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IEEE TRANSACTIONS ON SMART GRID, VOL. 0, NO. 0, MONTH YEAR 1

A Multi-timescale Scheduling Approach forStochastic Reliability in Smart Grids with Wind

Generation and Opportunistic DemandMiao He, Student Member, IEEE, Sugumar Murugesan, Member, IEEE, and Junshan Zhang, Fellow, IEEE

AbstractIn this study, we focus on the stochastic reliability ofsmart grids with two classes of energy users - traditional energyusers and opportunistic energy users (e.g., smart appliances orelectric vehicles), and investigate the procurement of energy sup-ply from both conventional generation (base-load and fast-start)and wind generation via multi-timescale scheduling. Specifically,in day-ahead scheduling, with the distributional informationof wind generation and demand, we characterize the optimalprocurement of the energy supply from base-load generation andthe day-ahead price; in real-time scheduling, with the realizations

of wind generation and the demand of traditional energy users,we optimize real-time price to manage opportunistic demandso as to achieve system-wise reliability and efficiency. Morespecifically, we consider two different models for opportunisticenergy users: non-persistent and persistent, and characterizethe optimal scheduling and pricing decisions for both modelsby exploiting various computational and optimization tools.Numerical results demonstrate that the proposed scheduling andpricing schemes can effectively manage opportunistic demandand enhance system reliability, thus have the potential to improvethe penetration of wind generation.

Index TermsReliability, wind generation, opportunistic en-ergy user, real-time pricing, demand response, multi-timescalescheduling, Markov decision process.

I. INTRODUCTION

A. Motivation

Wind energy is expected to constitute a significant portion

of all renewable generation being integrated to the bulk power

grids of North America [1]. High penetration of wind genera-

tion not only brings many benefits, economically and environ-

mentally, but also puts forth great operational challenges [2],

[3]. Unlike conventional energy resources, wind generation

is non-dispatchable, in the sense that wind energy can be

not harvested simply by request. Further, wind generation

exhibits greater variability across all timescales, which makes

it challenging for system operators to obtain the accurateknowledge of future wind generation. Therefore, variable and

uncertain wind generation can have a significant impact on the

Manuscript received January 13, 2012; revised July 31, 2012 and December11, 2012; accepted December 27, 2012. This work was supported in partby the US National Science Foundation under grant CPS-1035906, in partby the DTRA grant HDTRA1-09-1-0032, and in part by the Power SystemEngineering Research Center. Paper no. TSG-00499-2011.

The authors are with School of Electrical, Computer and Energy En-gineering, Arizona State University, Tempe, AZ 85287, USA (e-mail:[email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSG.2013.xxxxxxx

reliability of power systems, since the precise balance between

the energy supply and demand at all times is of paramount

significance to the reliable operations of power systems.

Recently, a significant amount of effort (e.g., [4][7]) has

been directed towards integrating wind generation into the

operations and planning of bulk power grids, in which wind

generation is usually treated as negative load, and auto-

regressive models (e.g., in [4]) or scenario trees (e.g., in [5],

[6]) are used to characterize the uncertainty of net load. Tocope with the uncertainty of net load, the approaches proposed

in [5][7] resort to operating reserve (the additional generation

capacities from on-line or fast-start generators) which is co-

scheduled with energy supply. However, as pointed out in

[1], the variability and uncertainty of demand is much less

significant compared to that of wind generation, and hence, the

cost of operating reserve would increases significantly when

the penetration level of wind generation is high.

With the objective to enhance the reliability of bulk power

grids and improve the efficiency of wind generation integra-

tion, we observe that operating reserve can be obtained from

the demand response of an emerging class of energy users,

namely opportunistic energy users, instead of conventionalgeneration. It is noted in [8] that over 10% daily electricityconsumption in U.S. is from residential and small commercial

energy users such as water heater, cloth dryers, and dish

washers. Traditionally, these energy users pay a fixed price per

unit of electricity that is established to represent an average

cost of power generation over a given time-frame (e.g., a

season). In smart grids with two-way communications, real-

time pricing programs can be implemented so that price is tied

with generation cost and can vary according to the availability

of energy supply. In this scenario, these energy users could be-

comesmartby receiving and responding to real-time price, and

are branded as opportunistic energy users, with the following

behaviors distinct from traditional energy users: 1) they accessthe energy market of smart grids in an opportunistic manner,

