Demand Response Contracts as Real Options: A ... - IEEE Xplore

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IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 2, MARCH 2016

Demand Response Contracts as Real Options: A Probabilistic Evaluation Framework Under Short-Term and Long-Term Uncertainties Jonathan A. Schachter, Student Member, IEEE, and Pierluigi Mancarella, Senior Member, IEEE

Abstract—This paper aims to set up a probabilistic framework to assess the value of a portfolio of demand response (DR) customers under both operational (short-term) and planning (long-term) uncertainties through real options (ROs) modeling borrowed from financial theory. In an operational setting, DR is considered as an RO contract allowing an aggregator seeking to maximize its revenue to sell flexible demand in the day-ahead market and balance its energy portfolio in the balancing markets. Sequential Monte Carlo simulations (SMCS) are used to value DR activation decisions based on market price evolutions. These decisions combine DR physical characteristics and portfolio scheduling optimization, whereby the aggregator chooses to exercise only the contracts probabilistically leading to a profit, also considering the physical payback effects of load recovery. Sensitivity of profits to changing market conditions and payback characteristics is also assessed. In an investment setting, subject to long-term uncertainties, the value of an investment in DRenabling technology is quantified through the Datar–Mathews RO approach that applies hybrid SMCS and scenario analysis. The results show how the flexibility value of DR can be highlighted by modeling it as RO, particularly in high volatile markets, and how realistic inclusion of payback characteristics significantly decrease the benefits estimated for DR. In addition, the proposed RO framework generally allows hedging of the risks incurred under long-term and short-term uncertainties. Index Terms—Demand response (DR), flexibility, real options (ROs), risk hedging, sequential Monte Carlo simulations (SMCS), uncertainty.

Ac C K Lα,c,t τα,c θi δα,c ρα,c ϕi α,c,t S(t) S∗ (t) Ft Xc c Ccontract c Cfee

φmin φmax

N OMENCLATURE Indices α c k t

Appliance group index. Customer group index. Simulation group index. Time group index.

Variables c − t,α,c + t,α,c

Parameters N

IC

Number of periods in the day.

Manuscript received September 29, 2014; revised January 18, 2015; accepted February 16, 2015. Date of publication March 6, 2015; date of current version February 17, 2016. This work was supported by the Engineering and Physical Sciences Research Council within the Autonomic Power Systems Project under Grant EP/I031650/1. Paper no. TSG-00967-2014. The authors are with the School of Electrical and Electronic Engineering, Electrical Energy and Power Systems Group, University of Manchester, Manchester M13 9PL, U.K. (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSG.2015.2405673

Total number of appliances α of customer c. Total number of customers c. Total number of simulations. Load of appliance α of customer c at time t [kWh]. Duration of the curtailment of appliance α for customer c. Each load reduction period ∀i = (1, 2, . . . , τα,c ). Delay before the appliance α for customer c is turned back on. Length of the recovery of appliance α for customer c. Each load recovery period, ∀i = (1, 2, . . . , ρα,c ). Matrix of load recovery redistribution of appliance α of customer c at time t. Spot electricity price at time t [£/MWh]. Imbalance price at time t. Forward electricity price. Exercise price of demand response (DR) contract for customer c [£/MWh]. Aggregator cost of DR contract of customer c [£]. Aggregator cost of availability fee for DR contract of customer c [£]. Minimum number of times the DR contract can be exercised. Maximum number of times the DR contract can be exercised. Investment cost [£].

 Vt φi Pk (t) Rc c Cpayback

Matrix of load changes for customer c [kWh]. Matrix of load reduction at time t of appliance α of customer c [kWh]. Matrix of load recovery at time t of appliance α of customer c [kWh]. Matrix of load changes for the entire portfolio [kWh]. Value of (real) option at time t [£]. Exercise status indicator; 1 if contract is exercised, 0 otherwise. Discounted payoffs of simulation k at time t. Aggregator revenue from selling load of customer c to the market [£]. Aggregator cost of load payback of customer c [£].

c 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 1949-3053  See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

SCHACHTER AND MANCARELLA: DR CONTRACTS AS ROs: A PROBABILISTIC EVALUATION FRAMEWORK

Sk (t) πc π IV XV xvk ROIV OP+ OP λ NPV DCF NPV RO

Simulated spot electricity price for simulation k at time t. Total profit of aggregator from exercising DR contract of customer c [£]. Total profit of aggregator form exercising DR contracts of the portfolio [£]. Intrinsic value of (real) option [£]. Extrinsic value of (real) option [£]. Extrinsic value of (real) option at simulation k [£]. Real option investment value [£/customer]. Operating profits leading to a positive net present value [£/customer]. Operating profits [£/customer]. Risk-adjusted probability of a successful positive net present value forecast [%]. Net present value from discounted cash-flow valuation [£/customer]. NPV from RO valuation [£/customer]. I. I NTRODUCTION

EMAND response has the potential to improve market efficiency, increase system security, and provide economic benefits to consumers, utilities, and system operators (SOs) [1], [2]. SOs can request DR from aggregators to balance the system load, thus reducing overall balancing costs, or in times of contingency to reduce system stress. In addition, DR can help SOs avoid or at least postpone large investment costs required in network reinforcements, both at the transmission- and distribution-levels. However, DR approaches such as based on real-time pricing [3], [4] may lead to greater exposure to electricity price risk caused by very high and volatile prices, particularly for small commercial and domestic customers. Active participation of domestic and small commercial consumers can be solicited on a contractual basis by an aggregator using price signals, as proposed in [5]. By offering to alter their consumption patterns, consumers give an aggregator access to flexible demand in real-time in the balancing market or to take advantage of price arbitrage in the day-ahead market. However, consumers require their service demand to be eventually satisfied, meaning that any load reduction will need to be recovered at a later time [6]. This physical characteristic of demand (“payback” effect) has been assessed in an operational setting, but has yet to be evaluated on planning decisions [7], [8]. Although [9] considers load recovery for minimizing the total economic and emission cost of DR for planning, the study focuses on current day-ahead market prices and ignores the effect of different market risks. On these premises, the aim of this paper is to develop a probabilistic framework to quantify the flexibility benefits of a portfolio of DR customers for both operational and investment decisions, while hedging against the risk of uncertain and volatile market prices and taking into account realistic payback characteristics. The proposed assessment framework is based on ROs modeling borrowed from financial mathematics, and makes a consistent use of sequential Monte Carlo

