Simulating price-aware electricity storage without linear optimisationK.R. Ward, I. Staffell⁎
Centre for Environmental Policy, Imperial College, London, SW7 1NE, UK
A R T I C L E I N F O
Keywords:Electricity storagePower systems modelElectricity market modelElectricity systemsProfit maximising
A B S T R A C T
Electricity storage could prove essential for highly-renewable power systems, but the ability to model its op-eration and impacts is limited with current techniques. Studies based on historic market prices or other fixedprice time-series are commonplace, but cannot account for the impacts of storage on prices, and thus over-estimate utilisation and profits. Power systems models which minimise total system cost cannot model theeconomic dispatch of storage based on market prices, and thus cannot consider large aggregators of storagedevices who are not perfectly competitive.
We demonstrate new algorithms which calculate the profit-maximising dispatch of storage accounting for itsprice effects, using simple functional programming. These are technology agnostic, and can consider short-termbattery storage through to inter-seasonal chemical storage (e.g. power-to-gas). The models consider both com-petitive and monopolistic operators, and require 1–10 s to dispatch GWs of storage over one year.
Using a case study of the British power system, we show that failure to model price effects leads to materialerrors in profits and utilisation with capacities above 100 MW in a ∼50 G W system. We simulate up to 10 GW ofstorage, showing dramatically different outcomes based on ownership. Compared to a perfectly competitivemarket, a monopolistic owner would restrict storage utilisation by 30% to increase profits by 85%, thus reducingits benefit to society via smoothing demand and output from intermittent renewables by 20%.
1. Introduction
“Energy storage is like bacon: It makes everything better” [1]. Itoffers the potential of ‘baseload renewables’ by managing their inter-mittent output, and of hyper-flexible large thermal generators so thateven nuclear reactors could match variable demand. This could re-volutionise grid management, facilitate deeper decarbonisation andsignificantly reduce the requirement for fossil fuels to provide flex-ibility. Electricity storage is therefore considered one of the most im-portant issues within the energy industry [2], with “the potential todictate the pace and the scale of the energy transition”. It is one of thenecessary foundations for clean energy according to the Global ApolloProgramme Report [3] and Bill Gates’s Breakthrough Energy deem low-cost storage to be “transformational” [4].
Realising the potential of storage requires continued technologicaldevelopment and cost reductions, and for sufficient revenue to existfrom providing bulk energy arbitrage to justify the large-scale invest-ments required. As with any emergent, disruptive technology, themodes in which storage will be operated, their effects on the widerelectricity system and their potential profitability are all uncertain. Itfalls to the modelling community to offer quantitative insights into
these issues and the wider implications for the electricity and energysectors.
The main purpose of this paper is to present a new method for in-cluding bulk electrical energy storage (EES) in electricity marketmodels without the need for an optimisation framework. The presentedalgorithm derives an optimal dispatch schedule that maximises profitsfor storage owners taking account of price-effects; that is, it includes theimpact that deploying large amounts of storage has on system price,due to the dispatch of other generators. The approach can be appliedwithin any market model formulation that produces a time-series ofwholesale prices. In this paper, the algorithm is coupled with a simplemerit order stack (MOS) pricing model, resulting in extremely fastcalculation times, allowing rapid testing of the storage algorithm acrossa wide scenario-space. We illustrate our description of this new ap-proach with a demonstration of the model on the British power system.
We present two variants of the algorithm, corresponding to twoextremes of storage market behaviour: perfect competition (i.e. anatomistic market comprising many small merchant storage operators)and perfect monopoly (i.e. a market in which a single large utilityowner or technology aggregator can exert market power). In the case ofperfect competition the optimised dispatch maximises the utilisation of
https://doi.org/10.1016/j.est.2018.08.022Received 14 December 2017; Received in revised form 31 July 2018; Accepted 28 August 2018
available storage and thereby achieving also maximum smoothing ofnational net demand and societal benefit. In the case of a marketmonopoly, the optimised dispatch maximises the operator’s profits byrestricting the utilisation of storage.
The remainder of the paper is set out as follows. Section 2 providesbackground on how the impact of storage on prices and how storagedispatch is modelled. Section 3 describes the algorithms we develop andthe simple electricity market model we use to demonstrate them. Theresults in Section 4 explore the differences between the algorithms interms of storage owners’ profits, storage utilisation, dispatch patternsand influence on smoothing demand; and explore the trade-off betweenspeed and accuracy that can be obtained with the algorithms. Section 5discusses the implications for energy systems modellers and policy-makers, and concludes.
In the interests of promoting transparent and reproducible science,the storage algorithms and their implementation in an exemplaryelectricity market model are released open source, in the form of VisualBasic code and an Excel spreadsheet model. Their generic nature shouldallow them to be easily reinterpreted into other languages.
2. Background
Storage of electrical energy has many potential revenue sources[5–7]:
• earning profits either from arbitrage or in the ancillary markets;• integration with existing infrastructure to reduce balancing costs;• time-shifting delivery or managing constraints for demand centres
(to reduce network service charges and peak demands); or• deferring costly upgrades to transmission and distribution systems.
The most commonly studied revenue source is arbitrage, whichunlike the other sources mentioned, exists and may be quantified solelythrough electricity price spreads within the market. Other revenuesources are linked to the precise set-up and compensation policieswithin specific markets. In this work we focus exclusively on arbitragerevenues.
A storage device discharging at peak time reduces the need forgeneration from the most expensive generators on the system, poten-tially reducing prices. Conversely, when storage is recharged, systemdemand is increased and prices should rise. In reality, this narrowing ofthe price differential means that the revenue from selling stored
electricity is lower, the cost of recharging is higher, and thus the profitsfrom arbitrage are lower than would be expected based on the pricesthat are observed without that storage. These so-called ‘price effects’become significant as the amount of storage within a system grows [8].
It is common within the storage literature to model the profits ofstorage based on a fixed time-series of prices, either emerging fromhistoric markets or based on future simulations [9–13]. Such studiescommonly refer to situations in which price effects are neglected as thestorage being a ‘price-taker’. This is incorrect and may lead to confu-sion. To economists, the price-taker/price-maker terminology refers tothe behaviour of individual firms within a market and their ability (orotherwise) to influence prices due to the levels of competition with themarket. Specifically, a price-taker is a storage operator which is toosmall to move prices by itself, but still takes account of the effect thatthe overall fleet of storage devices has on prices. It would be moreaccurate to refer to storage dispatch models in which price effects areneglected as having exogenous pricing (i.e. prices are determined ex-ternally), and those that include price effects as having endogenouspricing (i.e. storage is part of the price formation process).
As we discuss both price-effects and competition in this paper, werefer to exogenous pricing (commonly referred to as price-taker) as a‘fixed-price’ approach; endogenous pricing with price-taker firms as a‘competitive’ approach; and endogenous pricing with price-maker firmsas a ‘monopolistic’ approach.
