Risk Constrained Trading Strategies for Stochastic Generation with a Single-Price Balancing Market

Due to the limited predictability of wind power and other stochastic generation, trading this energy in competitive electricity markets is challenging. This paper derives revenue-maximising and risk-constrained strategies for stochastic generators participating in electricity markets with a single-price balancing mechanism. Starting from the optimal---and impractical---strategy of offering zero or nominal power, which exposes the participant to potentially large imbalance costs, we develop a number of strategies that control risk by hedging against penalising balancing prices in favour of rewarding ones. Trading strategies are formulated in a probabilistic framework in order to address asymmetry in balancing prices. The large-scale communication of system information characteristic of modern power systems is utilised to inputs for electricity price forecasts and probabilistic system length forecasts. A case study using data from the GB market in the UK is presented and the ability of the proposed strategies to increase revenue and reduce risk is demonstrated and analysed.


I. INTRODUCTION
E LECTRICITY markets were designed for dispatchable generation. Since energy liberalisation in Europe, the US, and elsewhere, supply and demand have been matched by centralised operation of transmission systems using a combination of connected markets for energy and ancillary services [1]. Stochastic generators, such as wind and solar, require power production forecasts to participate economically in these markets, and while high-quality forecasts are widely available and improving, they will never be perfect.
Much attention has been given to how stochastic generation can be integrated into electricity markets [2]. Strategies for their participation can benefit from use of information about forecast uncertainty in order to deal with asymmetric penalties for over-or under-producing. Most of this work has focused on wind since this technology is more established, but the principals are transferable to other stochastic generators, such as solar, or other smart grid actors such as aggregators [3]. Looking towards the future, new electricity market arrangements which facilitate the active participation of renewables in balancing markets are becoming a reality [4], and market designs that incorporate directly properties of stochastic generation by allowing probabilistic offers have been proposed [5].
Offer strategies for trading wind power in day-ahead markets are derived in [6], and for dual-price balancing markets the optimal quantile of a predictive distribution can be calculated based on forecasts of imbalance prices [7], [8]. This J. Browell is with the Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow, UK, e-mail: jethro.browell@strath.ac.uk.
Manuscript Submitted August 10, 2017 analysis has been extended in [9] to include a recourse offer closer to the time of delivery. Participation in an intraday power exchange (which facilitates anonymous bilateral trading) is considered in [10] by accepting available bids and offers which are deemed improve the participant's market position, though without a sophisticated offer strategy for the day-ahead market. The strategic behaviour of wind power as a pricemaker has also been studied [11], [12]. Much of this work has been presented from the perspective of the power forecaster; however, participants in electricity markets also require price forecast to inform their decision making, as the examples given above acknowledge. Electric utilities forecast prices from hours to months ahead in order to reduce risk or maximise profits, with day-ahead and intraday forecasts critical for effective bidding strategies [1]. An extensive review of electricity price forecasting can be found in [13] and results of the price forecasting track of the 2014 Global Energy Forecasting Competition in [14] give an overview of state-of-the-art practices. Electricity prices are driven fundamentally by supply and demand costs, and as such it is necessary to model these when making predictions [15], [16], [17]. Much attention has been given to short-term electricity prices, with familiar time-series models (ARX, ARIMA, etc) being popular. Forecasting balancing prices has received less attention but is considered in [18] and balancing volumes in [19], who use approaches based on ARIMA and exponential smoothing, respectively.
While a great deal has be learned about how stochastic generators can participate in electricity markets the majority of this work has focused on markets a with dual-price balancing mechanism. This is largely due to the high penetration of renewables in these markets, particularly in Europe. While the majority of electricity markets in Europe operate dualpricing systems, single-price markets dominate in the US and are operated in Germany, the Netherlands and, since November 2015, the UK. Large volumes of system data that are collected and shared in smart grids to enable efficient use of available assets and resources [20]. This data may also be utilised in the strategies of electricity market participants. Furthermore, since this information is available to participants electronically it can easily be incorporated into automated trading systems.
In this work, participation of stochastic generation in a dayahead energy market coupled with a single-price balancing balancing market is considered. It is observed that in this situation, electricity market forecasts are of primary importance and that power forecasts are required only to apply risk-constraints. A secondary result is the observation of weak incentives for variable generation to provide accurate forecasts in this scenario, as also observed in [21]. Strategies based on taking a long or short position in order to manage asymmetric imbalance costs are proposed. Probabilistic forecasts of system length (sign of the net system imbalance) are required in addition to forecasts of day-ahead and balancing prices. Revenue maximisation and risk-constrained strategies are both derived, with only the latter requiring forecasts of power production.
An introduction to day-ahead and balancing markets is offered in Section II followed by the formulation of the offer strategy problem and possible solutions in Section III. Probabilistic system length forecasts using logistic regression, and Price forecast using ARMAX models, both of which utilise wider system data, are described in IV. A case study using real data form the UK is presented in Section V. Forecast performance is compared to standard benchmarks and quantified in monetary terms based on the performance of trading strategies. Finally, concluding remarks are made in Section VI.