according to the availability of energy supply; 2) different from

the always-on demand of traditional energy users, the load

profiles of opportunistic energy users can be bursty and can

be either inelastic or elastic; 3) the demand of opportunistic

energy users responds to price on a much finer timescale, and

thus can be used to tune the balance between energy supply

and demand in a real-time manner (i.e., within minutes). The

prevalence of the new class of opportunistic energy users,

if utilized intelligently, makes demand side management a

promising solution to reducing the cost incurred by high

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2 IEEE TRANSACTIONS ON SMART GRID, VOL. 0, NO. 0, MONTH YEAR

penetration of wind generation.

B. Summary of Main Contributions and Organization

Aiming to tackle the challenge of integrating variable wind

generation into bulk power grids, we study multi-timescale

scheduling with two classes of energy users, namely traditional

energy users and opportunistic energy users, and address the

following challenges in maintaining the reliability of bulkpower grids: 1) the supply uncertainty as a result of variable

and non-stationary wind generation; 2) the demand uncertainty

due to a large number of opportunistic energy users and

their stochastic behavior; 3) the coupling between sequential

decisions across multiple timescales.

Building on the multi-timescale scheduling framework de-

veloped in our initial work [9], [10] and utilizing refined

system models, we rigorously characterize the solution to

the multi-timescale scheduling problem, and provide useful

insight into the impact of opportunistic demand on system

reliability. Motivated by the fact revealed in recent studies [11],

[12] that energy users can respond to high price by energy

conservation or by load shifting, we consider two types ofopportunistic energy users: non-persistent and persistent. Non-

persistent energy users leave the system when the current real-

time price is unacceptable, whereas persistent opportunistic

energy users wait for the next acceptable real-time price. We

obtain closed-form solution to the multi-timescale scheduling

problem when opportunistic energy users are non-persistent.

For the persistent case, the scheduling problem is formulated

as a multi-timescale Markov decision process (MMDP) which

we recast, explicitly, as a standard Markov decision process

(MDP) that can be solved using state-of-the-art techniques.

Further, we demonstrate via extensive numerical experiments

that when opportunistic demand is elastic, the proposed

scheduling and pricing schemes can effectively manage the

demand of opportunistic energy users and enhance the system

reliability, and thus has the potential to improve the penetration

of wind generation into smart grids.

The rest of the paper is organized as follows. In Section II,

we describe the multi-timescale scheduling framework and

provide the problem formulation. We study multi-timescale

scheduling for the non-persistent case in Section III. In Sec-

tion IV, we consider persistent opportunistic energy users and

formulate the scheduling problem as a multi-timescale Markov

decision process. Numerical results are presented in Section V.

We provide concluding remarks and identify directions for

future research in Section VI.

II. A MULTI-TIMESCALE S CHEDULING F RAMEWORK

In this paper, we study multi-timescale scheduling in a

framework as illustrated in Fig. 1. Specifically, system operator

procures energy supply from conventional generation and wind

generation, and manage the demand of both classes of energy

users via day-ahead/real-time price to achieve system-wise

reliability. Conventional generation, in turn, is drawn from two

sources: base-load generators (e.g., thermal units) and fast-start

generators (e.g., gas turbines). Energy supply procurement and

end-user pricing are performed in two stages, i.e., day-ahead

Wind Generation

(uncertain energy supply)

Base-load Generation

Traditional Energy Users

Opportunistic Energy Users(Non-persistent/Persistent)

(Multi-timescale

scheduling)

Fast-start Generation

day-ahead schedule

day-ahead retail prices

uncertain energy demand

real-time retail prices

real-time schedule uncertain energy demand

(a) Multi-timescale scheduling framework

1T

2T

2T

1T

Day-ahead scheduling with the distributional information of WandD atT1 scale

Real-time scheduling and balancing at T2 scale

(b) Scheduling horizon and timescales

Fig. 1. A multi-timescale scheduling framework with wind generation andtwo classes of energy users

scheduling and real-time scheduling, at different timescales.

In day-ahead scheduling, with the distributional information

of wind generation Wand traditional energy users demandDt, system operator decides the energy supply procurements from base-load generators and day-ahead price u, for thenext day. In real-time scheduling, upon the realization ofWand Dt, system operator decides the real-time price v foropportunistic energy users, and dispatchs fast-start generation

or cancels part of scheduled base-load generation, as needed,

to close the gap between demand and supply. It is worth

noting that the above framework that consists of day-ahead

and real-time scheduling is developed based on the state-of-

the-art scheduling schemes of power systems (e.g. [2], [3],

[13]), by explicitly incorporating opportunistic energy usersand the heterogeneous demand response of two classes of

energy users.

A. Problem Formulation

As illustrated in Fig. 1(b), a 24-hour period is divided into

M T1-slots of equal length, and eachT1-slot, in turn, consistsof K T2-slots1. The objective of system operator is to finda policy that dictates the multi-timescale decisions s, uand v, so that the overall expected profit across the next dayis maximized2. A general formulation of the multi-timescale

scheduling problem is provided below:

P : max

Mm=1

Rum(), (1)

whereRum() is the total net profit in a T1-slot, given by3:

Rum() =Kk=1

E

lk,m

Rlk,m(lk,m, ), (2)

1A T1-slot and a T2-slot can span an hour and 10 minutes, respectively.2Here, the scheduling problem is investigated from the perspective of an

(vertically integrated) utility which owns the wind generation asset. Accord-ingly, wind generation is utilized as much as possible (with no curtailment).

3The notation Ey

x denotes the expectation over x conditioned on y .

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HE, MURUGESAN AND ZHANG: A MULTI-TIMESCALE SCHEDULING APPROACH FOR STOCHASTIC RELIABILITY IN SMART GRIDS 3

where Rlk,m is the net profit in the kth T2-slot of the mthT1-slot (henceforth called the (k, m)th slot), and

lk,m is the

system state in the (k, m)th slot that is observable in realtime4. When opportunistic energy users are non-persistent,

system state consists of wind generation and the demand from

traditional energy users, i.e., lk,m={Wk,m,Dtk,m}. Whenopportunistic energy users are persistent, system state is

lk,m

={Wk,m,Dtk,m

,Plk,m

}, where Plk,m

denotes the number

of the persistent opportunistic energy users carried over from

the previous T2-slot to the (k, m)th slot.

B. Energy Supply and Demand Models

1) Stochastic wind generation: Wind generation forecasting

has been the focus of a significant amount of research and

industrial effort; see [14] for a detailed discussion on the var-

ious wind generation forecasting methods of practical power

systems. One key insight revealed by the survey [14] is that the

efficiency of wind generation integration would highly depend

on the accuracy of wind generation forecast. Therefore, fol-

lowing the modeling approach in [5], we directly characterizethe forecast error. Specifically, given a point forecast Wm, theamount of the wind generation in the (k, m)th slot in day-ahead scheduling is given by

Wk,m= Wm+ wm , (3)

where forecast error wm is closely tied to forecasting tech-niques, and hence can have arbitrary probability distribution

and can also be non-stationary across T1 slots. Another obser-vation from [14] is that the variance ofwm can be quite large(e.g., the mean absolute percentage error (MAPE) can be as

high as 20%) for existing forecasting techniques in practice.

2) Generation cost: To facilitate qualitative analysis, weadopt a linear generation cost model derived from the literature

[3], [13]. Specifically, base-load generation incurs a cost of

c1 per unit when dispatched in real-time; fast-start generationincurs a cost c2 per unit when dispatched in real-time as non-spinning reserves; base-load generation, scheduled day-ahead

but canceled in real-time, incur a cost ofcp per unit. Typically,c2>c1>cp. Further, we assume that wind generation is cost-free and utilized without curtailment.

3) Uncertain demand of traditional energy users: Based on

[15], we model the demand of traditional energy users in the

(k, m)th slot as a random variable with mean depending onday-ahead price um, as follows:

Dtk,m =tmumt + t, (4)

wheret is a zero-mean random variable that accounts for theuncertainty of traditional energy users demand, t is the priceelasticity that characterizes the demand response of traditional

energy users to price, and tm is a normalizing constant.The price elasticity of energy users (either traditional oropportunistic), is defined in [15] as the ratio of the percentage

change of the expected demand to that of price, i.e., for the

4The super-scripts u and l are used to distinguish between the upper-level quantities in a T1-slot and the lower-level quantities in a T2-slot.

case of traditional energy users,

t = umE[Dtm ]

dE[Dtm ]

dum. (5)

It is worth noting that price elasticity is typically negative.

