Optimisation models, horizons and algorithms

A storage revenue optimisation model in its simplest form can be modelled a Linear Programming (LP) problem; Denholm & Sioshansi (2009) used LP to evaluate arbitrage revenues in the PJM market. (Bradbury et al. 2014) similarly used linear optimisation to investigate arbitrage values across several storage technologies. In Austrian power markets, Kloess & Zach (2014) used an LP model to operate on arbitrage values taking into account variable costs. More recently McConnell et al., (2015) also used an LP model to determine arbitrage values in the Australian markets.

More complex models have been evaluated using Mixed Integer Linear Programming (MILP); storage optimisation problems often have binary variables at their core and hence the use of MILP. Drury et al., (2011) have used MILP to compare the performance of CAES vs AACAES under co-optimisation of revenues. He et al., (2011) used MILP to determine a business strategy for storage while Yucekaya (2013) used MILP, to evaluate a CAES revenue model with several binary variables. More recently Moreno et al., (2015) use MILP to evaluate co-optimised storage operation embedded in distribution networks whereas Chazarra et al., (2016) use an MILP model to schedule PHES output.

Storage has also been modelled through the unit commitment and economic dispatch problem; Foley & Díaz Lobera (2013) have used PLEXOS (Energy Exemplar n.d.) to model CAES participation in Ireland.

The objective function is the minimisation of the whole system electricity cost for each period, consisting of generation costs and uplift costs for each generator3_{. Similarly, Das et al., (2015)}

investigate the co-optimised value of storage in the US under different wind penetration scenarios. The model first solves the Unit Commitment problem of conventional generation based on wind and load forecasts. The next phase looks at determining the most economic dispatch solutions taking into account locational marginal prices and Market Clearing Prices (MCP) for wholesale and ancillary services respectively. The authors use an MILP technique to minimise within the transmission system, the cost of wholesale energy, spinning reserve, non-spinning reserve, upregulating reserve and downregulating reserve as well as penalty payments for not serving the load.

While these studies have clearly presented an optimisation model; other studies refer to the use of optimisation without the explicit description of an optimisation algorithm or mathematical notation. Rather the term ‘optimisation’ is used to refer to a profit generating regime rather than profit maximising one. For example, Ekman & Jensen (2010) refer to their arbitrage strategy as one that maximises revenues; the authors argue that by imposing the efficiency adjusted selling price to be greater than the buying price, maximum revenues are derived, shown in equation 2.1. Variables n and p represent the energy volume and prices respectively. Pbuy and Psell are price thresholds for buying and

selling, shown in figure 2.5. The use of price thresholds is sub-optimal as a true optimisation algorithm cannot use fixed price thresholds as a criterion in the presence of price variations.

Similarly Locatelli et al., (2015) refer to their approach as an optimisation method, however, it is not clear whether the model actually performs an optimisation. They propose that storage operates similar to a generator, discharging when efficiency adjusted prices are greater than the marginal cost, in their model this being the variable operating cost. This is also one of the conditions Ekman & Jensen (2010) used. While this is a necessary condition for maximum profitability it is not a sufficient one. As opposed to an optimisation algorithm, the use of price thresholds to determine charging and discharging results in missed opportunities for arbitrage trade. Under this approach, when the net selling price is greater than the threshold (any price above marginal costs), discharging will take place immediately irrespective if there is a better trade later. By contrast an optimisation models will pick the best trades over the optimisation horizon even if this means foregoing present discharge for a more lucrative discharge later.

For example, a storage system which has a choice between discharging 1 MWh of energy at a £25 profit now or a £75 profit in the next period will choose the former approach (that marginal revenues

3 _{Uplift costs are defined as additional costs to marginal costs such as start-up costs and other variable and fixed} costs.

is greater than marginal cost or an equivalent price threshold). However, an optimisation algorithm would defer discharge for the next period to maximise profits.

A further limitation of Locatelli et al., (2015) is the treatment of Round Trip Efficiency (RTE); it is not explicitly included in the model. Instead, efficiency is included as a fixed cost, calculated by the efficiency loss on the average price of electricity at £34/MWh. As an example, the authors show that an 85% RTE PHES would lose £5.1 per MWh (charged and discharged). This static treatment of efficiency, through the utilisation of a fixed cost, is likely to result in sub-optimal dispatch and hence sub-optimal profits.

(2.1); (Ekman & Jensen 2010, p.1146)

Figure 2.5: Example of a storage energy capacity scheduling (top) and trading strategy (bottom)

Source: Ekman & Jensen (2010)

Generally, an optimisation model returns the value of an objective function and the decision variables associated with this value. For a storage problem, the decision variables are usually charging and discharging volumes for discrete time periods. Hence the number of decision variables evaluated

determines the time period considered for the optimisation problem; for example, evaluating storage operations for two half-hourly periods implies the model is evaluated over a one-hour optimisation horizon.

In previous studies the length of the optimisation horizon has been chosen, without explicit justification; Chazarra et al. ,(2016) and Sioshansi et al., (2011) use a 1-week horizon. Sioshansi et al., (2009) and Drury et al., (2011) used a 2-week optimisation horizon in their model to allow for inter and intra-day trades. Kloess & Zach (2014) and Yucekaya (2013) used a 1-year optimisation horizon. Therefore, there is particular merit in exploring the impact of optimisation horizons on storage value and whether clear preferences exist.