Optimal Storage, Investment and Management under Uncertainty: It is Costly to Avoid Outages!

AuthorGeske, Joachim

    Electricity storage has the technical potential to increase the efficiency of electrical systems significantly--especially in the context of integrating intermittent renewable technologies. This is achieved by shifting energy from periods with a low valuation of electricity (due to low demand or high renewable generation) to periods of high valuation (high demand and low renewable generation). Thus, the share of electricity generated in expensive peak load power plants can be reduced and the share generated by cheaper medium-load power plants can be increased. Additional efficiency gains can come from adapting generation capacity to the post-storage load--with a higher base load and lower peak load share.

    As capacities are adapted to high levels of electricity storage, the available generation capacity will sometimes fall below peak load level, because the storage is a substitute for peak load generation and cutting capacity reduces system costs. But since electricity storage technologies are limited both in the amount of power (MW) they can discharge and the amount of energy (MWh) that they can store, a prolonged demand peak coinciding with low renewable generation raises the risk of outages. The per-MWh costs of unanticipated load-shedding are far higher than the per-MWh gains from adjusting generation capacity and scheduling. Furthermore, electricity demand and renewable generation are uncertain.

    Arbitrage from storage requires charging when prices are low and discharging when they are high. We show that it is optimal to limit discharging, or even to charge the store at times of high prices, if it is holding relatively low amounts of energy. This reduces the risk of a future power cut in the event of an extreme demand pattern, even if it forgoes the arbitrage profit and immediate cost saving that could be made in the short term. As an anonymous referee has pointed out, this could be seen as a strategy of patient arbitrage--holding back energy in the hope of the very high prices likely to accompany extreme demands--but we give it the name "precautionary storage", since it is a response to the uncertainty of electricity demand that would not be seen in a deterministic model. We show the relevance of this precautionary storage in a stochastic model that integrates short-term operations and long-term investment decisions. We model explicit technologies with a step function for marginal cost, even though this makes derivative-based approaches to the solution of stochastic dynamic problems impossible.

    While there are numerous stochastic modelling approaches in electricity systems (see overviews by Wallace and Fleten (2003), Kallrath et al. (2009), Most and Keles (2010) or Rebennack and Kallrath (2017)), relatively few deal with storage. Sioshansi et al. (2009) and Teng et al. (2012) apply the price taker assumption thus abstracting from the endogenous character of precautionary storage in the energy system. A literature overview of the value of storage can be found in Zucker et al (2013). Cruise and Zachary (2015) and Durmaz (2016) explicitly deal with precautionary storage. However, the endogenous capacity component is not considered. Powell et al. (2012) do consider capacity expansion in a stochastic model, comparing solution techniques. They find that precautionary storage is optimal when using Approximate Dynamic Programming, but is not seen when using a scenario-based approach and taking advantage of perfect knowledge within each scenario.

    We set up a stochastic welfare maximization model for the electricity system with storage and derive a jointly optimal strategy for storage, for the (perfectly coordinated) conventional generation outputs and for capacity decisions under residual load uncertainty. We solve the model numerically and analyse the optimal strategy and capacities. To keep things simpler, residual load is considered as the only stochastic driver.

    This residual load exhibits diurnal, weekly and seasonal patterns which make the stochastic optimization time inhomogeneous. These problems are almost insolvable with numerical techniques, not to mention analytical ones. Commonly-used approaches to the solution focus on numerical solutions taking either the short- or the long-run perspective. As these exclusive approaches are not well suited to this analysis we abstract from seasonal, (e.g. Simonsen et al., 2004) time inhomogeneity and unit commitment modelling and apply a Markov Decision Process framework with a diurnal structure to a merit order stack. The Markovian approach and its efficient handling of stationarity opens a feasible way to interpret the long term as the "sum" of many short periods thereby neatly unifying short and long-term perspectives.

