This chapter studied a novel problem in disaster management, the Single Commodity Allocation Problem (SCAP). The SCAP models the strategic planning process for disaster recovery with stochastic last-mile distribution. We proposed a multi-stage stochastic optimization algorithm that yields high-quality solutions to real-world benchmarks provided by Los Alamos National Labora- tory. The experimental results on the benchmarks indicate that the algorithm is practical from a computational standpoint and produces significant improvements over existing relief delivery proce- dures. This work, currently deployed at LANL as part of the National Infrastructure Simulation and Analysis Center (NISAC), is being used to aid government organizations such as the Department of Energy and the Department of Homeland Security in preparing for and responding to disasters.
Restoration of the Power Network
Restoration of the power grid during the first few days after a major disaster is critical to human welfare since so many services — water pumps, heating systems, and the communication network — rely on it. This chapter solves the following abstract disaster recovery problem: How can power system components be stored throughout a populated area to minimize the blackout size after a disaster has occurred? This case study has only one infrastructure system, the power network. But we will see that just one infrastructure system brings significant computational challenges. The restoration set problem and the restoration order problem from Section 1.3 are essential to solve problems of real-world sizes within the disaster recovery runtime constraints.
This problem is studied in two sections, the stochastic storage problem (Section 3.2) and the restoration routing problem (Section 3.7).
The contributions of this work are:
1. Formalizing the stochastic storage problem and restoration routing problem for power system restoration.
2. An exact mixed-integer programming formulation to the stochastic storage problem (assuming a linearized DC power flow model).
3. A column-generation algorithm that produces near-optimal solutions to the stochastic storage problem under tight time constraints.
4. A multi-stage routing solution for the power system restoration routing problem that produces substantial improvements over field practices on real-life benchmarks of significant sizes. 5. Demonstration that all the solution techniques can bring significant benefits over current field
practices.
The rest of this chapter is organized as follows. Section 3.1 reviews related work in power restoration and power system optimization. Section 3.2 presents a specification of the power system stochastic storage problem (PSSSP). Section 3.3 presents an exact MIP formulation using a linearized
DC power flow model. Section 3.4 presents the column-generation algorithm for solving PSSSPs, Section 3.5 presents greedy algorithms for PSSSPs aimed at modeling current practice in the field, and Section 3.6 reports experimental results of the algorithms on some benchmark instances to validate the stochastic storage algorithms. Then Section 3.7 formalizes the restoration routing problem, Section 3.8 presents the multi-stage approach based on constraint injection, and Section 3.9 reports the experimental results validating the approach. Section 3.10 concludes the chapter.
3.1
Related Work
Power engineers have been studying power system restoration (PSR) since at least the 1980s ([1] is a comprehensive collection of work) and the work is still ongoing. The goal of PSR research is to find fast and reliable ways to restore a power system to its normal operational state after a blackout event. This kind of logistics optimization problem is traditionally solved with techniques from industrial engineering and operations research. However, PSR has a number of unique features that prevent the application of traditional optimization methods, including:
1. Steady-State Behavior: The flow of electricity over a AC power system is governed by the laws of physics (e.g., Kirchoff’s current law and Ohm’s law). Hence, evaluating the behavior of the network requires solving a system of nonlinear equations. This can be time-consuming and there is no guarantee that a feasible solution can be found.
2. Dynamic Behavior: During the process of modifying the power system’s state (e.g., en- ergizing components and changing component parameters), the system is briefly subject to transient states. These short but extreme states may cause unexpected failures [2].
3. Side Constraints: Power systems are comprised of many different components, such as generators, transformers, and capacitors. These components have some flexibility in their operational parameters but they may be constrained arbitrarily. For example, generators often have a set of discrete generation levels, and transformers have a continuous but narrow range of tab ratios.
PSR research recognizes that global optimization is an unrealistic goal in such complex nonlinear systems and adopts two main solution strategies. The first is to use domain expert knowledge (i.e. power engineer intuition) to guide an incomplete search of the solution space. These incomplete search methods include Knowledge-Based Systems [89], Expert Systems [62, 3, 7], and Local Search [75, 76]. The second strategy is to approximate the power system with a linear model and solve the approximate problem optimally [108]. Some work has hybridized the two strategies by designing expert systems that solves a series of approximate problems optimally [77, 61].
Interestingly, most of the work in planning PSR has focused on the details of scheduling power system restoration [2, 3]. More specifically, what is the best order of restoration and how should sys- tem components be reconfigured during restoration? In fact, these methods assume that all network
components are operational and simply need to be reactivated. In this work, we consider both stock- piling of power network components before a disaster has occurred and recovery operations after a disaster has occurred. This introduces two new decision problems: (1) How to stockpile power- system components in order to restore as much power as possible after a disaster; (2) How to dispatch crews to repair the power-system components in order to restore the power system as quickly as pos- sible. There are strong links between traditional PSR research and our disaster-recovery research. In particular, finding a good order of restoration is central in the repair-dispatching problem. However, the joint repair/recovery problem introduces a combinatorial optimization aspect to restoration that fundamentally changes the nature of the underlying optimization problem. The salient difficulty is to combine two highly complex subproblems, vehicle routing and power restoration, whose objec- tives may conflict. In particular, the routing aspect optimized in isolation may produce a poor restoration schedule, while an optimized power restoration may produce a poor routing and delay the restoration. To the best of our knowledge, this work is the first PSR application to consider strategic storage decisions and vehicle routing decisions.
The ultimate goal of this work is to mitigate the impact of disasters on multiple infrastructures. Disaster response typically consists of a planning phase, which takes place before the disaster occurs, and a recovery phase, which is initiated after the disaster has occurred. The planning phase often involves a two-stage stochastic or robust optimization problem with explicit scenarios generated by sophisticated weather and fragility simulations (e.g., [30, 31, 102]). The recovery phase is generally a deterministic optimization problem that assumes, to a first approximation, that the damages to the various infrastructures are known. For the power infrastructure, the planning phase — determining where to stockpile power components under various disaster scenarios — is described in Section 3.2, while the recovery phase — how to repair and restore the power infrastructure as fast as possible given the stockpiling decisions — is described in Section 3.7.