4) Uncertain demand of opportunistic energy users: Under

real-time pricing, we assume that opportunistic energy users

have the following behaviors:

Opportunistic energy users arrive independently accord-

ing to a Poisson process with rate o, which is constantwithin a T1-slot but can vary across the T1-slots;

In each T2-slot, an opportunistic energy user i in thesystem decides to accept or reject the announced real-

time price v by comparing it with a price acceptancelevel Vi, which is randomly chosen and is i.i.d acrossopportunistic energy users;

The expected demand of opportunistic energy users has

a price elasticity o, which is defined similarly as in (5); Each active opportunistic energy user has a per-unit

power consumption Eo.

5) Day-Ahead and Real-Time Pricing: We consider thefollowing multi-timescale end-user pricing model:

Traditional and opportunistic energy users have separate

contracts: traditional energy users pay day-ahead price,

and opportunistic energy users pay real-time price;

Traditional energy users are informed, one day ahead, of

the day-ahead price u for each T1-slot; Opportunistic energy users receive the real-time price v

at the beginning of each T2-slot; Both day-ahead price and real-time price have price cap

ucap andvcap, respectively. The price caps are motivatedby the study [16] and intended to protect energy users by

hedging against the risk under variable price.

C. Net Profit in aT2-slot

Given the amount of conventional energy supply procure-

ment sm and the day-ahead price um settled for each T2-slotof themthT1-slot in day-ahead scheduling, together with therealizations of wind generation Wk,m and traditional energyusers demand Dtk,m , the net profit in the (k, m)th slot isgiven by

Rlk,m(lk,m, )

=umDtk,m + E

Dok,m[vk,mDok,m + (cpsm)1A

+ (cpk,m c1(sm k,m))1B

+ (c1sm+ c2k,m)1C], (6)

where Dok,m denotes the demand of opportunistic energyusers, and k,m=Wk,m+sm(Dtk,m+Dok,m) quantifies thegap between supply and demand. Indicator 1A corresponds

to the scenario when wind generation is sufficient to meet the

demand of both classes of energy users. Indicator 1B refers to

the scenario when wind generation is not sufficient but there

is energy supply surplus, which necessitates the cancelation

of part of scheduled base-load generation. Indicator 1C cor-

responds to the scenario when there is energy supply deficit

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and fast-start generation has to be dispatched. Formally, these

indicators are described as follows:

1A=

1 ifWk,m Dtk,m + Dok,m0 otherwise

1B =

1 if1A= 0 andk,m 00 otherwise

1C=

1 ifk,m< 00 otherwise

(7)

III. MULTI-TIMESCALE S CHEDULING WITH

NON-PERSISTENT O PPORTUNISTICE NERGY U SERS

The tight coupling between scheduling decisions, in the

sense that the energy supply procurement and the price in

day-ahead scheduling have a significant impact on real-time

price, and real-time pricing policy, in turn, affect the realized

net profit of day-ahead schedule, underscores the need for the

joint optimization of day-ahead and real-time schedules. To

this end, we take a bottom-up approach in solving the multi-

timescale scheduling problem. Specifically, we first solve the

real-time scheduling problem, conditioned on day-ahead deci-

sions. Then, we investigate the day-ahead scheduling problem

by taking into account real-time pricing policy.

A. Real-time Scheduling in TimescaleT2

Recall that non-persistent opportunistic energy users re-

sponse to high real-time price by leaving the system, thus

the total demand in a T2-slot only depends on the currentdecisions. Then, it suffices to examine the scheduling problem

in a T2-slot, which is formulated as5:

PRTnonpst : maxvRl(l, s , u, v). (8)

The optimal solution to PRTnonpst defines a real-time pricingpolicy s,u:lv, a mapping from the system state to a real-time price conditioned on the day-ahead decisions (s, u).