2.1. The price effects of dispatching storage
The issue of price effects only becomes important once the amountof storage capacity within a market is sufficient to cause significantchanges in price. Fig. 1 demonstrates this for a simple case with linearelectricity supply curve (diagonal line in panel 1a and 1b). Thick arrowsshow the impact of storage: in panel a, charge and discharge have noeffect on prices while in panel b the sloping supply curve is taken intoaccount. The difference between the area of the solid red and solidgreen bars signifies the profit made from arbitrage, the hatched areas inpanel b show the profit that is lost by storage influencing prices. Thiscreates the difference between the realised profits shown by the twolines in panel c.
The profits shown in Fig. 1a and in the exogenous line of 1c cannotbe obtained in the real world, but these are what a naive fixed-pricealgorithm would anticipate as greater amounts of storage are dis-patched. The economic reality is shown in Fig. 1b and the blue line of
Fig. 1. Schematic of the impact storage dispatch has on electricity prices and demand and thus on arbitrage revenue. Panels a and b show electricity price against thepower dispatched from storage charging (short bar) and discharging (tall bar). Q(Tmin) and Q(Tmax) denote the electricity supplied in the absence of storage at twotimes, Tmin and Tmax, while Q(Tmin)’ and Q(Tmax)’ denote how the supply from non-storage sources alters once storage is included. Panel c visualises how profits varywith the power supplied from storage, where ΔQ = Q(Tmax) – Q(Tmin). A negative profit indicates that more money was spent charging the storage device than wasobtained discharging it – i.e. a loss was made.
K.R. Ward, I. Staffell Journal of Energy Storage 20 (2018) 78–91
79
panel c: the fact that prices are pulled together reduces the realisableprofits, to the point where the storage device could lose money if itscharging pushed prices up above the price when it discharges (whichalso get pulled down by discharging). Storage that is incorporated intooptimisation models would cease to operate in this situation andbottom out at zero profit, but dispatch based on fixed prices would not.
If we consider the situation where we charge our storage deviceduring time period Tmin when prices are lowest, and discharge it whenprices are highest, during period Tmax, we can take two approaches tocalculating the profit that we earn. The first, shown in Fig. 1a neglectsthe effect that our operation has on system demand and price. Theprofit, Π (in £), is given by Eq. (1):
==
SS m
(P P ) t(Q Q ) t
T T
T T
max min
max min (1)
Where S is the power output from the 100% efficient storage device (inMW); P is the system price (in £/MWh); Q is the system demand (inMW); m is the gradient of the supply curve (in £/MWh per MW) and Δtis the length of time intervals Tmin and Tmax over which charging/dis-charging has occurred (½ hour in the British market). This shows thatprofit increases linearly with the power dispatched from storage, asindicated by the red line in Fig. 1c.
Fig. 1b accounts for the fact that operating the storage device leadsto a drop in prices during Tmax and an increase in prices during Tmin. Inthis case, we find that the profit is given by Eq. (2):
==
S P P tS m S t
( ´ ´ )(Q Q 2 )
T T
T T
max min
max min (2)
In which P’ represents the altered system prices. The profit varieswith S2, attaining a maximum at S equal to a quarter of the differencebetween the original demand levels during Tmin and Tmax, as shown bythe blue line in Fig. 1c. Once the price changes are accounted for, it canbe seen that increasing the amount of power dispatched from storagedestroys the price differential on which arbitrage profits are founded.For this case of a linear supply curve and simple transfer of energybetween only 2 time periods, ignoring price effects over-estimates theprofit by over 10% if the power dispatched from storage is 5% of thedemand difference.
2.2. Modelling arbitrage revenue from electrical storage
Interest in calculating and modelling the profits that accrue to theowners of electrical storage was initially driven by the desire to un-derstand the economically optimal operation of hydro. The arbitrageprofit that storage devices might earn in existing markets is commonlyestimated using linear optimisation techniques combined with time-series of historic market prices [9,10,14–16]. From the arguments setout in the previous section, these can only comment on the profitabilityof a small amount of storage capacity installed in a system. If storagedeployment becomes significant, their results no longer hold.
While much of the literature is concerned with describing resultsobtained by these historical analysis methods, a number of alternativeapproaches have also been described. For instance, Barbour et al. sug-gest a Monte-Carlo method for obtaining the upper bound for the profitavailable to a storage device; and Lund et al. describe an algorithm usedin the EnergyPLAN model to value the arbitrage earnings available to acompressed air storage device. This latter algorithm has been exploitedas a stand-alone approach to storage profit-optimisation, most notablyby Connolly [14], and also in other studies (e.g. [6,17]). This is theapproach we adopt and modify in this paper, and it is described furtherin Section 3.1.
In order to consider the possible future usage and profitability ofstorage, it is necessary to employ electricity market models; mostcommonly these are unit commitment (UC) models. Broadly, a unitcommitment problem seeks the optimal dispatch schedule for power
generation units within a particular generating system, subject to con-straints (e.g. ramp times, power outputs, transmission, capacity re-serves) and costs (e.g. fuel, start up and shutdown) [18]. There arecommonly three different types of UC model, which correspond to threedifferent optimisation objectives: cost-based UC models seek the sche-dule that delivers the minimum system cost; price-based UC (PBUC)models deliver the schedule that gives maximum profit to market par-ticipants; and security-constrained UC models prioritise security ofsupply [19].
The first and third of these are typically of most relevance to si-tuations in which there is centralised dispatch and an independentsystem operator; prices and the profitability of individual market par-ticipants are of secondary importance to the optimality of the system asa whole [20]. A treatment of bulk EES is included in most of thesemodels (see for a review). This framework is able to provide usefulinsights into the quantities of storage that might be optimal from asystem-wide perspective and the ways in which storage might be cen-trally deployed to reduce overall systems costs [21–24]. However, itoffers little insight into how an independent storage operator mightschedule its bids to operate. Shahidehpour et al. write “it is wrong toassume that maximizing the profit is essentially the same as minimizingthe cost” [19].
On the other hand, PBUC models approach the scheduling from theperspective of the generating company (or storage owner) who isseeking to plan the dispatch of their plant in order to maximise theirprofits. For this type of problem, the key determinant of whether a unitis scheduled to run is the anticipated system price at a given time [19].The PBUC problem is extremely complex, with a significant literaturededicated to the presentation of solution methodologies (see [18] for abrief summary). The addition of storage and the feedback from priceeffects, introduces non-linearity to the system, and the resulting pro-blems, where not intractable, are typically computationally demandingand slow to run [8]. This limits resulting studies to short time-spansand/or small parameter sets, which ultimately restricts their usefulnessas a tool for investigating the long-term business case for storage. A fewauthors have implemented PBUC approaches that approximate theprice effect, for example by assuming a linear supply curve [9,25],using published market data to produce a more authentic price response[8,26], or through residual inverse demand functions [13]. As Brijset al. [8] point out however, the results obtained in the majority ofthese studies produce non-optimal deployment strategies and profits,and result in an ex-post discrepancy between expected and realisedprofits/prices. The conclusion is that, while it is conceptually simple tounderstand the origin and significance of price effects, it is considerablymore difficult to incorporate them fully into structural market model-ling approaches.