II. DAY-AHEAD AND BALANCING MARKETS
Market structures can vary widely between regions but are typically made up of four main components: bi-lateral contracting between individual parties from days to years ahead of delivery, a day-ahead auction that determines the schedule for the activation of generators and large industrial consumers for the following day, an intraday market which allows participants to modify their position closer to delivery, and a balancing market utilised by the transmission system operator to balance supply and demand in real time [22]. Additional markets may also exist for ancillary services such as frequency response, reserve power and provision of reactive power, and financial products such as energy options and futures.
The most important markets for stochastic generators are the day-ahead and the balancing markets since the need to forecast generation makes trading further in advance impractical and because forecast errors result in imbalances. Intraday markets enable participants to modify their position closer to gate closure, but often suffer from low liquidity meaning that it is difficult for participants to find counter-parties to trade with.
Day-ahead markets are typically double-blind auctions into which generators and consumers submit anonymous offers to generate and bids to consume certain volumes of energy at a price they are willing to pay or be paid. Supply and demand are compared and a market price is calculated for each period of the following day. This price is applied to all accepted bids and offers and is an important reference for intraday and balancing markets since it gives an indication of the marginal price of energy for a given period.
Balancing markets are used by the transmission system operator to balance supply and demand and operate from from gate closure to the point of delivery. The cost of balancing incurred by the system operator is recovered through payments by those who are out of balance. Prices are calculated based on either a single-or dual-pricing system. All participants in a single-price system resolve their imbalance at the same imbalance price, whereas in a dual-price system participants receive different prices depending on the sign of their imbalance. Balancing prices represent the cost to the system operator of increasing or decreasing net-generation, and as such depend on whether the system has a net energy surplus or deficit.
In a single-price balancing market, whether the single price is greater or less than the day-ahead price depend on the system length, i.e. whether the transmission system operator has had to increase or decrease net-energy production during a given time period. If the system is short of energy, the balancing price will be greater than the day-ahead price the reflect the utilisation of more expensive or flexible generators (or demand reduction), and the converse if the system is long. The effect of this is to penalise market participants who are out-of-balance in the same direction as the system, and to reward those who are helping the system by being out-ofbalance in the opposite direction. This is different to the twoprice system where imbalances contributing to the system imbalance are penalised, and those helping receive a neutral reference price, which is usually similar to the day-ahead price. The importance of system length forecasting is clear: being out-of-balance the wrong way invites a penalty, whereas being out-of-balance the right way is profitable. However, forecasting the system length prior to submitting offers into the day-ahead market is challenging and, since the penalties and rewards for a correct forecast are not systemic, warrants a probabilistic approach. In the following section this problem is formulated and offer strategies for the day-ahead market are derived based on a probabilistic assessment of system length.

III. PROBLEM FORMULATION
Here we consider day-ahead offer strategies for a participant who is a price-taker in both day-ahead and balancing markets, and do not consider participation in intraday markets. For each settlement period t+k, a market participant will contract some volume of energy E C t+k at time t. The revenue R t+k for a participant contracting E C t+k but generating E t+k is given by where π C t+k is the contracted price for period t+k, and T C t+k is the cost associated with the energy imbalance d t+k = E C t+k − E t+k . In a single-price balancing market, each participant must buy the volume of energy equal to their deficit, or sell the volume equal to their surplus, at the imbalance price. The imbalance price π S t+k is a function of the balancing actions relating to period t+k taken by the system operator to maintain the balance supply and demand in real time and is calculated at the each settlement period. The imbalance cost T C t+k is given by It is useful to express a market participant's revenue in terms of the actual energy they generate and their imbalance as follows from which it is clear that in order to maximise revenue, balancing costs should be minimised. In order to reflect the dual-nature of the single imbalance price, we distinguish between the price resulting from net up-or down-regulation, which corresponds to the sign of the system net imbalance volume (NIV). Equation (4) then becomes where π + t+k > π C t+k is the up-regulation price, and π − t+k < π C t+k is the down-regulation price. The case NIV t+k = 0 is merged with NIV t+k < 0 for simplicity and without loss of generality since π + t+k = π − t+k = π C t+k in that situation. Assuming not participation in other markets or incentives to do otherwise, the aim of the market participant is to contract the volume of energy E C t+k that maximises revenue while managing risk.

A. Imbalance Minimisation
First we consider the simplest strategy for risk management: minimise exposure to imbalance charges by contracting the forecast generation for each period in the day-ahead market. This approach reduces exposure to penalising imbalance prices, but also reduces exposure to rewarding prices in the case where the sign of the participant's imbalance is the opposite that of the system. The bid in this case is given by whereÊ t+k|t is a forecast of E t+k made at time t set to minimise the mean absolute error. This strategy has the benefit of not requiring forecasts of market prices or system length.

B. Categorical Assessment of System Length
If the system length were known at the time of contracting, according to Equation (5), the optimal volume to contract would be ±∞! This is of course nonsense and in violation of the price-taker assumption since such offers would influence the clearing price of the day-ahead market and NIV.
Any participant with sufficient power would have to conciser their influence on the day-ahead price, NIV and the marginal price of balancing actions that the system operator would have to take, and the opinion of the market regulator. This situation is not considered here. We therefore proceed assuming that the capacity of the wind generators we consider is small relative to the magnitude of the NIV, and that the contracted volume is restricted to the range 0 ≤ E C t+k ≤ E max , where E max is maximum amount of energy the wind generator could deliver in a single settlement period.
If the sign of the NIV for period t+k is known, the optimal bid would be simply A deterministic forecast of the sign of the NIV is required to implement this strategy, but no power forecast is needed. Note also that bidding only extremes leaves the participant exposed to potentially large losses if the sign of the NIV is forecast incorrectly.

C. Probabilistic Assessment of System Length
As imbalance prices in periods of net up and down regulation are asymmetric about π C , it is desirable to formulate offer strategies from a probabilist perspective. Consider the energy generated during period t + k to be a random variable E t+k , and the probability at time t that the system will be short Pr t (NIV t+k > 0) = φ t+k|t . We include the possibility that the NIV is exactly zero in the chance that the system is long without consequence, so Pr t (NIV t+k ≤ 0) = 1 − φ t+k|t .
The subscripts t+k and t+k|t are dropped in the proceeding analysis to avoid notational clutter.
In this probabilistic framework the expectation of the imbalance cost T is given by Using the expectation operator E{·}, the optimal bid can now be calculated as Since the optimal bid depends only on the sign of the factor multiplying the expected imbalance, it is helpful to define the ratio and write the optimal bid as The ratio (12) can be interpreted as a cost/loss ratio defining the critical probability at which is becomes economic to bid as if the system is expected to be long or short. As the prices π C , π + and π − are unknown at time t they must be forecast along with φ. Note, however, that there is no need to forecast the level of wind generation.

