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Developing Bidding and Offering Curves of aPrice-maker Energy Storage Facility Based on

Robust OptimizationSoroush Shafiee, Student Member, IEEE, Hamidreza Zareipour, Senior Member, IEEE, Andrew M. Knight, Senior

Member, IEEE,

Abstract—This paper presents an algorithm to constructhourly bidding and offering curves to purchase and sell electricityfor a price-maker merchant energy storage facility participatingin a day-ahead electricity market. Hourly generation and demandprice quota curves (GPQCs and DPQCs) are used to model theprice impact of storage operation in the participation strategyproblem of a price-maker storage facility in the market. Thispaper introduces a max-min mixed-integer linear programmingmodel to present a participation strategy to manage the riskof uncertainty associated with forecasted GPQCs and DPQCsby robust optimization. The max-min formulation is convertedto its equivalent linear maximization formulation based on theworst case scenario during charging and discharging hours. Then,an algorithm is proposed to build hourly bidding and offeringcurves. To do so, the confidence intervals for GPQCs and DPQCsare divided into subintervals and the robust mixed-integer linearformulation is solved sequentially for each subinterval of GPQCsand DPQCs. Sequential constraints are applied to ensure thedecreasing and increasing nature of the bidding and offeringcurves, respectively. Then, hourly bidding and offering curves arebuilt based on the obtained scheduling and corresponding priceresults. The proposed bidding and offering strategy is presentedand validated using numerical simulations.

Index Terms—Energy storage facility, price quota curves,bidding and offering strategy, robust optimization.

NOMENCLATURE

Superscripts

c Charging mode.d Discharging mode.

indices

t Index for operation intervals running from 1 toT .

s Index for the steps of generation price quotacurves from 1 to ndt for hour t.

s′ Index for the steps of demand price quota curvesfrom 1 to nct for hour t.

S. Shafiee is with the Teshmont Consultants LP, Calgary, AB, Canada T2R1M1 (e-mail: sshafiee@teshmont.com)

H. Zareipour, and AM. Knight are with the Department of Electricaland Computer Engineering, Schulich School of Engineering, University ofCalgary, Calgary, AB, Canada T2N 1N4 (e-mail: h.zareipour@ucalgary.ca;aknigh@ucalgary.ca).

ParametersπNGt Natural gas price.πdt,s Forecasted price corresponding to step number s

of the GPQC at hour t.πct,s′ Forecasted price corresponding to step number s′

of the DPQC at hour t.πdt,s/π

dt,s Upper/Lower bound of price corresponding to

step number s of the GPQC at hour t.πt,s′/π

ct,s′ Upper/Lower bound of Price corresponding to

step number s′ of the DPQC at hour t.bd,maxt,s Size of step s of the GPQC at hour t.bc,maxt,s′ Size of step s′ of the DPQC at hour t.Esmax maximum energy level of air storage.

nd/ct

Number of steps of generation/demand pricequota curves.

P dmax maximum generation capacity of expander.P cmax maximum compression capacity of compressor.qd,mint,s Is the summation of power blocks from step 1 to

step s− 1 of GPQC for hour t.qc,mint,s′ Is the summation of power blocks from step 1 to

step s′ − 1 of DPQC for hour t.SOCmin minimum state of charge (SOC) of air storage.SOCmax maximum SOC of air storage.SOCinit Initial SOC level of air storage.V OMexp Variable operation and maintenance cost of ex-

pander.V OM c Variable operation and maintenance cost of com-

pressor.

Variables

πEt Electricity market clearing price at interval t.πdt,s/π

ct,s price corresponding to step number s of the

GPQC/DPQCs at hour t within the specifiedconfidence interval.

bdt,s The fractional value of the power block cor-responding to step s of the GPQC to obtaindischarging quota P dt in hour t.

bct,s′ The fractional value of the power block cor-responding to step s′ of the QPQC to obtaincharging quota P ct in hour t.

OCt Operation cost of the plant at time t.P dt Discharging power at time t.P ct Charging power at time t.

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SOCt Cavern state of charge at time t.uxt Unit status indicator in either modes x, i.e.,

discharging (d) or charging modes (c) (1 is ONand 0 is OFF).

xdt,s Binary variable that is equal to 1 if step s ofGPQC is the last step to obtain discharging quotaP dt in hour t and 0 otherwise.

xct,s′ Binary variable that is equal to 1 if step s′ ofDPQC is the last step to obtain charging quotaP ct in hour t and 0 otherwise.

A. Functionsπdt (P dt ) GPQCs, i.e, stepwise decreasing function that

determines the market price as a function of thedischarge quantity at time t.

πct (Pct ) DPQCs, i.e., stepwise increasing function that

determines the market price as a function of thecharge quantity at time t.