D

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simulations (SMCS) to deal with both short- (operational) and long-term (planning) uncertainties. Similarly to an option in finance, which gives an investor the opportunity to make an investment decision in a financial asset (a stock), a RO gives an investor the opportunity to make an investment in a real-asset, for instance, modifying a customer’s electricity consumption via a DR contract. It is an option that arises in relation to an investment decision and uses options theory to evaluate this real or physical asset [10]. A. Literature Review 1) Economic Benefits of DR: Several papers have studied the economic benefits from the optimal dispatch of household appliances under real-time pricing. For instance, Gottwalt et al. [11] suggested that investing in DR technologies results in very low benefits for individual households but the true value stems from having the flexibility to respond to hourly price changes, making DR attractive in supporting real-time balancing measures. Similar conclusions are reached in [12], indicating that greater benefits can be gained from aggregating the loads of household appliances for individual customers. However, the studies fail to consider a portfolio of different customers. A model for reducing procurement costs for an energy retailer using DR is proposed in [13], hinting at considerable financial benefits. However, the study fails to consider the physical payback effect, thus leading to a potential overestimate of the benefits. 2) Load Recovery: References [14] and [15] analyze various types of DR and highlight the fact that a load reduction is usually followed by a payback effect so as to eventually deliver the original service. This later increase in load comes at a cost since the energy recovered must be supplied by generators or alternative demand measures [16]. However, [14] and [15] ignored the potential financial implications arising from load recovery, while [16] does not address the possibility of hedging DR risk when exposed to different market prices. The economic effects of load recovery on the operational scheduling of DR is also modeled in [17]–[19], however the studies make no assessment of the risk exposure of an aggregator when trading in multiple markets. 3) Option Value of DR: The application of financial and RO modeling to the field of power systems extends from the planning of generation capacity [20], [21], transmission capacity [22], [23] to hedging risk in electricity markets using forward contracts and electricity options [24]. However, although these studies make only use of financial derivatives for hedging risk, DR can also be used as a physical risk-hedging tool. Indeed, DR provides an aggregator with the option to change a customer’s load profile when participating in different markets. According to [25], the value of this option can be quantified using similar methods as for financial options; for instance by using closed-form solutions derived from the Black–Scholes model [26]. However, this model assumes that decisions are exercised only at the time the option expires, thus ignoring the value in having the flexibility in exercising them throughout the life of the

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option. This paper also ignores the physical load constraints of individual customers due to payback effects and hence may overestimate the true value of DR. The use of financial and RO for demand-side electricity contracts was also studied in [27]. However, this paper ignores the effect of payback and aggregation and assumes that the electricity price follows a geometric Brownian motion, which disregards the seasonal, mean-reverting, and spiky nature of electricity spot prices [28], potentially leading to incorrect valuation. Numerical methods, like Monte Carlo simulations, should therefore be used for DR RO assessment, as discussed in [29]. B. Paper Novelty and Contribution The main contribution of this paper is the development of a novel comprehensive RO-based probabilistic framework to evaluate the benefits of contracting a portfolio of DR customers for an aggregator who is subject to both short-term (in day-ahead and balancing markets) and longterm (investment) uncertainties, while realistically taking into account the physical characteristic of DR (payback effect, primarily). The novelty of the proposed model, implemented in MATLAB, is the calculation of the RO value of a portfolio of DR customers. It quantifies the value that the aggregator would be willing to pay in order to have the option of altering the demand of its portfolio of customers in the day-ahead and balancing markets, thus allowing to precisely determine whether or not it is worth purchasing DR contracts, at what price, and with what risk (modeled through the value at risk), given the uncertainty in these relevant prices. The model applies a simulation-based RO approach to DR contracts, where the activation strategy is based on the highest expected price in the day-ahead market and on a price threshold in the balancing market. Both operational and planning aspects are dealt with consistently in a probabilistic manner through SMCS, freeing the model from a number of assumptions found in the option theory literature. In particular, the uncertainties (here, market prices) are not limited to specific stochastic processes, nor do they require the assumption of constant volatility and mean, which is unrealistic when applied to long-term investments. This is an important contribution and novelty of this paper. Further, a realistic planning model is proposed based on future cash-flow scenarios, making the analysis consistent with real decision-making practices. Implementation of “RO thinking” allows considering that an aggregator has the right, but not the obligation, to exercise a DR contract, and would exercise it only if it is worth doing so. Not only is this flexibility in decision-making ignored in the literature, but there is also currently no probabilistic model considering DR as a physical option for hedging against electricity price risk under both operational and planning perspectives. Finally, the economic benefits of DR under future market price conditions (higher volatility) that might occur, for instance, as a result of the integration of renewable generation and greater electrification [30], is analyzed so that a valuation of DR in the context of long-term investment planning can be performed. This combined effect of load recovery and market volatility is, for the first time in the literature, assessed on planning decisions under uncertainty.

IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 2, MARCH 2016

C. Paper Structure The remainder of this paper is organized as follows. Section II presents the market model in which the DR program is considered and builds a general mathematical formulation for correctly considering load recovery occurring after demand reduction. It allows the aggregation of a portfolio of several DR customers, which is crucial in order to truly assess the benefits of DR from a techno-economic perspective. Section III describes the SMCS-based models for considering DR as operational RO for hedging exposure to electricity price risk. How DR can be used for arbitrage and portfolio balancing is described and the implications, benefits, and risks for an aggregator are discussed. Using data from [31], the effect of load recovery and of changing market price evolution on a DR aggregator’s cash flows is studied in Section IV. The load recovery shapes in these case studies are of parametric form and assess the impact of different recovery shapes on DR profitability. Finally, Section V presents the SMCS-based RO model, elaborated from [32] [“Datar–Mathews” (DM) method], for assessing long-term investments in DR technologies.

II. M ODEL F ORMULATION FOR DR P ORTFOLIO A. Market Model Flexibility, in this paper, refers to the modification of consumption patterns in reaction to an external signal (price signal or activation) in order to provide a service. The aggregator uses flexible DR (modeled here as a RO contract) from its customers to play in different markets. Since the DR contract creates a cap on the electricity price paid by the aggregator, the aggregator is hedged against the risk of high and volatile spot prices. In the proposed DR program, the aggregator participates in the day-ahead energy market with the aim of making an economic gain. The aggregator forecasts day-ahead prices and exercise DR contracts in its portfolio of customers only if the expected price is greater than the DR contract cost, thus taking advantage of price arbitrage. Furthermore, grid-connected customers are balance responsible parties (BRPs), meaning that they are responsible for balancing their individual supply and demand imbalances. Household and small and medium size enterprise customers typically (and implicitly) outsource their balancing responsibility to the retailer. Retailers and BRP can also outsource their balancing responsibility to an aggregator who acts as an intermediary between customers who provide flexibility and procurers of this flexibility (retailers and BRP). By participating in the balancing market, an aggregator can balance its energy portfolio and reduce imbalance costs by exercising DR contracts from the time of balancing gate-closure until real-time, thus hedging its electricity price risk associated with real-time price volatility [33], [34]. However, imbalance prices are calculated after all balancing mechanism activity has stopped. In other words, they are not known until after the system has been balanced, thus making them difficult to forecast accurately. Based on the above, in this paper, we consider two business cases for an aggregator, which is also an

SCHACHTER AND MANCARELLA: DR CONTRACTS AS ROs: A PROBABILISTIC EVALUATION FRAMEWORK

energy retailer, participating in the day-ahead energy market and in the balancing market.

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TABLE I O PERATING R EGIMES FOR T WO C USTOMER L OADS W ITH A O NE P ERIOD L OAD R EDUCTION (τ _(α, c) = 1) AND A T WO P ERIOD L OAD R ECOVERY (ρ_(α, c) = 2)

B. DR Modeling A key feature of demand is the diversity in usage of appliances [7]. Many of them, such as thermal loads (heat pumps, hot water tanks, etc.) are interruptible and can be curtailed through peak clipping, while deferrable loads (e.g., washing machines) can be shifted throughout the day. Nonetheless, interrupting loads does not necessarily lead to an overall energy reduction [14] since the consumption, when the load is turned back on, may be even higher than without curtailment (for instance due to losses). This bears implications in the market, as this extra energy payback needs to be purchased somewhere. Yet, many DR studies (see [35], [36]) tend to ignore load recovery. Furthermore, when a DR request is sent to a customer, an aggregator needs to be able to predict what the final demand profile of its portfolios will be, at different times, taking into account the payback effect of different customers. To handle this, an N × N matrix c is created representing the changes in load occurring for each of the N periods in the day following a DR request for each individual customer c (with c = 1, . . . , C for C customers in the portfolio) with Ac appliances. Given the operating regime of a specific appliance α for each customer (α = 1, . . . , Ac ) and its load Lα,c,t (in kWh) at each time-period t, the load changes are a function of the following. 1) τα,c : The duration of the curtailment of appliance α. 2) θi : Each load reduction period, ∀i = (1, 2, . . . , τα,c ). 3) δα,c : The delay before the appliance is turned back on. 4) ρα,c : The length of the recovery. 5) ϕi : Each load recovery period, ∀i = (1, 2, . . . , ρα,c ). A matrix describing how the load recovery is distributed over the ρα,c periods is defined by the components α,c,t ∀t ∈ {ϕ1 , . . . , ϕρα,c }. The reduction matrix − t,α,c is an N × N × Ac × C variable matrix taking the value −Lα,c,t ∀t ∈ {θ1 , θ2 , . . . , θτα,c } when the load is reduced and 0 at all other periods    −Lα,c,t , t ∈ θ1 , . . . , θτα,c − = (1) t,α,c 0, else. The recovery matrix + t,α,c is an N × N × Ac × C matrix given by (2). Load recovery occurs in the periods t = (ϕ1 , ϕ2 , . . . , ϕρα,c ) after the load reduction periods s = (θ1 , . . . , θτα,c )    θτα,c −  · − , t ∈ ϕ , . . . , ϕ + α,c,t 1 ρ α,c s,α,c s=θ 1 (2) t,α,c = 0, else. Therefore, for each customer, the load change matrix c is  + c = − (3) t,α,c + t,α,c . Ac

This matrix representation has two advantages over the methods presented in [11]–[14]. 1) It models all possible load redistribution patterns and creates a general method for considering the load recovery.

Fig. 1. Aggregated portfolio  of two DR customers, 1 and 2, whose loads (in kWh) change respectively according to matrices 1 and 2 .