D. Risk Constrained Contracted Volume
So far we have only considered revenue maximisation. Next we consider a risk constrained approach for two main reasons: first, the risk associated with revenue maximisation is potentially large since imbalance prices are volatile and the strategies investigated so far require the participant to expose themselves to the largest imbalance possible; and second, participants with potential market power may be able to participate in a similar way by hedging smaller volumes, and this should be done in an informed way.
In this section alternative strategies are considered that restrict the size of the expected imbalance by adjusting the offer away from the forecast generationÊ = E{E} in order to hedge against penalising imbalance prices. Three options are considered: an additive adjustment where the offer is equal to expected generation plus/minus some parameter, ν; a multiplicative adjustment where the offer is equal to expected generation multiplied by some factor, 1±η;and finally, offering a quantiles of a probabilistic generation forecasts.
1) Additive Adjustment: In this strategy the contracted energy for a given settlement period is the expected energy plus/minus a fixed adjustment. In effect, the capacity is partitioned intoÊ − νE max and 2νE max with the latter part traded using the probabilistic forecast of system length. The final offer bound by 0 ≤ E C ≤ E max . This strategy can be written as The choice of ν is a trade-off between maximising revenue and reducing exposure to imbalance charges.
2) Multiplicative Adjustment: Here we consider a contracted volume proportional to the forecast generation. This strategy has the pleasing property that exposure to imbalance charges increases with expected generation, and therefore with expected revenue for a given period. Put differently, the participant is only exposed to risk when the expected revenue is already high, and is exposed to little risk when expected revenue is low. The contracted volume is equal tô E ± (η × 100)%, bound by zero and E max . The strategy given by where η ≥ 0.

E. Quantile Offer
The additive and multiplicative strategies result in an imbalance d equal to the wind power forecast error,Ê − E, plus or minus an adjustment, the aim being to increase the likelihood that this term is either positive of negative, depending on the values of φ and Φ. Probabilistic forecasts provide information about uncertainty associated with forecast errors. This information can be used to chose E C such that the probability of d > 0 is a specific value.
The predictive distribution of E can be described by a set of quantiles {q α , α ∈ [0, 1]} where Writing this in terms of d and E C gives Pr d < q α − E C = α. Therefore, the contracted volume E C which results in a probability α of d being negative is given by the quantile q α . This strategy is written as where α ′ is the probability that the realisation of d has the desired sign. This approach is attractive because it explicitly models the uncertainty associated with forecast errors allowing this riskfactor to be controlled explicitly. It is also more elegant since it removes the need to impose bounds on offers as quantiles are bound by [0, E max ] automatically.

A. Probabilistic System Length Forecast
The probability that the system is will be short, φ, is estimated using a logistic regression model. This approach allows φ to be estimated conditional on some set of explanatory variables X, formally, The logistic regression model is given by where the vector β contains the model parameters to be estimated. Solving for φ yields Explanatory variables are chosen from the wide range of power system and market data that are available to participants. In this work, the parameters β are determined by maximum likelihood estimation using R, specifically the function glm from the package stats. Deterministic system length forecasts are produced using the same method but rounding φ ≥ 0.5 to 1, and φ < 0.5 to 0.

B. Price Forecasts
For the purpose of this study, we employ the popular ARMAX-type models for price forecasting [13]. A separate ARMAX model is fit for each settlement period and type of day to capture the different dependencies between price and exogenous variables in each situation. The time index τ is used to indicate the position of price π τ in a sequence of prices corresponding to the settlement period and day-type. This approach regresses the price at time on its past values at τ − 1,...,τ − p, the the model error ǫ τ and exogenous variables X k,τ . The model is written where α i are the autoregressive coefficients, β j are the moving average coefficients, and γ k are the regression coefficients for the exogenous variables. The forecast of π τ is given bŷ The parameters α i , β j and γ k are determined by maximum likelihood expectation and the model order (p, q) by minimising the Akike Information Criterion implemented using the R package forecast [23].

