Two-stage temporal decomposition technique

A scheduling algorithm is only useful when it can generate an optimal strategy (or at least a near optimal strategy) within time restrictions imposed by the market horizon. All optimal strategies discussed till now were generated in a few hundred seconds using the optimization model. For cases where greater accuracy is desired (e.g. by increasing the linear segments defining a non-linear function) or when optimizing for more trading intervals such as for multiple days in the day-ahead market or for a balancing market (with 15 min trading intervals), the optimization routine is unable to provide solutions with zero optimality gap within reasonable time. In this section, a decomposition technique is developed to enable computation for these cases while delivering near-optimal solutions much quicker than the original program.

The decomposition method works by breaking the problem in the time domain into smaller tractable sub-problems. By tractable, it is implied that each of these sub-problems can be solved with a zero optimality gap in a reasonable time-frame with the computing resources at hand.

In the model structure, variable d^1C_t is computed as the difference of two consecutive time interval values of δ^1C where δ^1C is only a function of SOC (Equation 9.12). Thus, fixing an intermediate SOC value, effectively divides the problem into two independent sub-problems. This forms the basis of the temporal decomposition technique.

As for the value of the fixed intermediate SOC, the midway value of 50 % is a good heuristic as it hedges equally on the possibility of both charging and discharging. However, an even better guess for setting the intermediate SOC is possible. This is achieved by solving the original non-decomposed optimization problem for the entire market period for a fixed period of computation time or optimality gap, whichever comes earlier. The best feasible solution at this stage is used for fixing the intermediate SOC value. Typically, the optimization algorithm (such as CPLEX) finds good feasible solutions in little time initially but then has difficulty closing the optimality gap. It is demonstrated later in this chapter that fixing the intermediate SOC value in this way can lead to better solutions than fixing the intermediate SOC state at 50 %.

Once the intermediate SOC value has been fixed, the decoupled sub-periods,

9.7. Two-stage temporal decomposition technique 153

Figure 9.9. Two-stage temporal decomposition method

which are easier problems, are solved individually. The two-stage methodology is graphically shown in Figure 9.9. The method is not limited to fixing just one intermediate SOC. More intermediate SOC values can be fixed depending on the complexity of the problem.

Equivalence of the decomposition technique: the equivalence of the decomposition method to the original problem is first shown through the day-ahead market of the previous section of 48 trading intervals. The decomposition method works by fixing intermediate SOC values. For demonstration, the SOC at the end of the trading interval 24 is fixed at 50 %.

Optimizing for scheduling for the entire duration of 48 intervals is now equivalent to separately optimizing for the first 24 intervals and the last 24 intervals. This equivalence can be seen for the case ω = 0.4 in Table 9.2.

The solution obtained by fixing the intermediate SOC value to 50 % is sub-optimal compared to the full optimization over the 48-hour interval seen in Figure 9.7. This is because, the fixed SOC is not the optimal SOC state for that time interval. From Figure 9.8, the optimal SOC state after the 24th trading interval is seen to be zero. If the intermediate SOC at interval t₂₄ had been fixed at 0, instead of 50, it would have resulted in the optimal solution also using the two-stage temporal decomposition technique.

Table 9.2. Decomposition equality

(e) 102.40 8.50 93.90 102.40

Degradation

(mAh) 1.19 0.74 0.45 1.19

ζ 33.84 -1.01 34.85 33.84

Table 9.3. Progress of solving 336 interval as a whole Time (s) Value of ζ

Application of the decomposition technique: the application of the proposed decomposition technique for solving the optimization problem for more trading intervals is demonstrated by considering the day-ahead market prices for one week from 22 January 2018 to 29 January 2018 for the UK SEM. The number of trading intervals is 336. To make the problem even more difficult and highlight the utility of the two-stage temporal decomposition approach, ω = 0.3 case is chosen for demonstration. For this value of ω, the 48 interval problem could not find an optimal solution even after 3600 s.

When optimizing for the entire 336 interval as a whole, the multi-objective function ζ progress with time is as shown in Table 9.3. The incremental progress is very slow after 1500 s. Using the two-stage decomposition approach, 336 trading intervals are subdivided into 14 periods of 24 intervals each. The intermediate SOC values are set using the best solution case at 1500 s. Each sub-period is then individually optimized. The value of ζ after adding the individual sub-periods is found to be 116.89. The entire two-stage optimization routine runs in less than 2000 s and outputs a better scheduling strategy than the original program even after 36 000 s. This strategy can be seen in Figure 9.9.

In terms of the two objectives, this scheduling strategy yields 79 % of the maximum revenue (e497.15 vs. e629.45) while reducing the degradation by 81.6 % (4.61 mA h vs. 25.12 mA h). Degradation measured as capacity fade in one battery unit forming the storage system is also plotted in Figure 9.10. Sharp changes of SOC at high SOC values cause the most capacity fade in the battery.

There is a scope of further improvement in the total time taken by the