0.001

0.001

0.01

0.01

0.01

0.01

0.04

0.04

0.04

0.04

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.2

0.3

0.3

0.3

0.3

0.4

0.4

0.5

0.5

0.6

0.7

!1

! 2

x

x

x

x

500 1000 1500 2000 2500 3000 3500 4000 4500 50000

1

2

3

4

5

Fig. 13. The long-term expected fraction of time spent at a load-sheddingstate as a function of !1 and !2 for a fixed value of storage capacity B = 20MWh.

needed to guarantee specified levels of long-term expectedfraction of time at load-shedding. Finally Table II providesthe volume of three different storage systems that are capableof supplying B = 20 MWh [6], [16].

TABLE IAREA OF PV CELLS AND NUMBER OF WIND TURBINES FOR SPECIFIED

LONG-TERM EXPECTED FRACTION OF TIME AT LOAD-SHEDDING

FOT area of PV cells number of wind turbines0.1 105 ·103 m2 (25.9 acres) 2

0.01 105 ·103 m2 (25.9 acres) 30.01 17.5 ·103 m2 (4.3 acres) 50.001 38.5 ·103 m2 (9.5 acres) 5

TABLE IIVOLUME OF DIFFERENT STORAGE OPTIONS WITH B = 20 MWH

Type of storage Energy density#kWh/m3$ Volume [m3]

Pumped-hydro(100m of altitude) 0.28 & 5'117'117

& 41'41'41Compressed air 4.17 & 17'17'17

Lithium-ion battery 300 & 4.1'4.1'4.1

VII. CONCLUSIONS AND FUTURE WORK

Motivated by the challenges associated with integrationof renewable energy resources in to the electric grid dueto source intermittency, we considered a case study on anisland off the coast of Southern California, Santa CatalinaIsland, and investigated the feasibility of replacing dieselgeneration entirely with solar photovoltaics (PV) and windturbines, supplemented with energy storage. We used asimple storage model alongside a combination of renewablesand varying load-shedding characterizations to determinethe appropriate number of PV cells, wind turbines, andenergy storage capacity necessary to remain below a certainthreshold probability for load-shedding over a pre-specified

period of time and long-term expected fraction of time atload-shedding. Present work focuses on the validation theresults using more complicated storage models and load-shedding characterizations as well as data with smaller timeincrements. Neglected in this paper, but a natural extensionof this work is the incorporation of the effects of transmissionand distribution.

ACKNOWLEDGEMENTS

This work is partially supported by Southern CaliforniaEdison (SCE), the Boeing Corporation, and the NationalScience Foundation. The authors wish to thank Rui Huangfor the generating the demand profiles using SAM.

REFERENCES

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