B. Day-ahead Scheduling in TimescaleT1

The day-ahead scheduling problem can be re-formulated by

taking into account the real-time pricing policy s,u. Further,since W and Dt are independent and identically distributedacross the T2-slots of the T1-slot, the day-ahead schedulingproblem can be optimized by simply considering the snapshot

problem in a specific T2-slot, given by:

PDAnonpst : maxs,uEul

Rl(l, s , u, s,u(l))

. (9)

C. Approximate Solution

It is easy to see that a closed-form expression of s,u interms of(s, u)is unattainable. This observation, along with theconvolved nature of the uncertainties involved, makes a direct

joint optimization of PRTnonpst and PDAnonpst challenging.

We therefore take an alternative approach and obtain an ap-

proximate solution to the multi-timescale scheduling problem.

5In Section III, we drop the suffix (k,m) for notational simplicity.

In light of the characteristics of practical power systems, we

impose the following conditions.

Condition I: Wind generation is not sufficient to meet

the total demand in the system.

Condition II: The uncertainty in the demand of op-

portunistic energy users is significantly less than the

uncertainty of wind generation.

Condition I is motivated by the Renewable Portfolio Stan-dards of U.S. [17], in which most of the current state-by-state

projected penetration levels of renewable generation (wind

generation included) are below 30%. Under such a penetration

level, it is unlikely that wind generation is sufficient to meet

the overall demand. We then provide an explanation of Con-

dition II. Since opportunistic energy users arrive according

to a Poisson process with rate o, it follows that the numberof active opportunistic energy users that accept the current

price v, denoted as Na, is a Poisson random variable withmean oT2P(V v ). Further, it is clear that oT2 is large,and hence, Na can be approximated by a Gaussian randomvariable. Note that the demand of opportunistic energy users

is given by Do=NaEo, and it follows from (5) that

P(V v) ovo , (10)

whereo vominis a normalizing constant, and vmindenotes

the highest price that is acceptable to all opportunistic energy

users. Therefore, the demand of opportunistic energy users has

a Gaussian distribution N(qo(v), 2o(v)), with

qo(v) oT2ovoEo,

2o(v) oT2ovoE2o . (11)

Observe from (11) that the variance of the demand of oppor-

tunistic energy users is of the same order as its mean. Further,

wind generation and the demand of opportunistic energy usersare typically comparable regarding the mean, and it is observed

that the uncertainty of wind generation is so high that the

standard deviation is of the same order as its mean. Therefore,

we conclude that Condition II holds.1) Approximate Solution to the Real-time Scheduling Prob-

lem: It is easy to verify that 1A= 0holds under Condition I.Then, by using (11), (6) reduces to

Rl(l, s , u, v) = uDt c1s + c2Y + (v c2) qo(v)

(2)1/2co(v)exp

y2/2

c (Y qo(v))(1 Q (y)) , (12)

whereYs+WDt, yYqo(v)o(v) , ccpc1+c2, and Q() isthe tail probability of the standard normal distribution.

The demand of opportunistic energy users is said to be

relatively inelasticif 1o

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Remarks: The proof is provided in Appendix A. Note that

the above result is intuitive, since, with opportunistic energy

users demand being relatively insensitive (inelastic) to real-

time price, system operator can maximize the profit by simply

announcing the highest possible price, vcap.Proposition 3.2: Suppose Condition I and Condition II

hold. When the demand of non-persistent opportunistic energy

users is relatively elastic, i.e., o

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+ E{um,a

um}

Pum+1

Vum+1(um+1)

, (18)

wherem:{Wk,m,Dtk,m ,Plk,m}vk,m is a stationary mapping

within the mth T1-slot, and the various expectations used inthe MDP formulation are defined in (19).

E{um,a

um}

Pum+1

(.) = EPl1,m

Pl2,m

EPl2,m

Pl3,m

EPlK1,mPlK,m

EPlK,mPum+1

(.)

EPlk,m

Plk+1,m(.) =EWk,m

EDtk,m

ENk,m

E{Nk,m,P

lk,m}

Plk+1,m (.)

vk,m =P(V vk,m)

E{Nk,m,P

lk,m}

Plk+1,m

(.) =

Nk,m+Plk,m

Plk+1,m

=0

Plk,m+ Nk,m

Plk+1,m

(1 vk,m)Plk+1,m(vk,m)

(Nk,m+Plk,mP

lk+1,m)(.) (19)

A key step towards the above results is to view m as anaction taken in day-ahead scheduling. With this insight, we can

simplify the MMDP into a classic MDP, as discussed below.