At the other end of the electricity market modelling spectrum,economists use game theoretic approaches to consider the behaviour ofstorage operators in markets with different levels of competition (e.g.[27–29]). One finding amongst this literature is an incentive for mer-chant storage operators to restrict utilisation in order to maximiseprofits. There are also idealised and highly theoretical treatments of thisissue [30].
3. Methods
We present a suite of algorithms to generate optimal dispatchschedules for a storage device given a time-series of prices. Advancingon previous work we show how, by linking the storage dispatch modelto a price-generating model, it is possible to take into consideration thefeedback between storage deployment and system price.
The combined model is implemented in Microsoft Excel with VisualBasic, and solves in 1–10 s per 8,760 h on a 2015-era laptop. This ap-proach is orders of magnitude faster than the unit commitment modelscited above, enabling wide sensitivity analyses and Monte-Carlo simu-lations, and opening up the possibility of probing the multi-annual/
K.R. Ward, I. Staffell Journal of Energy Storage 20 (2018) 78–91
80
decadal impacts of storage. The coherent inclusion of price effects intostorage operating decisions offered by this approach gives an alter-native to the prevailing methods, and potentially sheds new light oninvestigations into scenarios involving markets in which there are sig-nificant quantities of storage.
Common to all the algorithms, the storage device is described by thefollowing parameters:
• maximum power output to the grid;• roundtrip efficiency of the charge-discharge cycle;• Energy to power (E/P) ratio – i.e. the number of hours for which it
can supply electricity at its maximum power rating (the inverse of c-rate);
• marginal costs associated with the charging and discharging pro-cesses.
A further model parameter, the precision (in MW), controls theprocess of dispatching storage and influences the accuracy of the finalanswer and the length of time the model takes to run.
We do not include:
• ramp rate limitations on generators or storage units;• device lifetime and the impacts that charge-discharge cycles may
have on this;• leakage of energy from the charged device (self-discharge);• the impacts of lack of ‘perfect foresight’.
Ramp rates are excluded because we integrate these algorithms withan electricity market model (described later in Section 3.2) with 30 mintime step resolution, whereas the ramp rates for electrochemical de-vices are typically < 1 s and thus superior to other power systemcomponents. Lifetime and self-discharge will primarily impact invest-ment decisions, but can also influence operating decisions if the cycle-life or state-of-charge have a binding impact on the overall systemlifetime. With continued research into storage durability these effectsare becoming less important. Pumped hydro and compressed air de-monstrate 50+ year lifetimes and negligible discharge, whilst lithiumion batteries are rapidly approaching 10,000 cycle life with greatertolerance to deep discharge [31–33]. These effects could be included inour operating decisions (non-optimally) through the use of heuristics,or could easily be studied as outputs from the model.
It is common to assume that the device owner has perfect foresight
of system electricity prices. The effect of this assumption is that thearbitrage values obtained will always represent an upper limit to whatmight be achieved in the real world, where inaccurate forecastingwould necessarily lead to less-than-optimal dispatch. Several authorshave investigated the size of the reduction in profit that might be ex-pected to arise. Estimates [9,10,14] suggest that forecasting errors re-duce expected revenues by 10–20% from those obtainable with perfectforesight. Results presented in [30] indicate that the reductions becomemore significant as the maximum discharge time of the battery in-creases.
Finally, as these algorithms do not employ an optimisation solversuch as CPLEX, they cannot be guaranteed to find the global optimumsolution. The mathematical logic presented in the following sectionsshows they will tend towards maximising arbitrage profits. However,when dispatching over several thousand periods, a large population ofpotential solutions develops as the constraints on battery charge levelsrestrict the available solution space. Different choices of step size andefficiency invoke these constraints at different time periods leading thealgorithm to find different local maxima dependent on model para-meter choices. However, from the results presented in Section 4.3 weestimate the variability among these maxima to be around ± 0.15%,and thus they do not significantly impact on results.
This section begins by introducing the fixed-price algorithm of Lundand Connolly in Section 3.1. Section 3.2 then describes two approachesto the optimisation of storage dispatch, taking account of price effects.Section 3.3 outlines the basis of the MOS model used to create thesupply curves.
Here we describe the basic algorithm that underpins the othermethods used within this paper. Given a time series of prices, it returnsthe dispatch schedule that will optimise the profits of a storage device.It was first proposed by [34] and further examined by [14]. A detaileddescription of its workings may be found both within these referencesand a worked example is presented in [6]. Here we present a briefsummary of the methodology. Fig. 2 presents a simplified flow diagramfor this basic algorithm.
The algorithm develops a deployment schedule within the limits ofthe overall energy storage capacity of the battery, its power input andoutput characteristics and its overall efficiency. It calculates the changein charge level during each period (dC/dT), subject to constraints on the
Fig. 2. Flow diagram for iteration of the exogenous routine. Other than using the demand as the optimising variable, this routine is identical to that used by Connolly[14].
K.R. Ward, I. Staffell Journal of Energy Storage 20 (2018) 78–91
81
maximum charge and discharge rates, and the maximum and minimumamount of energy that can be stored.
At the heart of the algorithm is the idea of pairing time periods tocreate notional charge-discharge pairs. The original algorithm usessystem price to choose periods for charging or discharging. In the workpresented here, we base the dispatch of the device on the residualsystem demand (that being the demand that is met by dispatchable,thermal plant). The period of maximum demand (Tmax) is identified,then a period of minimum demand (Tmin) is located that permits thedevice to recover the energy discharged during Tmax without violatingthe physical constraints of the system (i.e. we neither exceed themaximum storage capacity of the battery nor drain it beyond zero en-ergy content). These constraints mean the demand during Tmin mightnot be the global minimum demand. Once the final schedule of dispatchhas been determined, the profit is calculated as being the total revenuefrom discharging minus the total cost of charging, minus any marginalcosts of operation (which we have left as zero in our analysis).
This is shown schematically in Fig. 3. For the purposes of this il-lustration we consider a notional storage device with 100% efficiency,maximum power in/output of S and infinite energy storage capacity. Ashort time series of system demands (Fig. 3a) is placed into charge-discharge pairings. The curves in Fig. 3b show schematically how theprofit derived from each pairing would evolve with the power suppliedby storage. The larger the demand range of the charge-dischargepairing, the greater the price difference will be and therefore thegreater the potential profit. Since we are ignoring price effects, differ-ence in prices is maintained regardless of how much storage we use.Hence profits increase linearly with storage power (as indicated inFig. 1) with the gradient determined by the range of demand of thepairing. The maximum profit will be derived from utilising the fullpower capacity (also termed power capability) of the device, as denotedby the yellow diamonds. The difference in profits realised betweenthese three algorithms is explored further in the Appendix section 8.1.
3.2. Modelling price effects: competitive and monopolistic algorithms
As was indicated in Fig. 1c, the assumption of profits increasinglinearly with dispatched storage is incorrect. In Fig. 3c we includeconsideration of a linear supply curve. If we look at the actual profitsthat would be earned by the fixed-price algorithm under this assump-tion (yellow diamonds) we see that pairing 1 still delivers a profit:demand/price at T1 is still greater than demand/price at T2. For pairing2 the dispatch of storage equalises demand at T3 and T4, therefore thisoperation is cost neutral. For pairing 3, the use of storage inverts theordering of demands in T1 and T6. The fixed-price algorithm predicteda profit but, in fact, a loss was made (negative profit).
The blue and red circles in Fig. 3c illustrate two alternative ap-proaches that may be taken to choosing how to deploy storage. The bluecircles dispatch as much storage as possible without incurring losses: itsbehaviour is identical to the fixed-price algorithm for pairings 1 and 2.For pairing 3 however, it stops short of full deployment. The red circlesdispatch storage up to the point at which further usage starts to reduceprofits. In Sections 3.2.1 and 3.2.2 we describe how these two ap-proaches have been implemented using a MOS model to deliver thesupply curve. The supplementary material contains a further shortanalysis of this straight-line supply curve scenario, and details of afurther approach that is not included in the main paper.
The overall effect of dispatching storage via the blue circles methodis that (for 100% efficient storage1) we maximise the levelling of thedemand profile. In a real system this should mean reduced
requirements for peaking plant and fewer starts and stops and part-loading for low merit order plant (both of which lead to reducedcomponent lifetimes and reduced efficiencies); i.e. maximal systembenefit. However, reducing the demand/price spread also reduces ar-bitrage profit; as a storage operator we have maximised the benefit thatthe system may derive from our device, but in doing so we have de-stroyed our profits. As the red circle method shows, however, by doingless with the storage, we can make more money.
In practice the situation in which the red circle method might bedeployed would be limited to those in which storage within the system
Fig. 3. Variation of storage-owner profits with varying dispatch amounts for anotional set of demand across six periods. a) shows the charge-discharge pair-ings; b) shows how profits evolve for each pairing under a fixed-price as-sumption with diamonds indicating the storage power that the fixed-price al-gorithm would dispatch; c) shows how profits evolve assuming a linear supplycurve, with symbols indicating the amount of storage that would be dispatchedfor the three approaches we consider.
1 As the efficiency of the storage device falls, energy losses mean that theextent to which storage may draw together peaks and troughs in the demanddistribution without becoming unprofitable is reduced. In recovering the lossesdue to energy inefficiency, the overall system energy increases.
K.R. Ward, I. Staffell Journal of Energy Storage 20 (2018) 78–91
82
was owned by a single or small number of merchant operators. Hencewe refer to this as the monopolistic algorithm. The blue circles representthe more likely scenario, where storage ownership is distributed over alarge number of operators, and corresponds to a competitive storagemarket. This we refer to as the competitive algorithm.
3.2.1. Competitive dispatch: optimal smoothingNoting that dispatch of a small amount of storage by the fixed-price
algorithm does not lead to significant errors in estimating the profits, asimple workaround the issue of price-effects is to iterate use of thefixed-price algorithm. By dispatching the storage in small incrementalsteps (of size Δ) and recalculating prices after each iteration we canapproximate the price effects while ultimately deploying significantamounts of storage capacity. For example, 1 MW of storage may bemodelled using the initial prices from a power system model, then theoutput of this storage incorporated into the power model (e.g. nettingthe storage output from demand), which recalculates prices that arethen used to dispatch the second MW of storage. With a small enoughstep size, the errors in dispatch (i.e. charge/discharge pairs that yieldnegative profit due to their impact on prices) will be negligible.
We implement this approach using the MOS as a quick way to re-calculate prices. A flow diagram of the process is shown in Fig. 4.
This ‘breadth-first’ approach to maximising the smoothing effect canbe contrasted to a ‘depth-first’ approach, which where individualcharge-discharge pairs are selected in turn based on their price-spread.This is described further in the Appendix section 8.2, and was found toyield comparable results at the expense of much greater computationaltime.
3.2.2. Monopolistic dispatch: optimal profitsIn the both exogenous, fixed-price algorithm and the competitive
dispatch algorithm, charge-discharge time period pairings are foundand then the entire technically feasible storage power capacity is
dispatched. By contrast, the monopolistic algorithm selects the charge-discharge pairing (as per the exogenous algorithm) and then searchesthe entire range of possible deployment amounts in steps of size δ, re-calculating the price, via the MOS, and the profits earned each time.The dispatch power that yields the maximum obtainable profit is re-tained. The device is charged and discharged by this profit-maximisingamount in the final schedule, with minimum and maximum time per-iods being removed from further consideration if the maximum chargeor discharge power of the device has been reached. Successive pairingsare identified in the same manner as in the fixed-price algorithm, ac-counting for constraints imposed by charge/discharge rates and thestorage capacity of the device.
We distinguish between the step size Δ used in the competitive al-gorithms and δ in the monopolistic algorithm as they have differentmeanings: the competitive algorithms increase dispatch by Δ and stoponce a condition is reached; the monopolistic algorithm samples allpossible values of dispatch with a resolution of δ and selects the mostprofitable. A flow diagram is presented in Fig. 5.
3.3. Merit order stack model
In order to consider price effects, we couple the above algorithm(and variants thereof) with a MOS model programmed in MicrosoftExcel. The MOS minimises the cost of deploying a set of generators withthe simplification of neglecting inter-temporal constraints on generatoroperation [35,36]. This means each plant’s operating costs are pro-portional to its output with no consideration of start-up costs, andlimitations such as ramping rates and stop/start time are neglected. Theadvantages are that wholesale electricity prices can be obtainedwithout ambiguity as the variable cost of the marginal unit, and thesimplicity allows for model calculation times to be in the order of
Fig. 4. Flow diagram for iterative dispatch of the fixed-price algorithm, i.e. thebreadth-first competitive algorithm. The user specifies the size of dispatch in-crement (Δ); the algorithm then iterates between the process of dispatching thisstorage capacity and referring to the MOS model to recalculate electricityprices. The process is halted once all available storage capacity has been used.The coloured boxes are expanded in Fig. 2.
Fig. 5. Flow diagram for the monopolist algorithm. The coloured boxes areexpanded in Fig. 2.
K.R. Ward, I. Staffell Journal of Energy Storage 20 (2018) 78–91
83
milliseconds to seconds when considering a year at hourly resolutionwith tens or hundreds of plants [37].
We use a MOS model with price responsive demand, as described in[37–39]. Demand is assumed to fall with increasing price to modelprice-sensitive loads within industry and other forms of demand re-sponse. Electricity is priced either at the variable cost of the marginalgenerator, or at the level needed to constrain demand to be met by theavailable battery power. We assume that a price rise of £100/MWh isrequired to persuade 1 GW of demand to disconnect [37].
Only four technologies are modelled (nuclear, CCGT, coal andpeaking OCGT), but these are split into 5, 15, 15 and 5 tranches re-spectively which have a range of efficiencies and opportunity costs, andthus bid different prices into the market. The resulting supply curve isshown in Fig. 6.
Biomass, hydro and imports are neglected for the sake of simplicity(as the focus of this article is on modelling storage), but for referencethese supplied 9% of British supply over the last five years [40]. Windand solar output are based on historical data from, and are netted offdemand before it is provided to the model. The nuclear fleet is con-strained to run at a minimum of 90% load to match historic precedencein Britain. It is assumed that renewable power is spilled in preference ofturning down nuclear reactors, in which case the market price falls to–£50/MWh, in line with the cost of lost subsidies to wind generators
Fig. 6. Supply curve for the British electricity system in Q4 2016 assumed in themarket model.