I. INTRODUCTION

ENERGY storage systems are expected to be increasinglyintegrated into power systems worldwide due to their

various applications [1], [2]. A market study by CitiGroupin 2015 estimated a global market of up to 240 GW by 2030excluding pumped-storage hydroelectricity and car batteries[3]. From the system perspective, large scale energy storage(ES) systems such as batteries, compressed air energy storage(CAES), and pumped hydro storage systems are able toprovide time shifting and peak shaving, facilitate the fastgrowing penetration of the renewable resources into powersystems [4], [5]. Several studies have investigated the impactsof ES systems on the operation of power systems from a cen-tralized system viewpoint [6]–[9]. From another perspective,the ES facility can operate as a merchant investor-owned entity.From the investor’s point of view, the capability of storingconsiderable amount of energy brings business opportunitiesfor energy arbitrage by an ES facilities [10].

A large scale ES operator needs to optimize its participationstrategy in an electricity market to gain higher operationprofit. Thus, the operator needs a strategy for an ES facility,which consists of an offering strategy for selling electricity tomaximize the revenue and a bidding strategy for purchasingelectricity to minimize the cost. Over the past years, severalstudies have concentrated on developing optimal bidding andoffering strategy of a privately-owned ES facility in competi-tive electricity markets as described below.

The existing methods can be divided into two separategroups: price-taker storage facilities and price-maker ones. Aprice-taker storage facility refers to a facility, which cannotchange the market price by its charge/discharge actions. Thus,a forecast of day-ahead prices are used to optimize thecharge/discharge operation of the facility accordingly [10],[11]. In this group, the electricity market is modeled by theforecast of day-ahead prices. In the second group, a price-maker ES facility is large enough that its action can altermarket prices.

To optimize the scheduling of a price-maker market player,one method is to estimate the rivals’ marginal cost. Solving

the problem by predicting the rivals’ marginal cost leadsto a mathematical programming with equilibrium constraints(MPEC) problem. MPEC approach is used to address strategicbidding and offering of a storage facility [12]–[14]. MPECapproach imposes high computational burden due to its com-plicated mathematical formulation, which makes it difficultto be solved to the optimality specially for large systems[15]. Another method to solve the scheduling of a price-makerparticipant in an electricity market is to forecast the price quotacurves (PQCs) [16]. For a generation unit, the generation PQC(GPQCs) for an hour, is a stepwise decreasing curve that showsthe market price as a function of the total generation of theunit [17], [18]. GPQC models the producers interaction withother market participants. This method has significantly sim-pler formulation and consequently less computational burdencompare to the MPEC approach [15]. In the present paper,the concept of price quota curves is implemented to modelthe price impacts a price-maker ES facility. In addition tothe GPQCs, demand PQCs (DPQCs) are employed here tomodel the impacts of charging on market price to optimizethe scheduling of a price-maker ES facility. This issue is notaddressed in the literature.

Due to inevitable error in the forecasts of the GPQCs andDPQCs, the associated uncertainty needs to be incorporatedin the scheduling problem to minimize the risk. Therefore,an appropriate offering strategy for selling the electricity aswell as a bidding strategy for purchasing the electricity isrequired to maximize the arbitrage profit while managing therisk of forecasting uncertainty. Some studies do not considerthe uncertainty associated with the input data and develop arisk-neutral scheduling [16]. In order to deal with forecastinguncertainties when participating in electricity markets, differ-ent methods are applied such as stochastic programming [13],[14], [19], [20] or information Gap decision theory (IGDT)[21], [22]. As an alternative, robust optimization is used todevelop self-scheduling and bidding strategy of market par-ticipants such as thermal and hydro thermal generation com-panies [23]–[26], virtual power plants [27], and a wind farmcombined with ES [28]. Robust optimization is an intervalbased optimization method in which, instead of scenarios, theuncertainty is specified via intervals [29]. Thus, this method isapplicable when high level of uncertainty exists. It also has lesscomputational burden than stochastic programming [24], [25],[30]. Moreover, robust optimization makes no assumptionon the probability density function (PDF) of the uncertainparameter. This feature of the robust optimization is highlydesirable for the case of PQCs, since providing PDF for PQCsis not straightforward. Moreover, in the day-ahead electricitymarket, the storage operator can submit multi-step bids andoffers to the market for purchasing and selling electricity tomanage the risk of price uncertainty more effectively. Thus,constructing multi-step bidding and offering curves for an ESfacility is important.

In this paper, we develop a bidding and offering strategy fora large-scale price-maker independent ES facility to participatein day-ahead electricity market. While such facility couldstack multiple revenue sources by providing other services tothe grid, in this work, we focus only on energy arbitrage;

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developing bidding strategies for such cases is the focus ofthe authors’ future work. The impact of ES discharging andcharging operation on market price is modeled by meansof GPQCs and DPQCs. Then, we develop a robust-basedoptimization to model the uncertainty associated with GPQCsand DPQCs. In order to solve the min-max problem, it needsto be converted to a linear maximization problem. Robustoptimization has been used in the literature to model the priceuncertainty and develop participation strategies of a price-takergeneration company [24]–[26] as well as a consumer [31].In the case of a generation company and consumer, to solvethe robust optimization, the worst cases of price deviationare easily determined as the lower and upper bound of theconfidence interval, respectively [24], [31]. Unlike the caseof a generator or a load in which determining the worst casescenario of the uncertain parameter is straightforward, for thecase of a price-maker ES facility, the worst case scenario of theGPQCs and DPQCs depends on the charging and dischargingstatus of the facility, which needs to be properly modeled.Based on the developed robust scheduling formulation, wepresents an algorithm to construct hourly multi-step biddingand offering curves for the storage facility for participationin a day-ahead market. In [24],offer curves for a price-takerthermal unit are developed based on robust optimization usingprice subintervals [24]. For the case of a price-maker ESfacility, this becomes more challenging; since, incorporatingGPQCs and DPQCs to model the price impacts of the ESfacility, the developed algorithm in [24] needs to be modifiedand extended further to construct not only offering curve forselling but also bidding curve for purchasing electricity. To doso, the predefined confidence intervals of GPQCs and DPQCsare divided into subintervals, where the process of definingsubintervals for GPQCs differs from that of DPQCs. Then,the robust formulation is solved sequentially for each set ofsubintervals. Afterward, the obtained results are employed toconstruct the hourly multi-step bidding and offering curvesin order to take different levels of risk of forecast error intoaccount. It enables the operator to immune its decision againstunfavorable price deviations during charging or discharging.Meanwhile, it exploits favorable price fluctuation by incorpo-rating less conservative actions.

The main contributions of this paper can be summarized asfollows:• To propose an operation scheduling for an independent

price-maker ES facility based on robust optimization.• To develop an algorithm to construct hourly bidding

and offering curves for purchasing and selling electricity,respectively, considering different level of risk.

II. DETEREMINISTIC SCHEDULING OF AN ES FACILITY

In this section, a deterministic scheduling problem of anES facility participating in a day-ahead energy market isdeveloped. The storage facility aims to maximize its profitby purchasing and storing electricity during low price hours.The stored energy is used to generate electricity during peakprice hours. The ES facility is assumed to be a price-makerand can alter the market price by its operation. Compared

to [12]–[14], where MPEC model is applied to schedule aprice-maker ES facility, GPQCs and DPQCs are used in thispaper to schedule a price-maker ES facility, which has lesscomputational burden for a large problem. The GPQCs areused in [16] to address the self-scheduling problem of aprice-maker thermal producer. The method is extended in ourprevious work [32] to develop a self-scheduling frameworkfor a price-maker ES facility. The formulation in this paperis in line with that of proposed in [32]. In the previous work[32], it is assumed that the perfect forecasts of the hourlyGPQCs and DPQCs are available, whereas in the currentpaper, we aim to extend our previous work by incorporatingthe uncertainties associated with the forecasted GPQCs andDPQCs using robust optimization and propose a biddingand offering strategy. Therefore, the developed deterministicformulation proposed in our previous work needs to be explainhere as well.

max

T∑t=1

[P dt × πEt (P dt )− P ct × πEt (P ct )−OCt]

(1)Subject to:

Ψt(Pdt , P

ct ) ∀t ∈ T (2)

The objective function (1) consists of three terms. The firstterm is the revenue from selling the electricity to the market.The second term is the cost of purchasing the electricity fromthe market to charge the facility. The third term representsthe operating cost of the plant. Ψt represents the associatedoperation constraints of the ES facility related to the chargingand discharging power capacity and ES capacity. The set ofconstraints Ψt as well as the operation cost OCt are presentedin Appendix. This is a general formulation for an energystorage facility. Depending on the storage technology, theobjective function and the constraints Ψt could be adjusted.

A. Equivalent linear formulation

The developed self-scheduling formulation is non-linear dueto multiplication of two variables in the objective function.Considering stepwise GPQCs and DPQCs, as shown in Figs.1 and 2, the equivalent linear model is mathematically formu-lated as below:

max

T∑t=1

[ ndt∑

s=1

πdt,s(bdt,s + xdt,sq

d,mint,s )

−nct∑

s′=1

πct,s′(bct,s′ + xct,s′q

c,mint,s′ )−OCt

](3)

Subject to:(2)

P dt =

ndt∑

s=1

(bdt,s + xdt,sqd,mint,s ) ∀t ∈ T (4)

0 ≤ bdt,s ≤ xdt,sbd,maxt,s ∀t ∈ T (5)

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Fig. 1: Linearization process for GPQC

Fig. 2: Linearization process for DPQC

ndt∑

s=1

xdt,s = udt ∀t ∈ T (6)

P ct =

nct∑

s′=1

(bct,s′ + xct,s′qc,mint,s′ ) ∀t ∈ T (7)

0 ≤ bct,s′ ≤ xct,s′bc,maxt,s′ ∀t ∈ T (8)

nct∑

s′=1

xct,s′ = uct ∀t ∈ T (9)

πEt =

ndt∑

s=1

πdt,s × xdt,s +

nct∑

s′=1

πct,s × xct,s′ ∀t ∈ T (10)