Note that in (18), Vum(um) is the expected net reward from

slotm until slot M in day-ahead scheduling, and the terminalreward is given by

VuM(uM) = max

auM

RuM(uM, a

uM). (20)

We now proceed to explicitly characterize the immediate

reward Rum, for m{1, , M}:

Rum(um, a

um) = V

lk,m({Wm, P

lk,m}, a

um)|k=1, (21)

wherePl1,m=Pum by definition, and for k{1, ,K1}

Vlk,m({Wm, Plk,m}, a

um= {sm, um, m})

= EWmWk,m

EumDtk,m

Rlk,m(lk,m, a

um)

+ EPlk,mPlk+1,m

Vlk+1,m({Wm, Plk+1,m}, a

um), (22)

and V lK,m is given by

VlK,m({Wm, PlK,m}, a

um)

= EWmWK,m

EumDtK,m

RlK,m(lK,m, a

um). (23)

Note that Rlk,m and Vlk,m can be regarded as the immediate

reward and the net reward at the lower level MDP, respectively.

More specifically, (6) can be re-formulated as

Rlk,m(lk,m, a

um)

=umDtk,n + ENk,mE{Nk,m,P

lk,m,vk,m}

Nak,m[vk,mDok,m

+ (cpsm)1A+ (k,mcp (sm k,m)c1)1B

+ (c1sm+ c2k,m)1C], (24)

where Nk,m denotes the number of the opportunistic energyusers arriving at the(k, m)th slot, which is Poisson distributed.

Summarizing, we have rigorously formulated the multi-

timescale scheduling problem as an MMDP with special

characteristics, and shown that it can be recast explicitly

as an MDP with continuous state and action spaces. Using

3m

W3 mWmw

( )mw

f

Fig. 2. PDF of wind generation forecast error in themth T1-slot.

appropriate discretization techniques, we can reformulate it

as a classic MDP with discrete state and action space [19].

Then, the problem formulated in (18) can be efficiently solved

by using adaptive critics designs (ACD) [20]. Indeed, ACD

constitutes an effective solution technique, as suggested in the

literature [21], for a variety of dynamic, stochastic control and

optimization problems (e.g., [22][25]) in smart grids.

V. NUMERICAL R ESULTS

We now study, via numerical experiments, the performance

of the proposed approach in the multi-timescale schedul-

ing framework, through comparison with existing scheduling

schemes in a benchmark smart grid system where all the

energy users are assumed to exhibit traditional response to

the day-ahead price u with the same price elasticity t.Therefore, the scheduling and pricing decisions (s,u) ofthe benchmark system can be easily obtained from (15) by

neglecting the opportunistic demand and real-time pricing, i.e.,

s=tutW+F1Z (1cp/c) and u

=u. The performancemetrics are the per unit generation cost, which is defined as the

ratio of total generation cost to the total demand served, and

system reliability metrics including loss of load probability

(LOLP) and expected energy not served (EENS).

In this numerical study, we focus only on the key parame-ters and investigate their impact on the system performance,

including the penetration level of wind generation w, thepenetration level of opportunistic demand o and the uncer-tainty of wind generation. Specifically, as in literature, the

penetration level of wind generation w is defined as theratio of wind generation to the total energy supply over the

whole scheduling horizon. Similarly, the penetration level of

opportunistic demandois defined as the ratio of opportunisticdemand to the total demand. Data for other system parameters

are described as follows.

A. Simulation Data

1) Stochastic Wind Generation: Wind generation data arecollected from [26] and scaled according to the penetration

level w. The point forecast Wm in (3) is provided using theMarkov chain developed in [26]. For simplicity, the probability

distribution illustrated in Fig. 2 is used for w. It is easy tosee that is exactly the MAPE. For day-ahead forecast, isusually around 20% [14].

2) Generation Cost: We adopt the approach in [27], which

utilizes the heat rates of generators to compute the genera-

tion cost. Here, base-load generators of coal type and fast-

start generators of #2 oil type are considered. Specifically,

c1=30.45$/MWh,c2=228.51$/MWh and cp=15$/MWh.