Fig. 7. Exemplary results from the three storage algorithms, applied to a half-hourly simulation of the fourth quarter of the GB electricity system (4416 periods).Storage power was increased in 200 MW steps, with a unit size (precision) of 200 MW.
K.R. Ward, I. Staffell Journal of Energy Storage 20 (2018) 78–91
84
[37].This model was populated with historic UK data from the fourth
quarter of 2016, including demand, renewable output, day-ahead prices(required to calibrate the price-sensitivity of demand), and the cost ofcoal, gas and CO2 emissions. All data were specified at half-hourly re-solution, giving 4416 time periods.
4. Results
The output from the fixed-price, competitive and monopolistic al-gorithms were compared across a range of storage power capabilitiesusing a notional storage device with characteristics that approximate toa lithium ion battery: 95% round-trip efficiency and energy capacity tooutput at full power for 4 h (i.e. an energy to power ratio of 4). TheMOS model generates prices using market characteristics that areconsistent with the UK market in the fourth quarter of 2016.
Section 0 presents some key results for the different algorithmsbased on three months of operation in the British electricity system,highlighting the influence of storage power dispatched and operatingstrategy. Section 4.2 presents some sample dispatch schedules for thethree methods. Section 4.3 then examines the performance of the al-gorithms, and the available trade-off between speed and accuracy thatis available through altering the dispatch precision.
4.1. Influence of storage deployment strategy
Fig. 7 shows the quarterly profits earned by the storage operatorfrom the different dispatch approaches, the mean quarterly deviceutilisation, the model run time and the standard deviation in the re-sultant net demand following deployment of the device according toour schedule. This last metric is used as an indicator of the smoothingimpact of storage on system demand.
4.1.1. Storage profitFig. 7a shows that for small amounts of storage all three algorithms
deliver the same linearly increasing profit as maximum storage poweroutput is increased. The fixed-price algorithm, coupled with our elec-tricity market model only yields reasonable results (< 5% error onprofit and utilisation) for power capabilities up to about 400 MW in theBritish electricity system, which averages 34 GW of demand [35]. Be-yond this point, the profits generated by the fixed-price algorithm startto diverge from the two price-maker algorithms, as the failure to ac-count for the reduction in price spread starts to take effect. As expected,the profits from the fixed-price algorithm peak and then rapidly fallaway and, when simulating the 2.8 GW of storage present in the Britishsystem, this algorithm over-estimates arbitrage revenue by a factor of2.4. The amount of quarterly profit that may be extracted in the com-petition scenario peaks at around £33.5 M, achieved using 5.6 GW ofstorage. Beyond this point, the dispatch of additional storage causes theprofits to fall and eventually they are destroyed entirely. This requiresaround 256 GWh of storage – enough to flatten demand across theentire period – which is obviously an extreme proposition. In themonopolistic case, the more storage we are able to dispatch, the moreprofit we make. The marginal increase in profit drops as the amount ofstorage capacity available for dispatch increases, with profit reachingan asymptotic maximum value of £85 M only once 200 GW h of storageis installed (which is also outside the realms of rationality).
Fig. 8 shows how the normalised profit diminishes as storage powerincreases, albeit more slowly for the price-aware algorithms than forthe fixed-price dispatch. The absolute levels of profit suggest that sto-rage technologies such as lithium ion batteries have the potential togenerate reasonable financial returns from arbitrage, provided thattheir up-front costs come down sufficiently. For example, if the in-vestment price of stationary storage systems can reach £200/kWh for astorage system with an E/P of 4 [40], then Fig. 8 suggests up to 3 GW ofstorage could attain a 5% annual return on investment (based on
earning ∼£40/kW/year).This suggests a ‘target price’ for storage technologies of £200/kWh
(∼$300/kWh) to achieve commercial viability for pure energy arbit-rage (without providing other services or benefit stacking). For context,residential-scale storage systems currently average ∼$1,200/kWh[40,41] whilst the price-leader, the Tesla PowerWall 2 costs around$500/kWh [40,41]. With current rates of price reduction, several bat-tery technologies could attain this cost within the next 20 years [40]. Asthis study is based on a single quarter of data using stylised storageparameters, further analysis is required to properly assess the tech-nology- and situation-specific value of storage.
4.1.2. Storage utilisationFor the fixed-price dispatch schedule, the utilisation of the storage
device (Fig. 7b) does not alter as the available capacity of storage in-creases. The schedule calculated by the algorithm depends solely on theoriginal pattern of demand and is maintained regardless of the capacitydispatched. As is to be expected, the utilisation of a monopolisticallydispatched storage device falls as its size is increased. The extra powercapacity is used to squeeze out small additional amounts of profit attimes where battery capacity previously curtailed operation, ratherthan being deployed to increase the smoothness of the residual systemdemand. The competitive algorithm achieves higher levels of deviceutilisation than the monopolistic one, across all sizes of storage device,with the discrepancy growing as the installed storage capacity in-creases.
4.1.3. Net demand variabilityFig. 7c shows that there is no difference in the behaviour of the
three algorithms up to 2 GW of installed storage capacity: they all re-duce the standard deviation of the residual system demand. Once thesystem contains about 5 GW of storage capacity, the neglect of priceeffects by the fixed-price algorithm begins to convert the troughs inprice/demand into peaks and vice versa, and volatility starts to growagain. For the price-maker algorithms the variability falls with in-creasing storage amounts, with the competitive algorithm generatingthe smoothest residual profile, as can be seen in the sample systemdemand plots in Fig. 9.
4.1.4. Computation timesThe fixed-price algorithm shows no variation in run time with in-
stalled storage capacity (Fig. 7d). Fundamentally, the calculation isidentical regardless of how much storage is present: the schedule
Fig. 8. Normalised profits earned by storage per unit of installed power capa-city for the three approaches.
K.R. Ward, I. Staffell Journal of Energy Storage 20 (2018) 78–91
85
depends solely on the original pattern of demand and is maintainedregardless of the available storage capacity. The calculation time for themonopolistic algorithm also shows little sensitivity to storage capacity,increasing only slightly over the range of values considered. The com-petitive algorithm however has run times that increase linearly withstorage capacity: with increment size held constant, the number ofiterations required to dispatch it all is proportional to the total capacity
wavailable.
4.2. Sample dispatch schedules
Fig. 9 shows how the three algorithms choose to dispatch storageover the week beginning December the 3rd 2016.
From Fig. 9a we can see that, generally, the fixed-price scheduledoes one of only three things in each period: charges at 100% power,discharges at 100% power or does nothing. The only impediment to thisbehaviour is the constraint from the battery charge level. The action ofthe storage device can be seen to increase the volatility of the residualdemand over short timescales, and it is evident that short time-scalepeaks and troughs in original demand have been inverted.
