Computational performance

The computational performance of the non-flexible and flexible models is presented in Table 4-22. In the case of the flexible model, we also include the performance of the scenario- variable formulation. All runs have employed the contingency screening module and multi- cut Benders decomposition, where operational subproblems were run in parallel using 12 processors. Both models required only two iterations of the contingency screening module to identify the optimal N-1 secure investment and dispatch schedules.

Model Problem formulation Contingency screening iteration index Benders iterations Objective function (£m) CPU time (s) CPU Memory usage (GB) Non-Flexible Node-variable 1 7 193.2 67.3 0.31 2 7 435.6 254.1 0.54 Flexible Scenario-variable 1 8 110.7 613.7 1.04 2 - - >43,200 >56.0 Node-variable 1 8 110.7 86.4 0.45 2 11 376.0 1290.2 2.35

Table 4-22: Computational performance of the stochastic transmission expansion models.

The first thing to note is the increased complexity of the flexible model. Naturally, the computational cost of identifying the optimal investment strategy involving a large number of possible recourse actions is high when compared to finding a unique optimal expansion schedule.

The most important observation concerns the remarkable computational benefits of employing the node-variable over the scenario-variable problem formulation. Although the number of benders iterations taken to converge is the same, since the two formulations are equivalent, the increased number of operating points and investment decision variables as well as the explicit inclusion of non-anticipativity constraints in the scenario-variable approach, result in considerably longer CPU times and increased memory usage. For example, in this case study, the node-variable master problem contains 3,159 binary decision variables (N_MN_LN_W) related to the different line investment options. In addition, at each benders iteration, 2,700 operational subproblems are solved and as many Benders cuts are appended to the master problem. On the other hand, the scenario-variable formulation decomposes in a master problem containing 8,424 binary decision variables (N_SN_EN_LN_W) and 7,200 subproblems. The significant benefit of the reduced problem size becomes even

149 more apparent when taking into account the fact that problem complexity scales non-linearly with the number of constraints and decision variables considered.

As seen in Table 4-22, the node-variable model utilized a limited amount of RAM memory and converged in a total time of 1376 seconds, with the non-secure planning problem being solved in just over one minute. The scenario-variable formulation took more than 10 minutes to converge in the absence of security constraints. In the second contingency screening iteration, when constraints related to the identified binding contingencies were included in the operation subproblems, the model failed to converge within an acceptable time. The model was stopped at the 5th benders iteration, with the solver utilizing more than 56 GB of RAM to traverse the master problem’s branch-and-bound tree and failing to find the optimal MILP solution after 12 hours of processing.

Figure 4-13: Time spent in master and subproblems when using the scenario-variable (SV) and node-variable (NV) formulations in the first contingency screening iteration.

In Figure 4-13 we show the time spent in the master and subproblems under the two problem formulations in the first contingency screening iteration (intact system planning). With respect to solving the operational subproblems, the node-variable model needed an average time of 5 seconds, while the scenario-variable model needed more than 15 seconds; a three- fold increase which is roughly equal to the 2.67

M E S N N N

ratio. As far as the master problem runtimes are concerned, the scenario-variable model clearly has a substantially inferior

0 20 40 60 80 100 120 140 160 180 200 1 2 3 4 5 6 7 8 9 Second s Benders Iteration SV - subproblems SV - master problem NV - subproblems NV - master problem

150 performance, particularly in later iterations when a larger number of Benders cuts has to be considered.

From the above observations we conclude that when dealing with stochastic MILP problems consisting of multiple stages, operating points and decision variables, the utilization of a node-variable formulation leads to superior computational behaviour and is essential for achieving convergence in acceptable times. In addition, the utilization of Benders decomposition and contingency screening significantly reduce CPU time.