2) It allows to easily scale-up the changes in the load at an individual customer level to a larger aggregated level, by simply considering the portfolio matrix  = 1/C c c . As a result, an aggregator can schedule an entire portfolio of customers. For instance, focusing on aggregated customer loads, consider two customers over N = 4 periods in a day. They are requested to curtail loads for τ = 1 period given that the load recovery of customer 1 in terms of energy is 100% of the original load spread over ρ1 = 1 period without delay. For customer 2, the recovery is 100% of the original load but this recovery is spread equally over ρ2 = 2 periods, again without any delay. Here, the operating regime of each customer, described in Table I, differs in terms of recovery shape, which is eventually a function of the customer’s shifted appliances. Each customer matrix, 1 and 2 , shown in Fig. 1, is then aggregated as the portfolio matrix . DR activation can be manual, where the load adjustment is manually performed after receiving notification of an upcoming DR event. It can be semi-automated, where a local control system follows preprogrammed DR strategies following a notification. Finally, DR can be activated remotely using an event initiation signal to control loads directly. This fully automated DR activation is assumed in this paper. The customer also provides an availability matrix for each appliance informing the aggregator of specific periods in the day during which an appliance is available for DR. It specifies a start time and an end time during which the appliance is available. The load reduction and load recovery matrices are then defined based on this availability matrix. As mentioned earlier, load recovery may lead to greater costs during recovery hours, which need to be taken into account in the DR valuation. For a balancing responsible party, this could mean purchasing recovered loads at very high prices in the wholesale spot market.

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III. DR AS RO: S HORT-T ERM O PERATIONAL M ODELING Once the DR portfolio matrix  is built from aggregating the loads of individual customers, an aggregator can rank and schedule customers based on the cost of their DR contracts and participate in the market by providing flexible demand. To protect oneself from unexpectedly high prices, aggregators may typically purchase option contracts for a certain capacity on an hourly basis [13]. A call option, for instance, is a contract whose value depends on the spot electricity price S and gives the option holder the opportunity to purchase a preagreed capacity of electricity at some time in the future, at a preagreed price X called the strike price. If the spot price exceeds the strike price, the option holder exercises the option and purchases the electricity at the cheaper, strike price. If the spot price never exceeds the strike price, then the option is not exercised and the electricity is purchased at the cheaper spot price. The strike price represents the maximum price the option holder would pay in the spot market, thus providing a cap on the electricity price. Similarly to a spot price option, the strike price of a DR contract provides a cap on the price that the aggregator may experience, allowing the aggregator to maximize revenues while limiting the financial risk incurred from price uncertainty. However, this option takes into account the physical characteristics of DR and thus becomes a spot price RO. Therefore, below, DR RO are assessed using financial mathematics for pricing spot price options [37], subject to demand’s physical characteristics (Section I). This call option can therefore be used as a means to reduce or cut a customer’s load for a short period of time. In the same way, one could use a put option, which allows the option holder to sell a preagreed value of electricity at some time in the future, at a preagreed price X, as a means of increasing a customer’s load. The main focus of this paper is on call options but the model can be extended to put options as well.

IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 2, MARCH 2016

to the DR customer in return for their availability during a DR request or simply for their participation in the DR program, or via other forms of incentive (such as discount on monthly bill and so on) depending on the specific contract design. This eventually results in a cost reduction for the customer. In the planning model, the investment cost IC of setting up a DR contract with a DR customer takes into account the operational RO premium and the costs associated with the installation and automation costs for enabling DR technology. The option’s value Vt is shown in (6) and represents the value the aggregator would be willing to pay in order to have the option of altering the demand of its portfolio of customers in the day-ahead and balancing markets. The variable φi is binary and represents the exercise decision at each time ti , taking a value of 1 if the option is exercised and a value of 0 otherwise 

N  −r(ti −t) Vt = max Et e φi f (S(ti )) . (6) φ

i=1

The payoff function f , being convex [38], guarantees a global optimum, while the use of the risk-free discount rate allows trading decisions to be made regardless of individual risk preferences [39]. SMCS are used to assess (6), which also allow considerations of specific features of DR (Section I) and associated contracts. In fact, realistic contracts may have a limit on the number of DR exercises that can be made in a day, so that only the times that maximize the probability of capturing the most valuable hours are chosen. The number of exercises φi can, therefore, be bounded between a minimum (φmin ) and a maximum (φmax ) number of calls in a day φmin ≤

N 

φi ≤ φmax .

(7)

i=1

A. Model Description The option’s value Vt is calculated as the expectation Et of the option’s payoff f (S(ti )), given in (4) for a call option and in (5) for a put option, at each time ti , ∀i ∈ (1, . . . , N) dependent on the strike price X and the random variable S(ti ) representing the electricity spot price at time ti , discounted to today at the risk-free rate of interest r  max(S(ti ) − X, 0) for call option (4) f (S(ti )) = max(X − S(ti ), 0) for put option. (5) In finance, an option contract usually includes an option premium representing an amount paid to receive the opportunity to make an investment decision only when it is worth doing, thus taking advantage of the risk hedging characteristic of the option. What enables the opportunity of exercising a DR contract or not is the ability to have a DR contract available. In the operational model, parts of the aggregator’s profits may be for instance passed onto the customer via an availability fee, representing a fixed cost paid by the aggregator

The upper and lower limits on the number of DR exercises are subject to the requirements of the DR contract, of the DR customer and reflect the limitations of current DR technologies and perceived customer acceptance. Given that DR can affect a customer’s comfort level, customers may require an upper limit on the number of times they can be subject to disruptions in their consumption. In addition, a lower limit could be used to incentivize customers in partaking in DR schemes. If customers were never called and were never rewarded financially (e.g., given an availability fee), it could discourage them from signing up to such DR programs. On the other hand, if the contract comes with a fixed availability fee and a variable callout fee (a fee paid each time the customer is called), then a lower limit may not be required. The exercise decision and profit πc , shown in (8), are hence based on whether the revenue Rc from selling the energy that has been requested from each DR customer c is greater than c of contracting customer c with strike both the cost Ccontract c price Xc and a constant availability fee Cfee and the cost