The monopolistic algorithm smoothed the price considerably morethan the fixed-price algorithm (Fig. 9b). It also shows that the devicewas not used to output at full power at any point, and only charged atfull power during occasional half-hour periods. There are days when theprofit maximising constraint means that the device does not reach fullcharge (3rd and 9th December).
The competitive algorithm very obviously delivers the smoothestresidual system demand, with the peaks and troughs of the diurnalcycle removed (Fig. 9c).
4.3. Computational and numerical trade-offs
The unit size (or step size) parameter can be thought of as con-trolling the resolution of the algorithm. Smaller unit sizes mean pricesare recalculated more frequently with smaller changes to the storagedispatch, yielding a more accurate solution at the expense of longercomputational time. Fig. 10 examines the influence of storage unit size(step size) on the algorithms’ performance and accuracy.
A set of 4000 model runs were conducted to find the average per-formance of the algorithms across a range of storage parameters. Usingthe British power sector as of Q4 2016 as a case study, all 250 per-mutations of following input variables were considered:
• Algorithm: exogenous (breadth-first smoothing) and monopolistic.The endogenous / depth-first algorithm was excluded as it gavenear-identical results to exogenous;
• Total storage power capacity: 2, 4, 6, 8, 10 GW;• Device efficiency: 100%, 97.5%, 95%, 90%, 80%;• Storage E/P ratios: 3, 6, 12, 24, 96 h.
Each combination of parameters was simulated with sixteen levelsof precision, with unit sizes of: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200,250, 400, 500 and 1000 MW. These values were chosen to ensure aninteger number of units to dispatch in each situation. The results fromeach trial were compared against those for a 1 MW step size with thesame combination of parameters, to isolate the influence of precision onresults.
The increment size clearly influences the results obtained from thetwo price-maker algorithms. Fig. 10a and b show that as the algorithmprecision is reduced (step size increases) the Monopolist underutilisesand underestimates profit, while the competitive storage operator over-utilises and underestimates profit. Using a 200 MW step size in place of1 MW means profits are under-estimated by £0.23 ± 0.12 (competi-tive) and £0.26 ± 0.28 (monopolist) per kW of installed power, whichequates to a 2–3% relative error. The error on utilisation is similarlysmall: –1.0 ± 0.8% (competitive) and +1.0 ± 0.6% (monopolist) re-lative to the utilisation with 1 MW step size, meaning the absolute erroraveraged 0.15 percentage points across the range of trials.
Fig. 11 highlights the impact of algorithm precision on the output-duration curves for the storage devices. Lowering the precision dis-cretises the levels of power output that the storage device can dispatch,and thus introduces coarse-grained steps into the output duration curve,but it otherwise does not alter the shape.
Fig. 9. Example dispatch schedules from the three algorithms showing a singleweek (3–9 December 2016), with 5 GW of storage with 95% efficiency and E/Pratio of 4.
K.R. Ward, I. Staffell Journal of Energy Storage 20 (2018) 78–91
86
It should be noted also that the accuracy results will be influencedby the specifics of the market under study – its overall size, the size ofindividual power stations and the average gradient of the price demandcurve. In this case study, the average size of individual power stationunits (which have unique marginal costs) was 650 MW, with a range of170–1410 MW (5th–95th percentile). A national market comprisingsmall stations would require smaller storage dispatch increments than,for example, a pan-European where stations could be aggregated intolarger units.
Fig. 10c indicates that the competitive algorithm shows a 1:1 re-lationship between step size and calculation speed; for example thetrials with a 200 MW step size were 198 ± 16 times faster to completethan with a 1 MW step size. The speed gains are more limited for themonopolist algorithm as they approach the lower-bound time requiredto read data to and from the spreadsheet (approx. 1–5 s). This sameeffect can be seen earlier in Fig. 7b, the calculation time is almost in-variant with the amount of storage (and thus number of steps) beingsimulated.
The range of results observed across the different trials, depicted byerror bars on the points in Fig. 10a and b, are the result of small dif-ferences in the quantities storage dispatched triggering the constraintsof battery charge levels at different times. This leads to the algorithmfollowing a different path to its ultimate dispatch schedule. This rangein results shows no systematic relationship to any of the parameters thatwere varied (storage size, E/P ratio, efficiency).
For the sake of simplicity, we treat the results from 1 MW step sizeas being optimal. In an extended trial, we tested one combination ofparameters (2 GW of 95% efficient storage with E/P ratio of 6) with thecompetitive algorithm using step sizes from 0.1 to 10 MW in steps of0.1 MW. The 1 MW step size yielded a profit of £27.403 m, whereas themaximum achieved across all tests was £27.466 m (0.23% higher). Overthe set of 101 simulations (which required around 17 days to solve), themean and standard deviation of profit were £27.396 ± 0.029 m, givinga coefficient of variation of ± 0.11%.
We conclude that all results from these algorithms are thereforesubject to fine-grained noise caused by the locating of different localoptima, depending on the specific order in which time periods arechosen, utilised and removed from the solution. While this does notconclusively prove that the global optimum profit is significantly abovethe local optima found by these algorithms, it suggests that the varia-tion between results is trivial, and any value for the algorithms’ preci-sion under 10 MW is likely to yield results well within 1% of the bestobtainable.
5. Discussion and conclusions
In this article we introduce and demonstrate a computationally-ef-ficient method of simulating the profit-maximising dispatch of elec-tricity storage without the need for linear optimisation. The algorithmsdeveloped maximise the arbitrage profit of storage systems based on agiven set of market prices and, advancing on the work of Connolly andLund [14,34], account for the influence storage dispatch has on marketprices.
Our approach demonstrates that this price effect is a critical factorwhen modelling large quantities of storage operating in a power system.Based on a case study of the British electricity market, we find thatneglecting this effect introduces a 5% error on profits and utilisationwhen simulating 0.4 GW of storage (in a system with 34 GW averagedemand), and that this error increases as storage quantities grow. Giventhat mid-term scenarios see several GW of storage being deployed inBritain (e.g. up to 11 GW by 2030) [42], we argue that it is essential toincorporate this effect into models.
Modelling storage dispatch with an awareness of its impact onprices has until now required optimisation models, such as linear pro-gramming or unit commitment models. These are reasonably expensivein terms of setup and computational effort, and are not widely used tostudy multiple decades of operation, or wide sensitivity analyses [37].They also have a key constraint that prices equal marginal costs, andthus are limited to studying storage operation under perfect competi-tion.
Attempts to model the price-aware, profit-maximising dispatch ofstorage within price-based unit commitment models run into difficultiesas the feedback between price and storage utilisation introduces non-linearity into the problem formulation. There have been a few attemptsat tackling this but, as [8] point out, these methods produce non-
Fig. 10. The trade-offs between computation speed and accuracy when varyingthe storage unit size. Results are presented relative to calculations with a stepsize of 1 MW. The three panels give: (a) the relative error on device utilisation;(b) the absolute error on profit; and (c) the computation speed – defined as theinverse of calculation time (i.e. larger numbers are faster). Each line shows theaverage from a range of storage parameters (installed power capacity, effi-ciency and storage duration), and error bars show the standard deviation acrossthese parameters.