C. Wind Power Quantile Forecasts
Quantile forecasts for time t,q α,t are given by the function, q α,t = Q α (θ t ), of explanatory variables, θ t , that is the solution to the following optimisation problem (23) Here, gradient boosted machines are used to determine Q α for α = 0.01, 0.05, ..., 0.95, 0.99, inspired by the winning entry from the 2014 Global Energy Forecasting Competition using the R package gbm [24], [25].
V. CASE STUDY The performance of the proposed trading strategies is evaluated in a case study using historic data from the GB power system in the UK. There are two coupled auctions operated by APX and N2EX (Nordpool) which clear at the same price for each hour of the next day. The balancing market comprises half-hour settlement periods and is operated by the System Operator (SO) and Elexon. Trading in the intraday market can take place up until gate closure one hour before each settlement period begins, though participation in this markets is not considered here. Following the end of each settlement period, the single balancing price is calculated based on actions taken by the SO. This price is the volume-weighted average of the most expensive 50MWh of balancing actions taken relating to that period.
Electricity market data are available from Elexon [26], who operate the data service for the GB balancing mechanism. The data we utilise in this study are day-ahead and balancing prices, plus day-ahead forecasts of load, national wind and solar generation, and generation margin at peak demand. Halfhour resolution wind power and day-ahead power forecasts for five UK wind farm are provided by an anonymous GB wind farm operator and aggregated, since imbalances are calculated on an aggregate basis.
The period 06/11/2015 to 06/05/2016 is used in this case study covering the first six months following the switch from a dual-to single-price balancing mechanism. Due to the limited volume of data, all analysis is performed on a hold-out basis where a portion of the data are held-out and used for testing while models are fit to the remaining data.
Offer strategies have been implemented with benchmarks based on perfect and simple forecasts to demonstrate the relative value and limitations of each method.

A. System Length Forecast and Evaluation
The performance of the probabilistic forecast of system length is first evaluated in terms of the Brier score and its decomposition. As a benchmark, the historic proportion of occasion when each settlement period is short is used as a forecast, using the hold-out sample method.
The Brier score is a proper scoring rule for probabilistic forecasts of binary events and is given by where the observation o i = 1 if NIV i > 0, and 0 otherwise [27]. The Brier score rewards both reliability and confidence. The best score achievable is 0 if the either 0 or 1 is correctly forecast. Confident forecasts, i.e. those close to 0 or 1, are rewarded with a lower Brier score than cautions perditions, i.e. close to 0.5, if they are correct, and more heavily penalised if they are wrong side of 0.5. The Brier score can be decomposed into reliability, resolution and uncertainty [28]. Reliability is a measure of how close the forecast probabilities are to the proportion of positive outcomes, resolution is a measure of how much the forecast probabilities vary from the climatic average, and uncertainty measures the inherent uncertainty of the event being forecast. Mathematically these are given by where N is the total number of forecasts issued, K is the number of unique forecasts issued, and n k is the total number of times the k th unique forecast is issued. The termsō and o k are the mean outcome and the mean outcome conditional on the k th unique forecast being issued, respectively. Here, forecasts are grouped into 21 forecast bins centred on values from 0 to 1 in increments of 0.05. A separate model is fit for each settlement period. Forecasts of load, wind generation, and generation margin at peak are included as explanatory variables for all periods, while forecast of solar generation are only used during hours of daylight, specifically periods 12-41. Forecasts are produced out-ofsample for each day of the dataset using models trained on the all other data. The performance of this approach is tabulated in Table I, along with the performance of the benchmark model.
The performance of binary forecasts such as these can also be evaluated by examining their relative operating characteristic (ROC) curves [29], [30]. ROC curves depict the tradeoff between true-positive and false-positive forecasts across the full range of predicted probabilities. Loosely, a more skilful forecast method is that with a higher true-positive and lower false-negative rate than the competing method. ROC curves for system length forecasts are presented in Figure 1, which illustrates that forecasts produced by logistic regression consistently outperform the benchmark.

B. Price Forecast Evaluation
The exogenous variables available for price forecasting are the same as those used in the logistic regression for system length forecasting, namely day-ahead forecasts of load, wind, solar, and generation margin. Data are grouped into three day-types: weekdays, weekends, and holidays. It should be noted that because the day-ahead market requires offers to be submitted before 11am, balancing prices for times later than this will not be available as input to the forecast of balancing prices for the next day. For this reason, two-stepahead forecasts of balancing prices for periods after 10am are used, to allow for delays in the 10:00-10:30 and 10:30-11:00 balancing prices becoming available.
Models of order (1,1) and (2,1) are most common and account for over 25% of the models fit. Results are presented in terms of the critical probability, Φ = π C −π − π + −π − , and evaluated in terms of root mean square error (RMSE) and mean absolute error (MSE). These are given by and whereΦ t+k|t is the forecast of Φ t+k made at time t, and N is the total number of samples. The mean value of day-ahead and balancing prices from the same month and settlement period is used as a simple benchmark to asses the quality of the ARMAX forecasts. The MAE for the ARMAX and simple methods is 0.23 and 0.45, respectively; and the RMSE is 0.43 and 0.56, respectively. The ARMAX modelling approach clearly outperforms the simple method in terms of both error metrics. Forecasts from both methods will tested though implementation of the bidding strategies described in Sections III-B-III-D in order to quantify this improvement in monetary terms.

C. Offer Strategy Results
The revenue generated for the non-risk-constrained strategies described in Sections III-A-III-C are calculated using the half-hourly metered power from a portfolio of five UK wind farms and the forecasts described above. These results are tabulated in Table II along with results using perfect and simple benchmark forecasts for comparison. Any additional income from subsidies or other incentive schemes is not included, neither are the costs associated with securing access to the transmission system or electricity market membership.
These results indicate that strategies based on exploiting favourable imbalance prices and a probabilistic forecast of system length can generate more revenue than attempting to minimise imbalance volumes. The strategy based on a deterministic forecast of system length does not improve on imbalance volume minimisation except in the case of perfect foresight demonstrating the significance of imbalance price asymmetry. All strategies perform best when coupled with advanced rather than simple price forecasts. It is notable that perfect power forecasting does not increase revenue in the imbalance minimisation strategy.
The probabilistic forecast of system length based on logistic regression generates more revenue than that based on empirical proportions using simple price forecasts; however, the converse is true when using the advanced price forecasts, despite the logistic model having superior predictive performance.
Risk constrained strategies are evaluated in terms of revenue, the average size of imbalances, and value at risk (VaR α ). The α% VaR is a threshold value such that the chance of the revenue being below that threshold is α%. Here, it is calculated as the α-percentile of the empirical distribution of settlement period revenue. Mean absolute imbalance, given bỹ is also reported to compare the size of imbalance leveraged by each strategy. Revenue, VaR 1% , and mean absolute imbalance are calculated for the three risk-constrained strategies described in  Section III-D. The probabilistic forecast of system length from the logistic model is used along with ARMAX forecasts of the prices. Plots of these results are presented in Figures 2 and 3. Key results are tabulated in Table III. All strategies successfully reduce risk and increase revenue when offer volume adjustments are small, but tend towards the high-risk zero/max strategy for larger adjustments. This change in behaviour occurs at the highest revenue for the multiplicative strategy at the point where η = 1, where the offer is either zero or 200% of expected generation. When η > 1 increasingly extreme short positions are taken while long positions are restricted since offers below zero are not possible.
The additive and quantile strategies, on the other hand, are able to take short positions regardless of expected generation resulting in more frequent and extreme short positions, and therefore, more frequent losses and higher VaR. Since wind power generation is more likely to be close to zero than E max , the effect described above results in short positions being taken more frequently than long positions. This increases VaR since short positions can result in negative revenue, whereas long positions can only result in reduced revenue, unless the balancing price is negative.

VI. CONCLUDING REMARKS
Trading strategies for variable generation participating in electricity markets with single-price balancing mechanisms have been proposed and analysed. The problem is formulated as a decision-making problem under uncertainty and solved relying on a probabilistic forecast of system length, in the first instance to maximise revenue, and in the second to constrain risk. The trading strategies are based on simple analytics using robust and accessible forecasting methods making them adaptable and attractive to many players in the informationrich smart grid paradigm.  Table III. Crosses indicate the results from the revenue-maximising (zero/max) and imbalance minimising strategies tabulated in Table II. It is shown that by hedging against penalising imbalance prices, market participants both reduce imbalance charges and profit from increased exposure to favourable balancing prices. In the most extreme example, revenue is increased by over 10% before considering any subsidy, though this requires the participant to leverage large imbalances that may be considered unacceptable by risk-averse generators, and poor practice by regulators. However, a more conservative riskconstrained approach can increase revenue while simultaneously decreasing risk. While the problem is formulated with the UK electricity market in mind, the principal of positioning oneself favourably in any day-ahead market is applicable to other problems where the cost of correcting that position is reflected in a single price, be that a balancing price or some intraday contract.
Future work should consider extending the problem formulation to include probabilistic price forecasts in order to develop strategies based on the risk associated with specific settlement periods. Furthermore, the limits of the price-taker assumption should be established, and the price-maker scenario studied.