SCHACHTER AND MANCARELLA: DR CONTRACTS AS ROs: A PROBABILISTIC EVALUATION FRAMEWORK

c Cpayback paid for any load recovery, ∀t ∈ {θ1 , . . . , θτα,c }, ∀s ∈ {ϕ1 , ϕ2 , . . . , ϕρα,c }, 0 < t < s c c c πc = Rc − Ccontract − Cfee − Cpayback ⎞ ⎛ θτα,c θτα,c   c ⎠ − − = −⎝ t,c · S(t) − t,c · Xc + Cfee t=θ1

maximized. Day-ahead prices and load profiles, which can be forecasted to a reasonable accuracy [40], become inputs to the optimization. At each time ti , i ∈ (1, . . . , N), the value of the call option is approximated over all exercise times

t=θ1

 + t,c · max(S(s) − Xc , 0) .

(8)

s=ϕ1

φ

N 

e−r(ti −t) φi f (Ft ).

(9)

i=1

It reflects the option’s value if today the entire set of exercise strategies was decided based only on the information up to the time of the decision and the forward curve [39]. Since forward prices are known at the time of the decision, the problem becomes deterministic and IV represents a lower bound for the option value. This implies that a portion of the monthly payment can be hedged on the day the contract is created. On the other hand, the extrinsic value XV, in (10), depends on the volatility (and on the uncertainty) in future electricity spot prices, which are unknown 

N  −r(ti −t) e φi f (S(ti )) . (10) XV t = Et max φ

Since XV uses all the information on the future spot prices to optimize the value over the whole spot price path, it represents an upper bound for the option value. As a result, even if the intrinsic value today is positive because the spot price is greater than the strike price, there may be an even greater value in keeping the option alive and exercising it later. This characteristic allows quantification of the value from future price uncertainty, which is ignored using traditional valuation techniques. XV in (10) can be calculated by simulating k = (1, . . . , K) different paths for the spot price Sk (t) using SMCS and be approximated as shown in (11) and (12) for N time periods in a day K 1 xvk K

(11)

k=1

where xvk = max φ

In other words, the decision to exercise the DR contract is based on heuristic rules and is taken only when a profit is expected given that the day-ahead prices are forecasted and measured against an actual realization of day-ahead prices. As a result, the times of exercise are set in descending order of profit. Load is reduced when the prices are the highest in the day and recovered at forecasted lower prices. The profit is hence calculated as the difference between the revenue from selling the energy contracted to the market and the cost that the aggregator must pay the customers for their energy. The profit maximization objective of the aggregator’s expected profit πc received from each customer is a then function ϒ of the payoff in (4)   πc = max Et ϒ(f (S(t))) φ∈{0,1}

N  = max Et e−r(ti −t) φi φ∈{0,1}

i=1

⎛

× ⎝−

N 

  e−r(ti −t) φi f Sk (ti ) .

(12)

i=1

As suggested in [39], (12) can be solved as a linear problem for each simulated path such that the daily payoff is

θτα,c



− t,α,c · max(S(ti ) − Xc , 0)

t=θ1 ϕρα,c

−

c Cfee



−

 + t,α,c

· max(S(si ) − Xc , 0)

.

s=ϕ1

(14) The total expected profit π over all customers C is then

i=1

XV t =

(13)

i=1

Since Xc represents a cap on the customer’s price, the aggregator cannot charge the customer more than Xc . The aggregator then either pays S − Xc if S > Xc for the load recovery and incurs a cost, or pays S and charges the customer S if S ≤ Xc , thus incurring no cost or profit. Financial and RO are composed of two values: 1) an intrinsic; and 2) an extrinsic value [29]. The intrinsic value IV represents the difference between the customer’s strike price and the forward curve Ft and is given IV t = max

N  1  −r(ti −t)  e E f (S(ti )) . N

Vt =

ϕρα,c

−

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π =

C 

πc .

(15)

c=1

B. DR for Arbitrage In the day-ahead market (see Section II-A), future loads and prices for the next day can be forecasted [40] so that an aggregator can estimate in advance how much load will be available and how much profit π can be made by curtailing and shifting loads from peak periods, when electricity prices are high, to off-peak periods, when prices are low. The aggregator can calculate the discounted payoffs Pk (ti ) = e−r(ti −t) (Sk (ti ) − Xc ), rank them in descending order, and sum the first φmin payoff with all the payoffs up until φmax . Based on forecasted prices, the aggregator only exercises the contracts that may lead to a positive profit, hence the expected profit πC is either positive or limited to the availability fee   c . (16) πC = max ϒ(f (S(ti ))), −Cfee Here, a RO approach is essentially taken by guaranteeing a profit and avoiding losses since DR is activated only at times when the expected value of load reduction exceeds the expected cost of load recovery.

