Result analysis

In this section we compute a range of different metrics that allow us to further illustrate the advantages of the flexible stochastic model over using more naive approaches that do not consider scenario-specific recourse actions or the explicit modelling of the entire uncertainty characterizing the decision process.

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4.7.4.1 Expected Value of Perfect Information

The expected value of perfect information (EVPI) represents the price the planner would be willing to pay to gain access to perfect information concerning future generation developments and constitutes a proxy for the value of accurate forecasts [54]. It is a useful metric for quantifying the effect of uncertainty on system costs.

In the case that the planner had perfect foresight, knowing which scenario will occur with certainty, he could follow an optimal tailor-made plan for each realization. In Appendix C we list the optimal plan that would be optimal under each scenario, obtained by solving the corresponding deterministic problem ignoring uncertainty. The probability-weighted system cost of these plans is £277.7m, which is 26% lower than the expected system cost of the flexible stochastic solution. This system cost difference of £98.3m is the EVPI and reflects the high cost impact of the uncertainty present in this case study. For the inflexible planner, EVPI increases to £157.9m, illustrating that the inability to take advantage of managerial flexibility leads to an even greater value of acquiring an accurate forecast and an amplified impact of uncertainty.

4.7.4.2 Expected Cost of Ignoring Uncertainty

The expected cost of ignoring uncertainty (ECIU) is useful in quantifying the net benefit loss when ignoring uncertainty and basing first-stage investment decisions on a single scenario which is naively deemed certain to occur. This naïve approach can result in over or underinvestment in transmission assets, ill-conditioning the planner’s ability to effectively respond to an unfavourable realization in subsequent epochs. Thus, ECIU constitutes a proxy to the benefit of utilizing a stochastic approach over a deterministic one to determine the ‘here and now’ decisions [71]. In order to calculate ECIU, we first compute the expected system cost when first-stage decisions are fixed according to the optimal deterministic plan of each scenario. The cost of ignoring uncertainty is the difference between the acquired expected system cost and the optimal solution of the stochastic problem. The detailed results are shown in Table 4-20.

143 First-stage decision of scenario Expected System Cost [£m] Cost of Ignoring Uncertainty [£m] 1 379.4 3.4 2 379.4 3.4 3 379.4 3.4 4 379.4 3.4 5 379.4 3.4 6 434.3 58.3 7 379.4 3.4 8 434.3 58.3 9 434.3 58.3 10 398.8 22.8 11 398.8 22.8 12 398.8 22.8 13 398.8 22.8 14 398.8 22.8 15 398.8 22.8 16 398.8 22.8 17 398.8 22.8 18 398.8 22.8 Expected 20.5

Table 4-20: Costs of ignoring uncertainty.

The cost of ignoring uncertainty varies according to which scenario is used as a basis to draw the first-stage decisions. If the most probable eventuality is used (scenario 1), the cost is relatively low and stems from overinvesting in line 1 and prematurely committing to a line 6 upgrade. The highest costs occur when planning with respect to scenarios 6, 8 and 9 which correspond to a low projection of new generation additions. Under these scenarios, project A is chosen to reinforce the main exporting corridors of bus 3, significantly limiting the planner’s ability for recourse in the event of high-growth transitions. It is important to note that no deterministic plan results in the optimal first-stage decisions given by the stochastic model. For this case study, the ECIU is £20.5m, underlining the unsuitability of employing deterministic scenario analysis methods.

4.7.4.3 Value of the Stochastic Solution

The value of the stochastic solution (VSS) is a measure for quantifying the benefit of using a stochastic programming approach over a deterministic one, where random variables are replaced by their expected values [54]. VSS can be computed as the decrease in expected system cost between the extended stochastic formulation (£376.0m) and the objective

144 function of the stochastic problem when fixing first-stage decision variables to the optimal values provided by the equivalent deterministic model. The random variable expected values to be included in the deterministic formulation are shown in Table 4-21.

Generator Expected Capacity [MW]

Stage 1 Stage 2 Stage 3 Stage 4

G27 0 150 337.5 517.5

G28 0 100 225 345

Table 4-21: Expected value of future generation capacities.