    To be specific, the expected costs of generation with storage under capacity constraints to meet a stochastic residual load are approximated by an infinitely repeated "representative" 24-hour average cost minimization problem. This "representative" stage is scaled up to a year in length and the resulting total operating costs are combined with the annualised capacity cost. The model allows the expected system cost to be minimized under uncertainty, considering simultaneously the optimal generation investment and operating decisions for generation and storage.

    Unfortunately, the optimization problem as stated here is non-convex. To determine the global optimum we propose the following algorithm: In the short-term stage, the storage strategy and generation (by merit order dispatch of a set of conventional generation technologies) are optimised as a stationary Markov Decision Process given the generation capacities. The resulting expected load-duration curve is submitted to the long-term stage and capacities are incrementally updated according to a screening-curve approach. (1) These two steps are iterated until a fixed point is reached. A fixed point is a solution candidate for the global optimization problem; we compare it with any other fixed points and select the best. In this paper, we take the capacity of storage as fixed, but a further iterative process could find its optimal value, given costs and loads.

    This algorithm is applied to solve two case studies. First, we specify the model to a simple case with only two generation technologies. We compare a strategy that takes every opportunity for arbitrage with the optimal strategy. In some states of the world, it is optimal to keep more energy in the store than under full arbitrage, in order to reduce the risk of losing load once the store is fully discharged. We show how the optimal strategy depends on the Value of Lost Load. If power cuts are cheap, it may not be worth missing out on arbitrage profits to reduce the risk of running out of energy, while if losing load is costly enough, it will be optimal to keep enough generating capacity to always meet demand in full. This in turn would allow storage to follow an arbitrage strategy with no risk of causing power cuts.

    Our second case study represents Germany in 2011-15, with 300 GWh of storage capacity (6 hours average load), and five conventional generation technologies with empirical fixed and variable cost. The Markov process for the residual load is estimated from five years of hourly data. Under these more realistic conditions precautionary storage occurs as capacity is reduced below peak load. The strategies are qualitatively similar to those in the simple case but the quantitative extent could be determined more realistically. In comparison to perfect foresight analyses, uncertainty reduces the gain from storage by 27%, as energy is held back for precautionary reasons. This could be an overestimate of the difference, as some more information on future loads will be available due to weather forecasts, an issue we explore in follow-up work (Geske and Green, 2018); on the other hand, we ignore uncertainty over generator outages.

    The article is structured as follows: First in section 2 the general setting of a stochastic dynamic electricity system model with fossil generation technologies and storage as a Markov Decision Process is introduced and the solution algorithm is proposed. We show in Section 3 that the residual load can be estimated as a Markov Process, using hourly load data for 2011-2015 in Germany. Based on the modelling environment of Section 2 and a simplified version of the estimated Markov process in Section 3, in Section 4 a two-technology storage and generation model is presented. This analysis is empirically refined in Section 5 with our calibration to Germany in 20112015. The model is solved numerically and the optimal storage strategy is presented. Sensitivities with respect to storage capacities and a comparison to perfect foresight deepen the understanding of the storage strategy and the impact of the uncertainty. In Section 6 we draw conclusions for the implementation of the optimal strategy in a market environment.


    In the following section, welfare-maximizing capacities, outputs and storage decisions are determined to derive the social value of electricity storage. Welfare is interpreted in terms of minimising the annualised system cost of meeting the demand for electricity, or occasionally leaving some unserved at a cost of VoLL, the Value of Lost Load. The demand to be met, [D.sub.t], is the load net of the output of variable renewable generators such as wind and solar plants. Electricity can be generated by a portfolio of non-intermittent technologies, which we model as a merit order stack, deploying them in order of increasing variable cost without considering dynamic constraints. The vector [x.sub.t] [greater than or equal to] 0 describes production levels for each non-intermittent generation technology in hour t. [x.sub.t] is limited by capacities k that have to be set in advance, so that x

    Generators have fixed costs per unit of capacity...

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