4.8 Conclusions

In this section we have presented the multi-stage stochastic transmission expansion problem. Uncertainty has been expressed in the form of a discrete scenario tree representing the evolution of the generation background in the future. Under this approach, investment decisions are expressed in the form of a strategy, where the inter-temporal resolution of uncertainty is utilized to take more informed decisions. The techniques developed in the previous chapter have been incorporated in the formulation to improve computational performance. The combination of a multi-cut Benders decomposition scheme with a node- variable approach succeeds in rendering the large mixed integer linear problem tractable and allows the simulation of large systems. A case study on a three bus-bar system indicates how the lack of flexibility can prematurely lock planners into sub-optimal investment paths that lack the upgradeability required under some scenarios in the future. An additional case study is presented on the IEEE RTS, with results confirming that modelling the decision flexibility of a system planner results in further expected cost minimization than when adopting a fixed expansion schedule. This reduction highlights the strategic importance of planning with adaptability in mind. Of great interest is the difference in first-stage commitments that are taken in the absence of managerial flexibility. The stochastic planner is shown to adopt a ‘wait-and-see’ stance when appropriate, and proceed with projects on a conditional basis, subject to the unfolding uncertainty. On the other hand, the non-flexible planner exhibits a ‘jump-to-solutions’ behaviour, committing prematurely to projects that may prove to be unnecessary. Finally, the proposed method results in the minimum expected regret when compared to deterministic and inflexible approaches, illustrating that the optimal exercise of the planner’s inherent flexibility can constitute a well-founded approach to coping with system uncertainties.

151

5 Risk-Constrained Transmission Expansion

Planning

Abstract

In the face of uncertainty, the provision for limiting the impact of adverse scenarios is an indispensable part of robust decision-making. Although the optimal exercise of managerial flexibility can substantially reduce risk through contingent actions, it is only through the explicit introduction of a risk measure constraint that acceptable levels of risk exposure can be guaranteed under all realizations. In this chapter we include CVaR constraints to the flexible stochastic formulation, in order to limit the risk of experiencing excessive constraint costs. A novel node-variable formulation is developed and incorporated into the previously developed solution strategy. Computational load due to scenario-wise coupling is substantially reduced and large systems can be modelled. A case study on the IEEE RTS is undertaken in order to identify the optimal expansion strategies under different risk profiles.

152

5.1 Introduction

Risk is the possibility that a chosen activity or decision will lead to an undesirable outcome and is an integral part of any investment activity under uncertainty. It is essential that investors are well aware of the risk associated with their decisions and ensure that possible deviations from the expected outcome are acceptable. The stochastic transmission expansion formulation presented in Chapter 4 is risk-neutral, meaning that the planner’s objective is a straightforward minimization of expected system costs. However, there are aspects of the optimal solution that the planner may find unattractive. A characteristic example of such an undesirable outcome is excessive constraint costs being experienced. Investment decisions taken in the absence of risk considerations may limit the planner’s ability for recourse in the case of adverse scenarios materializing. Despite the minimization of expected system costs, the system can be left unable to adapt effectively to particular events, giving rise to a risk of high constraint costs. In many cases, the planner would wish to immunize his decisions against such an eventuality and ensure that the strategy being followed can guarantee with some degree of certainty that the realized constraint costs will always be within some allowable limit. Naturally, risk management always comes at a price, which in this context is an increase in transmission investment. By incorporating risk constraints in the developed stochastic optimization framework, it is possible to find the expansion strategy that minimizes expected system costs while bounding congestion risk according to the planner’s level of risk averseness.

The planner’s risk averseness has to be expressed in terms of a suitable risk measure. A risk measure is a function that associates a random variable (in this case constraints cost) with a single real number that characterizes the underlying risk. The most fundamental risk measure is the distribution variance. Other examples of risk measures may include the probability of being above a target value, or the threshold value whose probability of being surpassed is equal to a pre-defined confidence level. Risk management can be applied through the inclusion of the chosen risk measure in an optimization constraint that bounds it to be below an acceptable level. Some standard risk measures are: