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Weekly UpdatesINTRODUCTION
Many electric utilities in the United States were built in the early 1960s, and although highly reliable, the electricity infrastructure is old and aging (Brown and Willis, 2006; Li and Jiachun, 2006; Li et al., 2006; Endrenyi and Anders, 2006). Aging infrastructure has higher costs to operate and maintain and, more importantly, lower reliability. As equipment ages, the component outage rates increase, having an impact on the total system downtime and leading to increased costs. Typically, network upgrades and improvements have been made in an ad hoc or myopic manner given short-term budgets. Therefore, there is a need to develop models to systematically, and intelligently, upgrade the Electricity Transmission and Distribution Systems (ETDS) grid.
This article presents a new component replacement heuristic to solve replacement analysis problems for systems designed with a set of heterogeneous assets subject to annual budget constraints. In these types of problems, the objective is to plan for replacements over an extended planning horizon to minimize the total system cost. This is a unique problem since previous methods only consider systems composed of sets of homogeneous assets (Hartman, 2000, 2001: Karabakal et al., 1994; Hartman and Rogers, 2006). This new method can potentially be applied to obtain component replacement policies for many different types of systems composed with a large number of components, with different reliability behavior and different costs.
Replacement analysis is designed to minimize operating costs by identifying the optimal time periods to replace aging assets with a new or refurbished replacement. The performance of components within most operating systems deteriorates with age. As these assets are utilized over time, they age, become worn, and lead to increased operating and maintenance expenditures. Therefore, the timely replacement of these assets is necessary to assure economically efficient operations. Determining minimum cost replacement schedules requires the analysis of current and future costs over some time horizon.
Aging components associated with ETDS include transformers, power lines, circuit breakers, etc. Although routine maintenance can keep the equipment working, there comes a point in time when the repairs occur too frequently and are too expensive, and it becomes economically prudent to replace or completely refurbish the old component or system.
The present work proposes a method for solving capital replacement problems for a set of heterogeneous assets within ETDS subject to annual budget constraints. The modeling of a radial configuration is presented. The main objective of the replacement model is to identify where and when investments should be made in a more effective and efficient way in order to minimize the total system cost over a finite planning horizon.
BACKGROUND
Replacement Analysis Policies
Economic replacement analysis involves the determination of optimal time intervals to replace aging components. Given a level of output or service required from an asset over time, a decision is made periodically to either keep or replace the asset, as it wears with the aging process. This sequence of keep and replace decisions over the given time horizon is determined, such that some total cost function is minimized. Costs include capital or replacement costs (purchase costs and salvage revenues), operating and maintenance costs, and cost of unmet demand (referred to as opportunity costs).
In general, a replacement problem can be categorized as either serial or parallel replacement. Serial replacement problems consider a single asset or multiple independent assets. In serial replacement problems, it is assumed that there is no economic interdependence among the assets that provide the service together. Therefore, their replacement decisions can be made separately.
Parallel replacement analysis considers assets that are economically interdependent and operate in parallel. Economic interdependence may result from system-level budget constraints, demand constraints, or service requirements. For parallel replacement problems, the desired solution includes keep and replace decisions for each individual asset over the planning horizon, resulting in a difficult combinatorial optimization problem as the replacement of groups of assets must be analyzed.
Dynamic Programming Modeling of Replacement Analysis
In this work, a dynamic programming formulation is developed to obtain replacement policies for individual components (e.g., transformers) over a finite horizon. Dynamic programming provides a framework for studying problems where a sequential decision over time has to be made, as well as algorithms for computing optimal decision policies (Bertsekas, 1995). Bellman (1955) presented a dynamic programming formulation to solve the finite horizon equipment replacement problem with general costs. In his formulation, he considered a single challenger in each decision period. Later, Wagner (1975) solved a replacement analysis problem by using a dynamic programming formulation in which the state of the system is the time period and the main decision is the number of periods to retain an asset. Oakford et al. (1984) generalized Wagner's dynamic programming formulation to find an optimal sequence of replacements when there are one or more challengers. They modeled technological change with a variety of continuous functions. Sethi and Chand (1979) developed a dynamic programming forward algorithm and provided planning horizon results for several machine replacement models under an improving technological environment over time.
Hartman and Rogers (2006) presented two dynamic programming formulations, which were extensions of Wagner's (1975) and Bellman's (1985) models to solve the equipment replacement problem. Their formulation considered that assets available for purchase evolve over a finite time horizon according to continuous and discontinuous functions of technological change.
ETDS are heterogeneous assets because of the different components in the system, but even when the same component is used, the objective functions are different because of the opportunity costs considered (for example, an outage in New York City compared to rural Pennsylvania).
PROPOSED METHOD TO SOLVE THE REPLACEMENT PROBLEM FOR HETEROGENEOUS ASSETS
A method has been developed to solve equipment replacement problems for systems composed of sets of heterogeneous assets subject to annual budgetary constraints over a finite planning horizon. The proposed methodology is based on an integrated iterative dynamic programming and integer programming model. This methodology can potentially be applied to any replacement problem composed of sets of heterogeneous components subject to constraints imposed on the system. This is a new solution methodology that offers distinct benefits to previous methods that pertained directly to a system composed of homogeneous assets.
In the present methodology, a dynamic programming algorithm is developed and applied to the system components individually to obtain the optimal replacement policy for each asset in the system separately. The objective function is to obtain the minimum cost policy that minimizes the net present value (NPV) of maintenance, purchase, and opportunity costs for each individual component over the entire planning horizon. Once the time periods in which replacements should be made are identified, all the different solutions obtained from the dynamic programming model are used as inputs to a first integer program (IP1) to determine whether the individual replacement schedules can collectively form a feasible policy.
The IP1 model checks whether the replacement policies obtained satisfy the budget constraints for each time period. This is done by defining the sum of all constraints violations for the final problem as the IP1 objective function. If the first iteration of IP1 yields an objective function value of zero, then there are no budget violations, and the solution of IP1 gives the global optimal solution to the original problem. In this case a second integer program (IP2) is not required. In contrast, if the first iteration of IP1 yields an objective function value greater than zero, then constraint violations exist and additional component replacement schedules need to be generated but only for those components with replacement in the time periods having budget violations. Therefore, the dynamic programming models are run again, with the condition that it is now forbidden to make replacements in the periods where there are constraint violations due to exceeded annual budgets. This process is repeated until the optimal objective function for the IP1 is zero (a sufficient number of component replacement profiles have been generated to provide a feasible system-level replacement schedule), meaning that there are possible schedules with no constraint violations. Then, a second integer program (IP2) uses all the information generated from all the different individual replacement schedules generated. The IP2 is solved to determine the recommended system-level replacement schedule with the minimum NPV of the total system replacement analysis cost.
Dynamic Programming Formulation
In the dynamic programming formulation, the stages are the time or decision periods. At each time period (t = 0, 1, 2, ..., T), the decision-maker has the option to keep the existing asset or replace it with the purchase (or total refurbishment) of a new asset. An example diagram for each time period is shown in Figure 1. In this figure, the solid upwards arrow indicates a decision of keeping the asset one more year and a dashed downward arrow indicates a decision to replace the asset at the beginning of the time period. The objective is to find the minimum cost policy over the entire planning horizon of T time periods. In the figure, each node is labeled as [f.sub.t]([tau]), representing the NPV of all costs up until period t for an asset that is of age [tau] at time t (but may have been replaced periodically prior to t). At the beginning of the analysis, the asset has an age of [[tau].sub.0]. The minimum cost is defined to be the NPV of all costs over the T periods.
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