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C. DR for Portfolio Balancing DR may have even greater value as a portfolio-balancing tool (see Section II-A). A balancing-responsible aggregator can use DR in real-time to change the load and balance his portfolio if needed, which he cannot do through contracts after gate closure. If the aggregator’s position causes an imbalance in the system, this energy must be purchased at potentially high and volatile imbalance prices. However, with DR, the aggregator can estimate the cost of contracting a customer’s load reduction, and choose to exercise the contract each time the imbalance price hits a given strike price threshold, thus avoiding paying high prices for its imbalances and potentially minimizing imbalance volumes in its energy portfolio. Nonetheless, imbalance prices are generally not known in advance and hence the recovery cost can only be estimated. The aggregator becomes subject to imbalance price risk and may incur a financial loss, depending on both the imbalance price and the payback profile. The profit function in (17), therefore, differs from (16) in that it can now become negative even if a contract is exercised, with S∗ being imbalance prices   πC = ϒ f S∗ (ti ) . (17) IV. DR AS RO: L ONG -T ERM I NVESTMENT M ODEL In a long-term setting, DR may require typical investments, which the aggregator might be in charge of. These can for instance be advanced metering infrastructure such as smart meters at the customer level, IT systems to manage and coordinate data flow, smart appliances with DR capability to respond to signals, and so forth. Assessing the long-term value of DR is, therefore, needed to justify any associated costs. Yet, planning studies for DR investments are limited. As argued above, the ability of DR to exploit real-time, or close to real-time, price differentials in electricity markets requires valuation tools capable of incorporating planning flexibility and investment risk aversion. These cannot be explicitly modeled using traditional deterministic NPV analysis and DCF methods but require probabilistic, and in case RO-based, valuation techniques. The DM RO model [32] considers long-term uncertainties in projects via multiple scenarios, while shorterterm uncertainties are captured by SMCS of the project’s annual cash flows. Three operating profit cash flow scenarios (pessimistic, most likely and optimistic) are created each with an associated probability of occurrence. For each year, the three estimates represent a corner of a triangular distribution, which is randomized using SMCS. Using risk-adjusted discount rates and combining NPV analysis with SMCS result in a frequency distribution of the total range of values for the project’s operating profits. By assuming that the investor will terminate the project if losses are incurred, the opportunity to adjust decisions based on future market conditions is then captured. The RO investment value ROIV is calculated in (18), where OP+ represents all successful outcomes of operating profits leading to a positive NPV, IC is the investment cost, and K is the total number of simulations  1 ROIV = max OP+ − IC, 0 . (18) K

The resulting NPV distribution displays a risk-adjusted probability of a successful positive NPV forecast, λ%, and a mean value of successful outcomes, OP, which are then used in (19) to calculate the RO investment value [32] ROIV = λ · (OP − IC).

(19)

OP represents the best estimate today of the discounted future expected net profits, assuming that the project is only initiated if today a positive risk-adjusted NPV is forecasted. Besides its simplicity, generality, and flexibility (particularly useful to model physical features as in the case of DR), the DM model has the advantage, over other RO models, of not requiring the use of a risk-free discount rate, which is difficult to assume for real-life projects. Instead, the DM model applies two independent risk-adjusted discount rates, one to the more risky cash-flows and another to the less risky investment cost. This allows considering the effect of both marketand private-risk, and directly comparing the profits and costs of the project with similar investment opportunities. V. N UMERICAL A PPLICATIONS In this section, the value of DR for an aggregated demand of 1000 customers is analyzed for a typical winter day (January 12, 2010), considering modern residential dwellings of different types, namely, terraced, semi-detached, and detached houses and flats, with respective average peak loads of 4.59, 3.25, 3.09, and 2.40 kW, and average daily bills of £8.55, 5.90, 5.63, and 3.97, calculated as an average of 250 load profiles from each house type, totaling 1000 houses, for an average electricity price of 14 p/kWh for a standard rate [41]. Daily profits are calculated in £/MWh of reduced load, assuming a maximum of one call per customer per day. The profits from DR (considering the effect of load recovery) as a RO arbitrage and portfolio-balancing tool are assessed on an aggregated portfolio of loads. This paper’s case studies discuss a general formulation for load recovery shapes, modeled in parametric form, thus avoiding that conclusions are device- and appliance-specific. These load recovery shapes are for exemplificative purposes and may represent either physical paybacks from specific appliances or controlled paybacks whereby the aggregator schedules its load portfolio so as to have the desired shape. Five cases are studied, summarized in Table II. A base case where the loads in the DR portfolio are turned off for one halfhour period and turned back on in the next half-hour period without any load recovery; case 1 has a 100% load recovery in the half-hour period following curtailment; case 2 has a 50% load recovery in the next period following curtailment; and case 3 has a 70% load recovery in the first period following load reduction and a 30% load recovery in the second period. Case 4 has a 30% load recovery in the first half-hour period and a 20% load recovery in the second half-hour period. Profit sensitivity to changing market conditions is also assessed, before analyzing a long-term DM investment model. As the availability fee is directly related to DR contract design, which is outside the scope of this paper, without loss of generality and in order to focus on the general aspects of

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TABLE II C HARACTERISTICS OF DR L OAD R ECOVERY FOR E ACH C ASE S TUDIED

Fig. 3. Box plot of daily profits (£) from DR, day-ahead market, 50% increase in price volatility.

Fig. 2.

Box plot of daily profits (£) from DR, day-ahead market.

the proposed framework and on the specific impact of price volatilities and load payback shapes on DR profitability, the availability fee is set to zero in all cases. However, future work is intended to assess the impact of contract designs and different call-out and availability fees on DR profitability. A. DR for Day-Ahead Arbitrage For the day-ahead market, the optimal decision to exercise a DR contract is made based on the maximum profit achieved in the day (Section III-B), assuming that forecasts for dayahead prices are run and load profiles are known. Extrinsic and intrinsic values are compared at each time period and the decision to exercise the DR contract is taken. The contract is exercised only if the profit resulting from load reduction and later recovery is positive. Results of profits π are shown as a box plot in Fig. 2 and correspond to a distribution function built from 10 000 price simulations. A black diamond marker designates the average profits or loss for each case, a line in each box shows the median, and T-bars below and above each box represent the 95% and 5% value at risk (VaR95% , VaR5% ), meaning a 95% probability that profits will be above VaR95% and a 5% probability that profits will be above VaR5% . As expected, the base case has the largest potential profit, as no load is bought after load reduction. The lowest profits occur in case 1, where 100% of the load is recovered in the first period after reduction. Here, recovering the load over one or two periods makes very little difference to the profits. 1) Impact of Price Volatility: Given that future markets are expected to become more volatile [30], the sensitivity of profits is studied using possible future day-ahead prices with higher volatility compared to today’s prices. This was