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0 5% 10% 15% 20% 25% 30%26

28

30

32

34

36

38

Penetration level of wind generation

Perun

itgenerationcost($/MWh)

Multitimescale scheduling (o=30%)


Benchmark system

Fig. 3. Per unit generation cost (= 20%).

0 5% 10% 15% 20% 25% 30%150

250

350

450

550

650


Dispatchedfaststartgeneration(/MW)

Multitimescale scheduling(=30%)

Multitimescale scheduling(=10%)

Benchmark system

Fig. 4. Dispatched fast-start generation (= 20%).

3) Uncertain Demand:The hourly demand data in Table. 4

of [28] with peak value 8550 MW is used for both systems.

For given total hourly demand D, we set E[Dt]=D for the

benchmark system; for the multi-timescale scheduling system,

traditional and opportunistic demand are properly scaled by

setting E[Dt]=(1 o)D and E[Do]=oD, respectively. Fort, a zero-mean normal distribution with standard deviationt that is 3% of the expected demand E[Dtm ] is used bytruncating over(3t,3t). Further, we uset=0.5,o=2and 0.05$/kWh as the price cap for both u andv .

B. Performance Evaluation and Discussion

We use the case of non-persistent opportunistic demand and

the hour with peak demand of 8550 MW as an illustrative

example. Note that the metrics are computed via Monte-

carlo simulations by choosing wind generation and demandrandomly according to their distributions specified earlier.

1) Per Unit Generation Cost: In Fig. 3, the per unit

generation cost is plotted against various w. In the multi-timescale scheduling system, the per unit generation cost

decreases significantly withw; in contrast, in the benchmarksystem, the per unit generation cost is not really reduced.

The reason is revealed in Fig. 4, i.e., much more fast-start

generation is dispatched in the benchmark system.2) Reliability: In practice, usually, a fixed amount of active

reserve is scheduled. One practical rule is the X+Y rule[29], in which active reserve is R=X%D+Y%W for given

0 5% 10% 15% 20% 25% 30%0

5%

10%

LOLP

0 5% 10% 15% 20% 25% 30%

100

300

500

700


EENS

(MWh)

Multitimescale (o=3 0%) Multi time sca le (

o=10%) Benchmark system

Fig. 5. System Reliability (= 20%).

10% 15% 20% 25% 30%

100

200

300

400

500

600

MAPE of wind generation forecast ()

Faststartge

neration(MW)



Benchmark system

Fig. 6. Non-spinning reserve requirement (w = 20%).

forecasted demand D and wind generation W. As discussedearlier, part of s is scheduled as spinning reserve withan amount of SR=sE[DW]. Therefore, adhering to a3+10 rule, we consider N S=RSR fast-start generation asnon-spinning reserve, and investigate the system reliability,

which is quantified by LOLP, i.e., E[1{DsW>NS}], andEENS, i.e.,T1E[(DsWN S)+]. It is observed from theresults illustrated in Fig. 5 that, for given penetration levels

of wind generation, the proposed multi-timescale scheduling

approach achieves higher reliability with the same reserve

requirement.

3) Impact of Forecast Accuracy: We control the uncertainty

of wind generation by varying , and investigate the impactof forecast error on the fast-start generation that is necessary

to maintain the system reliability. It is observed in Fig. 6

that the fast-start generation requirement increases with inall systems. For multi-timescale scheduling, this is because vkeeps fixed and the opportunistic demand can not be reduced

any further in the supply deficit case of Proposition 3.2.

Therefore, it is imperative to develop advanced models and

techniques to improve the forecast accuracy.

Summarizing, the above results suggest that the proposed

multi-timescale scheduling schemes have the potential to en-

hance the system reliability and improve the efficiency of wind

generation integration. Note also that, the benefit brought by

multi-timescale scheduling can be very limited if o is low,e.g., when o10%, as observed in Fig. 3- Fig. 6.

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VI . CONCLUSION

We study multi-timescale scheduling and pricing with tradi-

tional and opportunistic energy users to address the challenge

of integrating variable wind generation into smart grids. The

optimal scheduling and pricing decisions are characterized

rigorously for both the non-persistent case and the persistent

case. Through numerical experiments, we demonstrate the

potential benefit of the proposed multi-timescale schedulingapproach, when compared with existing system and schemes.