K.R. Ward, I. Staffell Journal of Energy Storage 20 (2018) 78–91
87
optimal deployment strategies and profits, and result in an ex-postdiscrepancy between expected and realised profits/prices. Coupling theincremental dispatch approach developed here with a unit commitmentmodel offers the potential to robustly explore this issue. Owing to theircomplexity, previous implementations of storage models which accountfor the price effects appear to be constrained in the number of timeperiods considered in a single window (e.g. 7–14 days of operation[9,30]).
The algorithms presented here are demonstrated operating on 3months of half-hourly resolution data (4416 periods) on a standarddesktop computer. Computational precision can be adjusted to achievedifferent trade-offs between numerical accuracy and calculation speed.An assessment of this trade-off suggests that for the given case study,the algorithm can be operated with a step size in the 100 s of MWs,giving run times in the order of 1–10 s with negligible (1–2%) influenceon results, which may prove acceptable for many purposes. The speedof this particular implementation of the approach makes it suitable foruse both in studies with lengthy timeframes and for Monte-Carlo ana-lyses.
We quantify the influence of market power in storage ownership bycomparing two variants of the algorithm, simulating the dispatch ofstorage under perfect competition and perfect monopoly. These twocases are extremes, and so the results provide the lower and upperbounds on the expected profits, utilisations, and benefits to the widersystem operation.
Many parts of the energy industry are an oligopoly, such as Britishelectricity generation and retail, oil production, turbine manufacture.Such industries fall closer to the monopolist extreme, having a smallnumber of large companies dominating the market. Smart home auto-mation and electricity storage systems could conceivably follow suit,becoming dominated by tech giants (e.g. Amazon, Google) and/orstorage companies (e.g. Tesla). If these firms became sufficiently largeand concentrated in a country’s market for electricity storage, they willface imperfect competition where each firm can strategically withhold
capacity to drive up prices. Exactly where the results (such as utilisationand profit) would lie between the competitive and monopolistic casedepends on the strength of the price effect, and would require addi-tional modelling (e.g. via computing the Nash Equilibrium) to bequantified.
Using a simplistic case study of Britain, we find the difference inresults from competitive and monopolistic ownership widens as moresignificant quantities of storage are installed, to the point where amonopolist would earn nearly double the profits when controlling10 GW of storage, by restricting its utilisation by 30%. Qualitatively,this backs up the findings of others (e.g. [27] and [28]) suggesting thatunder some circumstances regulation may be required in order to alignmerchant storage operators objectives with optimal societal benefits.Or, conversely, it suggests there is a requirement for regulatory inter-vention to support the profitability of storage devices in situationswhere the maximum reduction in residual demand variability is de-sired.
The storage algorithms are implemented in Visual Basic (VBA) andare coupled to a merit order stack model in Microsoft Excel. We releasethe algorithm code and a working implementation as open-source to thecommunity to help foster further research in this area. The algorithmscould be translated into other languages; implementations in a moreperformance-oriented language would likely improve the algorithm’sspeed, and would form an interesting avenue of further research.Similarly, the breadth-first competitive algorithm can be coupled toother forms of power systems model, allowing for a more complex re-presentation of the power system, incorporating flexibility constraintson thermal plant, ancillary markets, forecast uncertainty, and so on.
Acknowledgments
This work was funded by the Engineering and Physical SciencesResearch Council under grant EP/M001369/1. We thank Robert Staffellfor proof reading and comments.
Appendix A
Analysis of profits with linear supply curves
Following on from the illustrative analysis of Fig. 3, Fig. A1 shows the variation of the profits derived from each charge-discharge pairing as thegap in demand between them varies (for a straight line supply curve). For the fixed-price algorithm, when the difference in demand is less than thepower capacity of the storage, a loss is incurred: it costs more to charge the battery than is gained from its discharge. For demand differentials greaterthan the power capacity we see a linear increase in profits, consistent with the linear supply curve. The situation for the competitive algorithm is thatit will dispatch storage to equalise the demand in the charge discharge pair, but no more. Therefore, up to its power capacity, it makes no profit.
Fig. 11. Output duration curves for the three dispatch algorithms when considering 10 GW of storage operating over the three-month period. The left panel shows aprecision of 1 MW, the right panel shows 200 MW.
K.R. Ward, I. Staffell Journal of Energy Storage 20 (2018) 78–91
88
Above this point it behaves identically to the fixed-price algorithm. The monopolist restricts use of the storage device so as to earn profits at all times.Once the difference in demand between the charge-discharge pairings exceeds twice the power capacity of the storage, however, its usage will mirrorthat of the fixed-price and competitive algorithms. The monopolist’s excess profits are plotted in Fig. A1. For a storage device of efficiency η the pointat which the competitive algorithm will start to earn a profit will increase to S/η and the point at which the profits from all 3 algorithms willconverge will increase to 2S/η.
Fig. A1. Profit delivered by each algorithm for different charge-discharge demand spreads. Storage device has power capacity of S, 100% efficiency and infiniteenergy storing capacity (hence there is no need to consider capacity constraints.
Fig. A2. Flow diagram for the depth-first utilisation maximising algorithm. The coloured boxes are expanded in Fig. 2.
K.R. Ward, I. Staffell Journal of Energy Storage 20 (2018) 78–91
An alternative, intuitively obvious approach to smoothing the demand is to systematically ‘shave the peaks to fill the troughs’: reducing the needfor peaking plant and raising the level of baseload, and to prioritise this over smoothing the demand at other times. This is achieved by finding theoptimal charge-discharge pairing (i.e. the pairing with the maximum spread in demand that may be linked without violating the physical constraintsof the storage system) and dispatching an incrementally small amount of storage at this time, before recalculating the demand and the price andsearching again for a new optimal pairing (Fig. A2).
In contrast to the breadth-first algorithm, this approach acts on a single charge-discharge pairing of periods then recalculates prices during thatpair; rather than acting on the entire series of demand at once. The effect is still to dispatch storage so that it always reduces demand spread.
We find that this method yields comparable results to the breadth-first approach, in terms of the profit obtained, device utilisation, and impact onreducing the variability of demand. However, it is considerably more time consuming to run. A schematic of the difference in behaviour between thisand the competitive algorithm is given in Fig. A3.
Appendix B. Supplementary data
Supplementary material related to this article can be found, in the online version, at doi: https://doi.org/10.1016/j.est.2018.08.022.
References
[1] J. Cavada, Will renewables plus energy storage compete against gas? Presented atFuture of Energy Global Summit (2018).
[2] World Energy Council, World Energy Issues Monitor, (2017).[3] D. King, J. Browne, R. Layard, G. O’Donnell, et al., A Global Apollo Programme to
Combat Climate Change, (2015).[4] Breakthrough Energy, Powering the Way Ahead, URL: (2017) http://www.b-t.
energy/landscape/electricity/.[5] E. Drury, P. Denholm, R. Sioshansi, The value of compressed air energy storage in
energy and reserve markets, Energy 36 (8) (2011) 4959–4973.[6] I. Staffell, M. Rustomji, Maximising the value of electricity storage, J. Energy
Storage 8 (2016) 212–225.[7] A. Malhotra, B. Battke, M. Beuse, A. Stephan, T. Schmidt, Use cases for stationary
battery technologies: a review of the literature and existing projects, Renew.Sustain. Energy Rev. 56 (2016) 705–721.