done by fitting a model combining a deterministic component of sine functions representing daily peaks and troughs, and a stochastic auto-regressive moving average [42] model to add a stochastic element to the current U.K. market prices, taken from the Elexon portal of U.K. market index price and volume in 2010 [43]. The relevant volatility component is then changed accordingly. More specifically, in 2010, the U.K. dayahead price daily standard deviation was around 7.75 £/MWh, or 20.85% relative to an average price of 37.15 £/MWh. Increasing price volatility by 50% (standard deviation equal to 11.63 £/MWh, or 31%), increases the likelihood that a very high price will follow a very low price, and as a result, not only is there a higher probability of making a profit, but also a higher probability of making a larger profit. This reduces the probability of making low profits, which would decrease the VaR95% . Larger profits therefore have a greater probability of occurring causing the entire profit distribution to shift right (it increases) and so does the VaR95% . As a result, average DR profits for all cases increase as seen in Fig. 3. Average profits for cases 1 and 3 increase by 350% and 270%, while cases 2 and 4 increase by 37% and 34%, respectively. Still, the difference in profit between each case is greatly reduced. A 50% increase in profits is realized when compared with 64.73£ under current prices. The potential spread in profits is also much larger. Fig. 3 also shows that, for the base case, VaR95% = 134£ and VaR5% = 187£, while these values are £134 and 147 under current market prices. Despite the VaR95% being the same, the potential for greater profits (VaR5% ) is 27% larger. Similarly, for case 1, there is a 90% probability that profits are between £31 and 98, while these values are between £8.50 and 19.90 under current prices. In this case, the lower limit increases by 265%, while the upper limit is multiplied by a factor of 5. Similar results apply to the other cases. Since the probability of making a profit increases as volatility increases, while losses are limited to the option premium, the value of the option increases as a result. This highlights the advantage of a RO approach as it allows taking advantage of upside conditions while limiting losses on the downside.

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Fig. 4.

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Box plot of daily profits (£) from DR, balancing market.

B. DR for Real-Time Portfolio Balancing In the balancing market, imbalance prices are not known in advance. Since the aggregator exercises the contract without knowing the prices in the next periods, a negative profit can result, as seen in (17). Fig. 4 shows that cases with 100% recovery (cases 1 and 3) now incur a negative VaR95% , respectively, £−3.36 and −0.30. The thick black line marks the breakeven point (£0). In addition, these cases have very low average profits, respectively, £0.85 and 2.92. The effect on average profits is not as important for the cases with lower recovery. Nonetheless, for all cases the VaR95% value is much lower than in Fig. 2, indicating lower potential profits. 1) Impact of Price Volatility: The effect of a 50% increase in imbalance price volatility is shown in Fig. 5. The cases with 100% recovery, cases 1 and 3, incur slightly larger losses than in Fig. 4, which go as low as £−8.97 for case 1 and £−4.51 for case 3. Their average profits, however, increase significantly to £14.38 and 12.93, respectively, while the potential for larger profits is also considerably increased. The effect on average profits is not as important as for the other cases, but greatly affects the top 5% profits (VaR5% ), as these are much higher than in Fig. 4. Although, higher volatility makes it difficult to predict when and how to best recover curtailed loads, a RO approach allows estimating future prices given their volatility and reduce the risk of downsides, leading to larger profits. Furthermore, there is, in both Figs. 4 and 5, a slight difference in profits depending on whether one recovers the load over one or two periods. Indeed, recovering the load over two periods (cases 2 and 4) increases the VaR95% compared with recovering the load in one period (cases 1 and 3). This is due to the risk of recovering the load at a high price being spread across several periods instead of one. Instead of recovering the entire load at one price, some load will be recovered at that price while the rest of the load might be recovered at a lower price, with a positive probability, thus causing a reduction in the overall load recovery costs. Still, an aggregator needing to curtail loads due to a portfolio contingency must keep in mind that prices in the next hours may lead to a loss. The payments to DR customers and the creation of appropriate DR contracts should, therefore, consider the expected value of future imbalance prices

Fig. 5. Box plot of daily profits (£) from DR, balancing market, 50% increase in price volatility. TABLE III DR O PERATIONAL B ENEFIT C ASH F LOWS FOR C ASE 1 ( IN £ PER C USTOMER )

that would need to be paid by the aggregator in the event of a portfolio-balancing measure. This fundamental feature resulting from considering physical payback effect is typically neglected in most DR studies, thus leading to optimistic benefits for DR. C. Long-Term Investment Model This section illustrates the use of the DM RO method [32] to quantify DR investment. Four different cases of investment costs IC per customer are considered, namely, IC = {£150, 300, 450, 600}. Investment costs were assessed against operational benefits, calculated as in Section V-A over ten years, whereby price volatility was increased yearly by 5% up to 50% volatility in year 10. The resulting cash-flow benefits, exemplified for case 1, are presented in Table III. As discussed in Section III, in the DM approach the three DR profit estimates in Table III (describing a pessimistic, equal to the VaR95% , a most likely, equal to the mean value, and an optimistic, equal to the VaR5% , forecast, respectively) represent the corners of a probability distribution, taken here as triangular, which are used to randomly generate the entire distribution by SMCS. In addition, a 10% weighted average cost of capital is applied to discount operating profits, and a different 5% interest rate to discount the investment cost, which is much less risky since the investor can choose whether or not to exercise the investment decision based on whether the project appears to be profitable. A first step is thus to calculate the traditional NPV under the most likely scenario, by discounting the sum of its cash flows at 10% discount rate and subtracting the present value of the investment cost: NPVDCF = £118.1 − £150 = £ − 31.90.