In essence, upcoming smart grids are expected to be char-

acterized by the sophisticated temporal dynamics arising from

various factors such as the temporal correlation in wind power

output, the demand response from opportunistic energy users,

the ramp constraints of conventional generators, to name a

few. We believe that the studies we initiated here on multi-

timescale scheduling for integrating variable renewal energy

into smart grids, scratch only the tip of the iceberg. There are

still many questions remaining open to improve the penetration

of renewable energy into power grids, such as wind curtail-

ment and the impact on the penetration of wind generation

[30], more accurate forecasting techniques [31], the tradeoffbetween interruptible load and reserves in wind generation

integration [32]; and our ongoing studies [31], [32] are along

this avenue.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers

for their valuable comments and suggestions to improve the

quality of the paper. They are also grateful to Dr. Lei Yang

for his insight and discussion.

VII. APPENDIX

A. Proof of Proposition 3.1

Usually the forecast error t and w usually have continu-ous, symmetrical and unimodal probability distributions (e.g.,

Gaussian distribution used in [15]). Let2Ydenote the varianceofY, then it follows from Condition IIthat 2Y

2o . Then,

there exists a finite constant c0 0 such that

P(|Y qo(v)| c0o(v)) 1, (25)

and6

Q(c0) 1, Q(c0) 0, exp(c20/2) 0. (26)

WhenY qo(v) c0o(v), (12) boils down to:

Rl(l, s , u, v) = (v (c1 cp)) qo(v) + (c1 cp)Y

+ uDt c1s; (27)

WhenY qo(v) c0o(v), (12) simplifies to

Rl(l, s , u, v) = (v c2) qo(v) + uDt c1s + c2Y. (28)

It is clear that (27) and (28) are unimodal for v [vmin, vcap],both with peaks at v = vcap. This yields the real-time pricingpolicy: s,u(

l) = vcap.

6It is well-known that (26) is valid for c0 3.

B. Proof of Proposition 3.2

When the demand of non-persistent opportunistic energy

users is relatively elastic, i.e., o

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Proposition 3.1 that the optimal real-time price is a constantvcap. Letting v0 denote this constant real-time price for boththe elastic and inelastic cases, respectively, we have

f2(s) = (c2 c1)s

+ (v0 c2)qo(v0), s s0. (36)

Therefore, f2(s) f2(s0), s s0, and (u

, s) indeedlies in the feasible region Fand hence optimizes the day-ahead

scheduling problem in (30). The optimal day-ahead decision,s, can now be computed using s and u.

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Miao He(S08) received the B.S. degree from Nanjing University of Posts andTelecommunications, in 2005, and the M.S. degree from Tsinghua University,in 2008. He is currently pursuing the Ph.D. degree at Arizona State University.

Sugumar Murugesan(S10-M11) received the B.E. degree from the Collegeof Engineering, Anna University, India in 2004 and the M.S. and Ph.D. degreesfrom The Ohio State University, USA, in 2006 and 2010, respectively.

Junshan Zhang (F12) received his Ph.D. degreefrom the School of ECE at Purdue University in2000. He joined the EE Department at ArizonaState University in August 2000, where he has beenProfessor since 2010. His research interests includecommunications networks, cyber-physical systemswith applications to smart grid, stochastic modelingand analysis, and wireless communications. His cur-rent research focuses on fundamental problems in in-formation networks and network science, includingnetwork optimization/control, smart grid, cognitive

radio, and network information theory.Prof. Zhang is a fellow of the IEEE, and a recipient of the ONR Young

Investigator Award in 2005 and the NSF CAREER award in 2003. He receivedthe Outstanding Research Award from the IEEE Phoenix Section in 2003.He served as TPC co-chair for WICON 2008 and IPCCC06, TPC vicechair for ICCCN06, and a member of the technical program committees ofINFOCOM, SECON, GLOBECOM, ICC, MOBIHOC, BROADNETS, andSPIE ITCOM. He was the general chair for IEEE Communication TheoryWorkshop 2007. He was an Associate Editor for IEEE Transactions onWireless Communications, and an editor for the Computer Network journaland IEEE Wireless Communication Magazine. He co-authored a paper thatwon IEEE ICC 2008 best paper award, and one of his papers was selected asthe INFOCOM 2009 Best Paper Award Runner-up. He is TPC co-chair forINFOCOM 2012.

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