The first-stage optimal solution of the corresponding deterministic model is the reinforcement of line 2 by 200 MW using project A. This results in substantial ill-conditioning of the system, by locking in a very specific investment path which may limit subsequent recourse actions. By utilizing this first-stage investment decision and solving the flexible stochastic transmission expansion problem, the expected system cost is £438.9m. As a result, the VSS is £62.9m, indicating that if the planner was to base his first-stage commitments on the expected value of future generation additions, he would be exposed to a 16.7% increase in expected costs. Solving the equivalent deterministic problem may initially appear as a practical proxy for modelling the underlying uncertainty, particularly due to the reductions in problem size and modelling complexity. However, failing to properly model uncertainty leads to the so- called ‘flaw of averages’ and negatively impacts the quality of decisions. The substantial VSS highlights the importance of taking into account the whole range of possible eventualities and fully utilizing the information available from the scenario tree, instead of relying on average values. Again, the ill-conditioning that may occur from relying on deterministic methods is evident.

4.7.4.4 Value of Flexibility

The value of decision flexibility can be defined as the expected system cost difference between the flexible and non-flexible approach. By definition, the net benefit gain of modelling decision flexibility is positive. This is because decisions are constrained only by the non-anticipativity dictated by the scenario tree instead of being forced to be identical across all realizations. For this case study, the value of flexibility is £60.6m, meaning that flexible stochastic planning results in a 13.8% reduction of expected system costs. The extra cost experienced under the inflexible planning paradigm is the result of eliminating the option to ‘wait and see’ until uncertainty is resolved while also depriving the planner from considering scenario-specific recourse actions. The value of flexibility becomes even greater

145 when versatile congestion management measures, such as the installation of quadrature boosters, are at the planner’s disposal. Due to their small construction delay, these devices can be deployed in a ‘just-in-time’ manner, according to the unfolding uncertainty. When considering investment in QBs, the value of flexibility increases to 29.8% of the stochastic solution. The detailed optimal expansion plans and system costs are presented in Appendix D.

The most important implication of ignoring decision flexibility is the sub-optimality of the first stage investment decisions, which is the implementable part of the solution. In the flexible stochastic problem formulation, the system planner adopts a ‘wait-and-see’ stance towards investing in line 6. The decision to upgrade this line is postponed to the second epoch, when locational uncertainty has been fully resolved and more informed decisions can be made. The inflexible planner resorts to a more conservative first-stage planning decision, where investment in line 6 is undertaken on a non-conditional basis. We can quantify the sub- optimality of this premature commitment by calculating the difference between the optimal flexible stochastic solution and the expected system cost that arises when committing to this first-stage decision, while allowing for scenario-specific recourse in subsequent epochs. The difference is £3.5m, representing the expected welfare loss due to the failure to consider flexibility when identifying the optimal ‘here and now’ decisions.

4.7.4.5 Regret Analysis

Further insight on the suitability of the proposed model can be obtained by conducting a regret analysis. The regret associated to a specific plan under a scenario realization is defined as the difference between the system cost experienced and the system cost that would have been obtained if the optimal course of action had been taken. With respect to each scenario, the best possible system cost is obtained through solving the corresponding deterministic planning problem (Table 4-21). In this analysis, we focus on the regret associated to the first- stage decisions obtained when employing different planning approaches, while allowing for scenario-specific recourse in subsequent epochs. The regret matrix (Figure 4-12) presents the regrets associated with the 21 planning approaches that have been considered. The first 18 rows relate to deterministic models (presented in Appendix C), where the planner bases his first-stage decisions on the assumption that a single specific scenario will occur. We also calculate the regret associated to the first-stage decisions obtained through the equivalent deterministic, non-flexible stochastic and flexible stochastic approaches. By taking the

146 probability-weighted average of the regret experienced over all scenarios, we can calculate the expected regret of each planning approach.

As can be seen in the regret table, the realizations that lead to the highest regret levels are scenarios 1 and 18. The former constitutes a high-growth eventuality that requires significant reinforcements. Premature commitment in projects capable of providing only small capacity additions (project A) severely ill-condition the systems and prohibits the planner from properly accommodating the arising power flows. The latter requires no reinforcements and thus any transmission investment undertaken is unnecessary. The largest expected regret is experienced when fixing first-stage decisions to the ones obtained through the equivalent deterministic model. High expected regret is also experienced when planning deterministically for scenarios 6, 8 and 9. All these approaches lead to project A commitments for the lines exporting power from bus 3, thus leading to very high costs if high-growth scenarios materialize. The expected regret is minimized under the flexible stochastic approach, since minimizing expected costs, while modelling decision flexibility, is equivalent to minimizing the expected regret [15]. Although adopting this approach results in positive regrets under all scenarios, with the greatest regret experienced if all phases of G28 are successfully commissioned (scenario 10), it performs best on average. This highlights the superior performance of first-stage decisions provided by the proposed model.

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Figure 4-12: Regret matrix when using first stage decision dictated by deterministic (Ds, where s is the scenario considered), equivalent deterministic (EB), non flexible stochastic (NFS) and flexible

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