SCHACHTER AND MANCARELLA: DR CONTRACTS AS ROs: A PROBABILISTIC EVALUATION FRAMEWORK

Fig. 6.

NPV distribution (per customer) for case 1 for £150 investment.

TABLE IV DM I NVESTMENT A NALYSIS ( IN £/C USTOMER ) F OR DAY-A HEAD M ARKET PARTICIPATION F OR A LL C ASES S TUDIED

Taking case 1 as an example, one obtains the distribution of Fig. 6, where positive and negative NPVs are in dark and light gray, respectively, after subtracting the present value of £150 investment cost, as in (18). According to Fig. 6, there is a 20% probability that NPVs are positive, with an average value of £172.76. The investment option value is then calculated using (19): ROIV = 20% × £172.76 = £34.55. While the traditional NPV DCF calculation gives a negative NPV equal to £−31.90, a RO approach thus gives a positive project value of NPV RO = £ − 31.9 + £34.55 = £2.65. All investment cases for day-ahead market participation are summarized in Table IV. For the base case, the project is profitable up until IC = £600, in which case, even the NPV RO is negative. Case 1 is profitable only under a flexible RO calculation for an investment of £150. Cases 2 and 4 are profitable up to an investment cost of £300, while case 3 never proves to be profitable. The RO investment analysis thus confirms that even considering all DR flexibility potential, realistic accounting for load payback characteristics greatly reduces DR profitability estimate. VI. C ONCLUSION This paper has introduced a comprehensive probabilistic framework for assessment of the value of DR through RO

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methods that make use of SMCS. The framework takes into account uncertainty and flexibility for both short- and longterm decisions as well as realistic physical features of demand. Case studies have demonstrated how DR can be modeled as a spot price RO for maximizing an aggregator’s profits both for arbitrage and portfolio balancing. The implications of physical paybacks have been quantified, together with sensitivity of the aggregator’s profits to future price scenarios with higher volatility. As shown, the proposed RO approach increases the potential for greater profits by taking advantage of upsides while limiting losses on the downside, thus allowing a hedge against price risk. High volatility in prices creates a higher potential for profits than for losses, making DR a profitable business case for an aggregator. Nonetheless, the payments to DR customers and their associated contracts must consider the expected value of future imbalance prices that need to be paid by the aggregator when balancing a portfolio as load recovery can create losses in revenue. This payback feature of DR is often neglected, resulting in overestimating DR value. Finally, a consistent RO model, based on the DM methodology, has been applied to the problem of long-term investment decision in DR infrastructure, which is subject to uncertainty in cash flows over the planning horizon. The results show how ignoring the uncertainty and flexibility in planning decisions, as is the case in traditional DCF methods, causes projects to be undervalued and can make profitable projects seem unprofitable. As a final comment, by consistent use of SMCS for both operational and planning studies, the model introduced in this paper relaxes the assumptions needed under traditional RO methods, derived from finance, for practical application in engineering. In particular, long-term uncertainties are scenariobased and no longer require a stochastic process to represent price evolution over time, and using different discount rates to reflect the risk of different cash flows makes the analysis transparent and more consistent with real decision-making practices. ACKNOWLEDGMENT The authors would like to thank eight anonymous reviewers for their insightful comments to improve this paper. R EFERENCES [1] S. Kiliccote, M. A. Piette, E. Koch, and D. Hennage, “Utilizing automated demand response in commercial buildings as non-spinning reserve product for ancillary services markets,” in Proc. IEEE Conf. Decis. Control Eur. Control, Orlando, FL, USA, 2011, pp. 4354–4360. [2] D. S. Kirschen, A. Rosso, and L. F. Ochoa, “Flexibility from the demand side,” in Proc. IEEE Power Energy Soc. Gen. Meeting, San Diego, CA, USA, 2012, pp. 1–6. [3] S. Juneja, Demand Side Response: A Discussion Paper, Regulatory Energy Econ. Sustain. Develop. Divis., London, U.K., 2010. [4] G. Barbose, C. Goldman, and B. Neenan, “A survey of utility experience with real time pricing,” Lawrence Berkeley Nat. Lab., Berkeley, CA, USA, Tech. Rep. LBNL-54238, 2004. [5] P. Palensky and D. Dietrich, “Demand side management: Demand response, intelligent energy systems, and smart loads,” IEEE Trans. Ind. Informat., vol. 7, no. 3, pp. 381–388, Aug. 2011. [6] E. Karangelos and F. Bouffard, “Towards full integration of demandside resources in joint forward energy/reserve electricity markets,” IEEE Trans. Power Syst., vol. 27, no. 1, pp. 280–289, Feb. 2012. [7] G. Strbac, “Demand side management: Benefits and challenges,” Energy Policy., vol. 36, no. 12, pp. 4419–4426, Dec. 2008.

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Jonathan A. Schachter (S’12) received the B.Sc. (Hons.) degree in international business and the M.Sc. degree in mathematical finance from the University of Manchester, Manchester, U.K., in 2009 and 2011, respectively, where he is currently pursuing the Ph.D. degree from the Electrical Energy and Power Systems Group. His current research interests include power systems economics, electricity markets, stochastic and econometric models, risk management, optimization, and network investment under uncertainty.

Pierluigi Mancarella (M’08–SM’14) received the Ph.D. degree in electrical engineering from Politecnico di Torino, Turin, Italy, in 2006. He was a Research Associate at Imperial College London, London, U.K. He is currently a Reader in Future Energy Networks with the University of Manchester, Manchester, U.K. His current research interests include modeling of integrated energy systems, business models for distributed energy resources, and network investment under uncertainty.