The objective function (3) states the operating profit of thestorage facility. The first and second terms are the linearizeddischarging revenue and charging cost, respectively. Using thebinary and continuous variables as shown in Figs. 1 and 2, thedischarging revenue and charging cost are linearly expressed in(3). The shaded area in Figs. 1 (Figs. 2) represents the revenue(cost), which is the discharging (charging) power multipliedby the market price at that discharging (charging) level. Thedischarging power and charging power are linearly formulatedin (4) and (8), respectively. Equation (6) states that at each hourof discharging period, only one instance of the variable xdt,sis nonzero, which shows the corresponding step of GPQC thestorage is operating at that hour; all instances of the variablexdt,s are zero at time t if storage is not in discharging mode atthat hour. Similarly, (9) determines the corresponding step ofthe DPQC during charging operation.

These equations are used to calculate the discharging rev-enue and charging cost, as formulated in the objective function.The market price, which is a function of discharging or charg-ing power, is calculated in (10) based on the defined binary

variables. Based on the step of GPQCs during discharging orDPQCs during charging that the storage is operating at, themarket price is determined.

III. PROPOSED ROBUST-BASED SCHEDULING MODEL

In the previous section, a deterministic scheduling opti-mization method for a price-maker ES plant was developed.It is assumed that the perfect forecasts of hourly GPQCsand DPQCs are available for all hours of the schedulinghorizon. However,it is difficult to accurately forecast PQCs,as predicting the market price based on the accepted amountof units bids or offers is complicated. Hence, forecastingerror should be taken into account in ES facility’s operationbidding/offering strategies to manage the risks.

Robust optimization has been used widely in power systemscheduling problems [23]–[28]. In robust optimization, thevariation ranges of uncertain parameter, known as confidenceinterval [29], is defined assuming that after the fact parameterfalls into this interval. Then, the worst case of uncertaintiesoccurrence inside of the interval is evaluated; and, the decisionvariables are determined base on this worst case analysis [25].

In this paper, we propose a robust-based self-schedulingmodel for a price-maker ES facility, in which the uncer-tainty associated with the PQCs is modeled using robustoptimization. In the case of modeling uncertainty of hourlyPQCs in (3), a confidence interval is considered for thecurve. Publicly available data of electricity markets alongwith different forecasting method can be used to estimatethe confidence interval of the PQCs for upcoming hours.Compared to [17], [18], where scenario-based methods areused to model the uncertainty of GPQCs for a generationcompany, robust optimization, which is used in this paper tomodel the uncertainty of GPQCs and DPQCs for a ES facility,has less computational burden.

The daily optimization scheduling problem of (3) can berewritten as (11), by including a robust-based model of thePQCs.

maxΩ

minπdt,s,π

ct,s

T∑t=1

[ ndt∑

s=1

πdt,s(bdt,s + xdt,sq

d,mint,s )

−nct∑

s′=1

πct,s′(bct,s′ + xct,s′q

c,mint,s′ )−OCt

](11)

Subject to:(2) and (4)− (9)

πdt,s ≤ πdt,s ≤ πdt,s (12)

πct,s ≤ πct,s ≤ πct,s (13)

πEt =

ndt∑

s=1

πdt,s × xdt,s +

nct∑

s′=1

πct,s × xct,s′ ∀t ∈ T (14)

This is a max-min optimization problem. The profit ismaximized with respect to the decision variable to find theoptimal scheduling. The profit is minimized with respect tothe uncertain parameters, πdt,s, π

ct,s to find the worse case of

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Fig. 3: An example of worst case for a confidence interval of a)GPQC, b) DPQC

πdt,s, πct,s within the interval of [πdt,s, π

dt,s] and [πct,s, π

ct,s].

Note that, the intervals are defined such that the decreasing(increasing) nature of the GPQCs (DPQCs) are retained. Forexample, one way is the upper and lower bound of the intervalto be proportional to the forecast (e.g. 20% above and belowthe forecasted PQCs).

In order to solve the max-min optimization problem of(11) with commercial solvers, it should be converted to amaximization problem. In (11), the optimization problem islinear with respect to the uncertain variables. Hence, theworst case of these variables would occur at the beginningor end of the intervals. During discharging, the worst caseof price happens in the lower bound of GPQCs interval,i.e., πdt,s = πdt,s. Conversely, the worst case of price duringcharging is the upper bound of the DPQCs interval, i.e.,πct,s = πct,s. For example, Fig. 3 shows the worst case fora GPQC and DPQC confidence interval. Therefore, the finalformulation of robust optimization in our problem could beexpressed as follows:

maxΩ

T∑t=1

[ ndt∑

s=1

πdt,s(bdt,s + xdt,sq

d,mint,s )

−nct∑

s′=1

πct,s′(bct,s′ + xct,s′q

c,mint,s′ )−OCt

](15)

Subject to:(2) and (4)− (9)

πEt =

ndt∑

s=1

πdt,s × xdt,s +

nct∑

s′=1

πct,s × xct,s′ ∀t ∈ T (16)

IV. DEVELOPING BIDDING AND OFFERING CURVES

The operator of a ES facility needs to submit hourly offersand bids for selling and purchasing electricity in order tomanage the risk of forecasting errors. In studies focusing onthe bidding strategy of an energy storage facility in electricitymarket [14], [28], [33], [34], only single block hourly bidsand offers are constructed. In day-ahead electricity markets,there is the possibility of submitting multi-step bids andoffers [24]. Thus, the storage operator should submit multi-step bids and offers to the market for purchasing and sellingelectricity considering different levels of risk to incorporateforecasting uncertainties. Moreover, compared to [27], [28],[30], [34], [35], in which the facility is assumed to be a price-taker, we proposed a bidding strategy for a price-maker ESfacility. Based on the proposed robust scheduling, we developa bidding and offering strategy for a price-maker ES facility.The proposed method is in line with the approach used in [24]

to construct offering curve for a price-taker generation unit.However, However, for the case of a price-maker ES facility,this becomes more challenging; since, incorporating GPQCsand DPQCs to model the price impacts of the ES facility,the developed algorithm in [24] needs to be modified for aprice-maker facility and also extended further to construct notonly an offering curve for selling but also a bidding curve forpurchasing electricity for a price-maker energy storage systemconsidering its impact on the price. Moreover, the bidding andoffering curves are interrelated due to the time dependency ofthe charging and discharging of the ES facility and cannot bebuilt independently; In addition to the constraints (22) that thestorage device cannot charge and discharge simultaneously ,the constraints (26) causes the time dependency of charge anddischarge scheduling. This equation dynamically calculates theSOC at time t+1 as a function of current SOC and the currentcharge and discharge schedule. Thus, the scheduling decisionsare timely interdependent.

To construct the bidding and offering curves, a set of robustoptimization problem are solved sequentially. The proposedalgorithm is essentially a sequential process in which the de-veloped robust-based scheduling model is solved sequentiallyto determine the pair of price-quantity of each step of thebidding and offering curves. Thus, the number of sequences isequal to the predefined desired number of steps of the curves.The algorithm is as follows:

1) The confidence interval for the hourly GPQCs andDPQCs are divided into several subintervals, based onthe desired number of steps the operator wants to submitto the market. For example, Figs. 4 and 5 show theconfidence interval and the subintervals for a sampleGPQC and DPQC with four subintervals.

2) Starting from the first subinterval, i.e., subinterval (a), therobust optimization scheduling is solved for each subin-terval sequentially to obtain the hourly pairs of chargingand discharging level and the corresponding prices foreach subinterval k, i.e, (P c,kt , πE,kt ), (P d,kt , πE,kt ). Basi-cally, step k of the curves corresponds to the sequence k ofthe algorithm. In order to have a increasing offering curveand decreasing bidding curve, the following constraintsare added to the robust scheduling when solving theproblem for step k:

P d,k−1t ≤ P d,kt ∀k (17)

P c,k−1t ≤ P c,kt ∀k (18)

πE,kt ≥ πE,k−1t if P d,k−1t ≥ 0 (19)

πE,kt ≤ πE,k−1t if P c,k−1t ≥ 0 (20)

Equations (17) and (18) force the charge/discharge levelof the current step be higher or equal than the previousstep. Since the price is a function of charging or discharg-ing level based on the PQCs, equations (19) and (20)result in an increasing offered price for offering curveand decreasing bid price for bidding curves.The process is repeated until the whole confidence inter-vals are covered.

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Fig. 4: An example of GPQC intervals to build offering problemsand the process of constructing offering curve

Fig. 5: An example of a DPQC intervals to build bidding problemsand the process of constructing bidding curve

3) The robust scheduling is solved considering the upperbound of GPQCs and lower bound of DPQCs as the lastsequence to find the maximum charging and discharginglevels and the corresponding price profile.

4) based on the obtained operation scheduling and corre-sponding price at each sequence, the offering and biddingcurves are constructed. In each step k, a charge/dischargeschedule and the corresponding price profile are deter-mined. For each hour of discharging, the set of prices forall steps and the discharge schedule result in an increasingoffering curve for that hour. For each hour of charging,the charge schedule and the corresponding price result ina decreasing bidding curve. Figures 4 and 5 illustrate theprocess of constructing offering and bidding curves basedon the subintervals of the GPQCs and DPQCs, and theobtained results.

Figure 6 shows the flowchart summarizing the process ofsequentially constructing the steps of bid and offer curves.

V. NUMERICAL RESULTS

For the case study, the compressed air energy storagetechnology is considered as the storage technology, since thistechnology can serve grid-scale applications, on the orderof 100s of MW for tens of hours [36]. Thus, due to itslarge power capacity, it is reasonable to be considered as

Fig. 6: The proposed algorithm for the cosntruction of bidding andoffering curves

a price-maker facility. Note that, the formulation presentedin this paper is general and can be modified based on thestorage technology. The scheduling formulation in this paperis modified for a CAES facility based on the CAES schedulingformulation proposed in our previous study [37]. Numericalsimulations are performed for a CAES facility with ratedcapacities of 100 MW discharging, 60 MW charging and10 hours of full discharge storage capacity. For a CAESfacility, since the charging and discharging components areseparate components, their capacity can be selected differentlydepending on the application [10], [11]. The details of thescheduling formulation, associated operation constraints andefficiency parameters can be found in [37]. The proposedmixed integer linear optimization problem is solved usingCPLEX in GAMS. GAMS and MATLAB c© are used to solvethe model sequentially to construct the offering and biddingcurves and also to perform after-the-fact analysis for sequentialdays. For the case studies, the historical GPQCs and DPQCsof the Alberta electricity market in year 2014 are used. Theprocess of constricting GPQCs and DPQCs based on hourlysupply curves and system demand can be found in [32], [38]

A. Robust scheduling: a demonstrative case

The hourly GPQCs and DPQCs of the Alberta electricitymarket at 18 February 2014 are used as the forecasts tobuild the confidence interval for the PQCs. For the illustrationpurposes, the GPQCs for the hours 17 to 21 are modified toobtain more distinguishable results than those of original ones.As some examples, Fig. 7 shows the DPQCs for hour 1 and6, and the modified GPQCs for hours 17, 18, and 20.

The robust self-scheduling, proposed in section III is solvedfor different length of confidence intervals from 5% to 50%for the GPQCs and DPQCs. For instance, for 20% confidenceinterval, the intervals are considered to be 20% below and

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Fig. 7: (a) DPQCs fpr hours 1 and 6, and (b) GPQCs for hour 17,18, and 20

Fig. 8: The guaranteed profit versus the length of confidence interval

20% above the hourly PQCs of this day. The guaranteed levelof profit versus length of the confidence interval is plottedin Fig. 8. Observe from this figure that the guaranteed levelof profit decreases for wider intervals. A wider confidenceinterval corresponds to a more conservative case in whichhigher range of forecasts error is considered at the cost oflower profit expectation. For example, based on this figure,for 10% confidence interval, the guaranteed profit is $7,946.0;the forecasting error must be less that 10% in order to gain thislevel of profit. The profit level is $4,633.0 for 20% interval;in this case, higher forecasting error up to 20% guarantees alower profit of $4,633.0.

For the case of 20% confidence interval, without loss of gen-erality, the intervals are divided into four subintervals, similarto that shown in Figs. 4 and 5. Then, the proposed methodto construct the bidding and offering curve, solving sequentialrobust scheduling, are applied. Figure 9 shows the obtainedscheduling for each sequence from one to five. The schedulingof the first sequence has the lowest charge/discharge level, asit is corresponded to the worst subinterval, i.e., the largestsubinterval in which the lower bound of the hourly GPQCs isthe lowest among all the subintervals and the upper bound ofthe hourly DPQCs is the highest among all the subintervals.Thus, the price differences between on peak and off peakhours are already the low without storage operation and thesebecome even lower as storage charge and discharge withhigher power due to the impacts on market price based onthe GPQCs and DPQCs. Therefore, the robust schedulingmodel decides to charge and discharge partially in most ofthe operation hours and retain some of the capacity to preventhigh impacts on market prices as defined by the PQCs andmake the arbitrage profitable. For the higher sequences, thecharge/discharge levels increases; since they are a better casecompared to the previous sequence, i.e, a higher sequencehas smaller confidence interval with higher GPQCs boundand lower DPQCs bound, which leads to a higher pricedifference between on peak and off peak hours. Thus, thefacility participate with higher level of power in the market

Fig. 9: scheduling of CAES facility for each sequence correspondedto each subintervals

Fig. 10: Obtained price profile for each sequence corresponded toeach subintervals

and retain less charge/discharge capacity in spite of higherimpact on market prices, since the arbitrage is still profitable.

Figure 10 also shows the corresponding resulting priceprofiles for each sequence. As can be seen in this figure, highersequence leads to higher prices during discharging hours andlower price during charging hours; since the correspondingsubinterval for higher sequence is a better case compared tothe previous one, i.e., smaller confidence interval with higherGPQCs bound and lower DPQCs bound; it leads to a higherprice difference between on peak and off peak hours thanprevious sequences. Moreover, the sequential constraints (17)-(20) ensure to obtain higher discharging prices and lowercharging prices for higher sequences to keep the increasingand decreasing nature of the offering and bidding curves,respectively. In another word, the lower sequence correspondsto a more conservative case where a flatter price profile isexpected, while higher sequence leads to a more volatile priceprofile.

The proposed method in section IV is applied to constructthe bidding and offering curves. Basically, to construct thecurves, the results, presented in Figs. 9 and 10 are used; thepairs of charging power and prices as well as dischargingpower and prices are sorted to build the bidding and offeringcurves. As an example, Figures 11 and 12 show the obtainedbidding curves for hours 1 and 6, and offering curves for hours18 and 20, respectively. In hour 1, as shown in Fig. 9, inthe first sequence, storage unit is charging with 5.6 MW ofpower and the resulted price is $44.8/MWh. Thus, these valuesare submitted as the first step of the bidding curve. The pairof charge power and price for the second and third sequenceis (18.3MWh,$41.4/MWh) and (60MWh,$38.1/MWh), respec-tively. These pairs are corresponded to second and third stepsof curve. Since the charging power for the fourth and fifth

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Fig. 11: Resulting bidding curve for a) hour 1, b) hour 6

Fig. 12: Resulting offering curve for a) hour 18, b) hour 20

sequence is also 60 MW, these two steps are zero. Biddingand offering curves for the other hours are build in the sameway.

B. The impact of uncertainties through a one-year analysis

In order to show the performance of the proposed biddingand offering strategy in managing the risk of forecastinguncertainties, the proposed robust optimization based strategyis compared with the risk neutral strategy. The historical PQCsof the Alberta electricity market during 2014 is used as thedeterministic forecasts. The proposed bidding and offeringstrategy is applied sequentially in a daily basis for the year us-ing the historical GPQCs and DPQCs of the Alberta electricitymarket as the forecasts considering 20% confidence intervalbelow and above the forecasts to find the robust scheduling foreach sub-confidence interval and consequently build the hourlybidding and offering curves. In order to show the impacts ofconsidering uncertainties in the actual gained profit and theninvestigate the performance of the strategy, in each day, afterthe fact analysis is conducted. To do so, the actual GPQCsand DPQCs for that day, i.e, the after the fact GPQCs andDPQCs, are artificially generated by adding up to 30% levelof random error to the forecasted GPQCs and DPQCs. Inother words, based on the forecasts of GPQCs and DPQCs,simulated PQCs are generated, which has up to 30% error.Then, based on the constructed bidding and offering curvesand the simulated GPQCs and DPQCs, the accepted bids andoffers, the market price, and consequently the gained profitis calculated. For the sake of comparison, another case isalso considered, in which the ES facility applies risk-neutralstrategy, i.e., employs forecasted PQCs to the detereministicscheduling model to schedule the facility. After determiningthe deterministic risk-neutral schedule based on the forecasts,the after the fact gained profit based on the actual GPQCs andDPQCs is calculated similar to that of the robust schedulingexplained above. The process is repeated sequentially for alldays of the year 2014. Furthermore, in order to further assessthe performance of the proposed strategy, the potential profitthat could be gained if the perfect forecasts of the GPQCsand DPQCs were available and there was no uncertainty is

Fig. 13: Monthly gained profit for risk-neutral, the proposed strate-gies, and the potential profit gained in case of perfect forecastavailability

calculated. To do so, the actual after the fact GPQCs andDPQCs are used as the input PQCs of the deterministicscheduling model developed in Section II-A and the process isrepeated sequentially in a daily basis to calculate the potentialprofit for each month. Then, the potential monthly gainedprofit is compared with that of robust strategy in the presenceof 30% uncertainty to explore how much of the potential profitcan be captured by each strategy.

Figure 13 presents the monthly gained profit for risk-neutral,the proposed strategies, and the potential profit. It illustratesthat for all months the gained profit of the proposed robust-based strategy is higher than that of risk neutral strategy. Thisfact shows a significantly better performance of the robuststrategy compare to the risk neutral method; It should be notethat there might be some days that the risk neutral strategyleads to a higher profit that the robust strategy. For example,based on the obtained result, during the months of Februaryand March, there are 17 days that the risk-neutral strategyled to a higher profit than that of the proposed strategy. Thereason is that for some days that the forecasting error isquite low during the charging or discharging hours, the robuststrategy might act conservatively while risk-neutral’ strategyis more optimistic; as a result, the robust strategy will losesome profitable opportunities whereas risk-neutral strategyleads to a higher profit. However, in the presence of inevitableforecasting errors, this case does not happen frequently andas presented in Fig. 13, in the long-run, the robust-strategyoutperforms the risk neutral, i,e, higher gained profit. since theproposed multi-step strategy makes the scheduling decisionsimmune against undesirable price variations. At the same time,dividing the confidence interval into subintervals and includingmultiple steps enables the strategy to take advantage of high(low) prices during discharging (charging). This comparisonshows the effect of forecast uncertainties on the company’sprofit, and also the importance of incorporating uncertaintieswhen participating in the electricity market.

Moreover, based on potential profit and the actual profitgained by each strategy illustrated in Fig. 13, table I showsthe percentage of the potential profit captured by each strategyin each month. Based on this table, the proposed robust-basedstrategy is able to capture a 76% up to 95% of the monthlypotential profit. Overall, the strategy captures 81.7% of thetotal potential profit during the year. However, the capturerate is significantly lower for the risk-neutral strategy. This

9

TABLE I: Percentage of the potential profit captured by each strategy in each month of 2014 for the case of 30% forecasting error[%]Strategy Jan. Feb. Mar. Apr. May. Jun. Jul. Aug. Sep. Oct. Nov. Dec. TotalProposedStrategy 83.8 84.1 76.0 83.7 83.1 84.4 81.4 76.8 95.1 82.0 84.6 83.5 81.7

Risk-neutralStrategy 74.1 67.6 57.5 69.5 58.2 70.8 55.7 56.9 71.7 59.7 73.3 64.5 61.8

Fig. 14: Percentage of the potential profit captured by each strategyin 2014 for different levels of forecasting error

rate ranges between 55.7% and 74.1%. Overall, 61.8% of thepotential profit could be gained by the risk-neutral strategy,which is 20% lower than that of proposed strategy.

C. Impact of Different level of Forecasting Error

In order to investigate the impact of forecasting errors onthe performance of the proposed strategy in comparison withthe risk-neutral strategy, the one year analysis is conductedfor the forecasting error of 10% and 20% in addition to the30% forecasting error. Figure 14 shows the percentage of thepotential profit captured by these two strategies for these cases.As illustrated in this figure, for all the cases, the proposedstrategy outperforms the risk-neutral due to considering 20%forecasting uncertainty and incorporating different level ofrisk in the robust-based strategy. Moreover, for the higherlevel of forecasting error, lower amount of the potential profitis captured by the both strategies. However, the risk-neutralstrategy experiences higher rate of decrease in the capturedprofit from 85.65% to 61.8%. This is because it ignores theuncertainty associated with the forecasts, whereas the capturedprofit for the case if 30% compared to the case if 10% errordrops by almost 10% to 81.7%.

VI. CONCLUSION

This paper develops a robust-based bidding and offeringstrategy for a price-maker ES facility in a day-ahead electricitymarket. The impact of ES operation on market prices ismodeled by means of GPQCs and DPQCs. The uncertainty inthe GPQCs and DPQCs is also modeled by robust optimizationin this work. Then, a sequential algorithm is proposed todevelop multi-step bidding and offering curves for the fa-cility to participate in the market. The applicability of thedeveloped strategy is verified in the numerical results. Theone-year analysis shows significantly better performance ofthe proposed strategy compared to the risk-neutral one. Theproposed strategy has the capability of capturing 81.7% of the

potential profit, while this rate is 61.8% for the risk-neutralstrategy.

In this paper, robust optimization is applied to take therisk of forecasting errors into consideration. Depending onthe application and the size of the problem, one might usestochastic optimization instead of robust optimization. Thecomparison of the performance of these two approaches needsto be addressed and is left for the future work. Moreover, forthe proposed robust-based strategy, it is assumed that the con-fidence intervals are pre-defined and are given to the facilityoperator as inputs. A key feature of the proposed method is thatit can tie probabilistic forecasting to optimal scheduling. Theconfidence interval forecasts are basically the outcome of anyprobabilistic forecasting approach. The forecasting approachgenerates a forecasting interval with a certain confidence level.These intervals can be used as the input of our proposedbidding and offering strategy. Furthermore, In our simulations,the level of the confidence intervals were selected arbitrarily.The main purpose of this paper is to develop a bidding andoffering strategy for a large-scale price-maker energy storagefacility participating in energy market. However, determiningthe optimal size of the confidence intervals is not in the scopeof this paper. This is an ongoing project in our research group.

The developed model in this paper is a general model foran energy storage system. For a specific storage technology,the developed formulation can be modified to incorporate itsspecific characteristics (e.g. degradation, operation cost) moreproperly.

APPENDIX

In the following, the operation cost of an ES facility, statedin (1), as well as the associated set of constraints Ψt areexpressed.

OCt = P dt × V OMd + P ct × V OM c ∀t ∈ T (21)

uct + udt ≤ 1 ∀t ∈ T (22)P cmin ≤ P ct ≤ P cmax.uct ∀t ∈ T (23)

P dmin ≤ P dt ≤ P dmax.udt ∀t ∈ T (24)SOCmin ≤ SOCt ≤ SOCmax ∀t ∈ T (25)

SOCt+1 = SOCt +P ctEmax

− P dt × EREmax

∀t ∈ T (26)

SOC1 = SOCinit ∀t ∈ T (27)

The operation cost is expressed in two terms in (21).The first term shows the operation cost during discharging,which is the variable operation and maintenance cost duringdischarging. The second term is the VOM of charging. TheES facility operates in either charging or discharging mode at

10

a time, which is stated in (22). The charging and dischargingpower limits of the ES are specified by (23) and (24). Theminimum and maximum state of charge of the cavern isexpressed in (25). The effect of charging or discharging onthe SOC of the facility for the next hour is stated in (26).The initial level of SOC is specified by (27). Note that,depending on the storage technology, the above formulationcan be modified to represent the characteristics of that storagetechnology.

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