[8] T. Brijs, F. Geth, S. Siddiqui, B.F. Hobbs, R. Belmans, Price-based unit commitmentelectricity storage arbitrage with piecewise linear price-effects, J. Energy Storage 7(2016) 52–62.
[9] R. Sioshansi, P. Denholm, T. Jenkin, J. Weiss, Estimating the value of electricitystorage in PJM: arbitrage and some welfare effects, Energy Econ. 31 (2) (2009)269–277.
[10] D. McConnell, T. Forcey, M. Sandiford, Estimating the value of electricity storage inan energy-only wholesale market, Appl. Energy 159 (2015) 422–432.
[11] P. Denholm, R. Sioshansi, The value of compressed air energy storage with wind intransmission-constrained electric power systems, Energy Policy 37 (8) (2009)3149–3158.
[12] E. Barbour, I.A.G. Wilson, I.G. Bryden, P.G. McGregor, et al., Towards an objectivemethod to compare energy storage technologies: development and validation of amodel to determine the upper boundary of revenue available from electrical pricearbitrage, Energy Environ. Sci. 5 (1) (2012) 5425–5436.
[13] J.A.M. Sousa, F. Teixeira, S. Faias, Impact of a price-maker pumped storage hydrounit on the integration of wind energy in power systems, Energy 69 (2014) 3–11.
[14] D. Connolly, H. Lund, P. Finn, B.V. Mathiesen, M. Leahy, Practical operation stra-tegies for pumped hydroelectric energy storage (PHES) utilising electricity pricearbitrage, Energy Policy 39 (7) (2011) 4189–4196.
[15] R. Walawalkar, J. Apt, R. Mancini, Economics of electric energy storage for energyarbitrage and regulation in New York, Energy Policy 35 (4) (2007) 2558–2568.
[16] K. Bradbury, L. Pratson, D. Patino-Echeverri, Economic viability of energy storagesystems based on price arbitrage potential in real-time US electricity markets, Appl.Energy 114 (2014) 512–519.
[17] E. Barbour, I.A.G. Wilson, P. Hall, J. Radcliffe, Can negative electricity prices en-courage inefficient electrical energy storage devices? Int. J. Environ. Stud. 71 (6)(2014) 862–876.
[18] A.K. Bikeri, C.M. Muriithi, P.K. Kihato, A review of unit commitment in deregulatedelectricity markets, Presented at Sustainable Research and Innovation Conference(2015).
[19] M. Shahidehpour, H. Yamin, Z. Li, Market Operations in Electric Power Systems:Forecasting, Scheduling, and Risk Management, Wiley-IEEE Press, 2002.
[20] T. Li, M. Shahidehpour, Price-based unit commitment: a case of Lagrangian re-laxation versus mixed integer programming, IEEE Trans. Power Syst. 20 (4) (2005)2015–2025.
[21] S. Pfenninger, A. Hawkes, J. Keirstead, Energy systems modeling for twenty-firstcentury energy challenges, Renew. Sustain. Energy Rev. 33 (2014) 74–86.
[22] F. Cebulla, T. Naegler, M. Pohl, Electrical energy storage in highly renewableEuropean energy systems: capacity requirements, spatial distribution, and storagedispatch, J. Energy Storage 14 (2017) 211–223.
[23] Y. Scholz, H.C. Gils, R.C. Pietzcker, Application of a high-detail energy systemmodel to derive power sector characteristics at high wind and solar shares, EnergyEcon. 64 (2017) 568–582.
[24] H.C. Gils, Y. Scholz, T. Pregger, D.L. de Tena, D. Heide, Integrated modelling ofvariable renewable energy-based power supply in Europe, Energy 123 (2017)173–188.
Fig. A3. Schematic illustration of the difference in approach of the peak-shaving and competitive algorithms. The peak shaving method prioritises the reduction ofmaximum system demand over the smoothing of demand at other times. The competitive approach seeks to smooth the demand at all times.
K.R. Ward, I. Staffell Journal of Energy Storage 20 (2018) 78–91
[25] R. Sioshansi, P. Denholm, T. Jenkin, A comparative analysis of the value of pure andhybrid electricity storage, Energy Econ. 33 (1) (2011) 56–66.
[26] X.A. He, E. Delarue, W. D’Haeseleer, J.M. Glachant, A novel business model foraggregating the values of electricity storage, Energy Policy 39 (3) (2011)1575–1585.
[27] R. Sioshansi, Welfare impacts of electricity storage and the implications of own-ership structure, Energy J. 31 (2) (2010) 173–198.
[28] W.P. Schill, C. Kemfert, Modeling strategic electricity storage: the case of pumpedhydro storage in Germany, Energy J. 32 (3) (2011) 59–87.
[29] C. Crampes, M. Moreaux, Pumped storage and cost saving, Energy Econ. 32 (2)(2010) 325–333.
[30] J.R. Cruise, R.J. Gibbens, S. Zachary, Ieee, Optimal control of storage for arbitrage,with applications to energy systems, 2014 48th Annual Conference on InformationSciences and Systems, (2014).
[31] B. Zakeri, S. Syri, Electrical energy storage systems: a comparative life cycle costanalysis, Renew. Sustain. Energy Rev. 42 (2015) 569–596.
[32] B. Nomark, How can batteries support the EU electricity networks? Insight_E 11(2014) 1–32.
[33] X. Luo, J.H. Wang, M. Dooner, J. Clarke, Overview of current development inelectrical energy storage technologies and the application potential in power systemoperation, Appl. Energy 137 (2015) 511–536.
[34] H. Lund, G. Salgi, B. Elmegaard, A.N. Andersen, Optimal operation strategies ofcompressed air energy storage (CAES) on electricity spot markets with fluctuatingprices, Appl. Therm. Eng. 29 (5-6) (2009) 799–806.
[35] D. Kirschen, G. Strbac, Introduction, in Fundamentals of Power System Economics,John Wiley & Sons, Ltd., 2005, pp. 1–9.
[36] R. Green, Competition in generation: the economic foundations, Proc. IEEE 88 (2)(2000) 128–139.
[37] I. Staffell, R. Green, Is there still merit in the merit order stack? The impact ofdynamic constraints on optimal plant mix, IEEE Trans. Power Syst. 31 (1) (2016)43–53.
[38] R. Green, I. Staffell, Electricity in Europe: exiting fossil fuels? Oxf. Rev. Econ. Policy32 (2) (2016) 282–303.
[39] R. Green, I. Staffell, N. Vasilakos, Divide and conquer? k-Means clustering of de-mand data allows rapid and accurate simulations of the british electricity system,IEEE Trans. Eng. Manage. 61 (2) (2014) 251–260.
[40] O. Schmidt, A. Hawkes, A. Gambhir, I. Staffell, The future cost of electrical energystorage based on experience rates, Nat. Energy 2 (2017) 17110.
