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A new replacement plan is developed and demonstrated to determine component replacements methodology for power distribution system. The proposed model is generic algorithm to afford optimal replacement policies over finite planning horizon for power distribution systems subject to budget constraints.
Generic algorithm is applied on radial configuration a simple series configuration. The method developed can potentially be applied to replacement problem composed of sets of assorted assets subject to constraints imposed on the system. This thesis presents the research efforts and the software implementation of a reliability analysis algorithm for electrical power distribution systems.
The economic and social effects of loss of electric service have significant impacts on both the utility supplying electric energy and the end users of electric service. The function of an electric power system is to satisfy the system load requirement with a reasonable assurance of continuity and quality. Aging infrastructure has higher costs to operate and maintain, and, more importantly, lower reliability.
As equipment ages, the component outage rates increases, 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 systematically, and optimally, upgrade the Electricity Distribution Systems. 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. 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. 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.
Power distribution system assessment for reliability:
The function of an electric power system is to satisfy the system load requirement with a reasonable assurance of continuity and quality. The ability of the system to provide an adequate supply of electrical energy is usually designated by the term of reliability. The concept of power-system reliability is extremely broad and covers all aspects of the ability of the system to satisfy the customer requirements.
Distribution system is a vital link between bulk power system and customers. In the past, the distribution segment of power system received considerably less attention in terms of reliability planning compared to generation and transmission segments. This is the reason that generation and transmission segments are very capital intensive and outages in these sections can cause widespread catastrophic economic consequences of society.
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.
Radial system configuration
Billinton and Allan (1983, 1984) and Billinton and Li (1994) present mathematical expressions to obtain different metrics for radial systems, where all the components are connected in series. Equations are used to obtain the total system outage rate, the average repair time and expected system downtime respectively.
Figure: Radial configuration
The outage rates are in outages per year, the repair rates are in hours per outage, and the expected system downtime is in hours per year.
4.jpg = Outage rate of component l at age l during period t
5.jpg = Repair time of component l during period t (rl, t= rl when repair times do not
Change with t)
6.jpg = System Outage rate during period t
7.jpg = System repair time during period t
8.jpg = System downtime during period t
The objective function can be formulated as
Subject to budget constraints
Maintenance cost (C:\Documents and Settings\shiri\Desktop\12.jpg): The cost required in maintaining a particular component in the power grid, during maintenance schedule.
The Maintenance Cost is calculated using
Moreover the maintenance schedule reduces the effective age of the component by a stated percentage of its actual age but it does not affect the failure note of the component.
Purchase cost (C:\Documents and Settings\shiri\Desktop\13.jpg): A cost of new asset or a component (when the old asset is replaced with a new asset) at a beginning of period t.
The Purchase Cost is a fixed cost depending upon the cost of the component to be purchased.
Unavailability cost (C:\Documents and Settings\shiri\Desktop\14.jpg): cost associated with the unavailability of the power to customers, due to the network shutdown during the system upgrade.
This is the cost increased due to the losses faced by the government when they fail to supply the power to the customers.
Non-homogeneous Poison Process (NHPP):
Non-homogeneous Poison Process (NHPP) models for repairable systems with non-constant
failure intensity function have often been used to model the reliability of aging or deteriorating
assets. In the present work, the Crow/AMSAA (Army Materiel System Analysis Activity) model is used to model the aging (increasing outage rates) for the different components in the ETDS.
The component mean repair times are assumed to remain constant through all periods in the replacement analysis model. The components in the system follow a minimal repair policy,
which means that once the components have an outage they are restored to the condition they
were just before the outage.
Crow AMSAA Model is used to model the ageing (increasing outage rates) for the different components in power distribution systems
¬l and ¢l are parameters of the NHPP for component l.
C:\Documents and Settings\shiri\Desktop\12.jpg = maintenance cost for component C:\Documents and Settings\shiri\Desktop\18.jpg of age C:\Documents and Settings\shiri\Desktop\16.jpg in period j
C:\Documents and Settings\shiri\Desktop\14.jpg =Purchase cost of component C:\Documents and Settings\shiri\Desktop\18.jpg in period j
C:\Documents and Settings\shiri\Desktop\13.jpg = unavailability costs associated with component C:\Documents and Settings\shiri\Desktop\18.jpg during period j
C:\Documents and Settings\shiri\Desktop\15.jpg = cost of minimum repair for component C:\Documents and Settings\shiri\Desktop\18.jpg of age C:\Documents and Settings\shiri\Desktop\16.jpg
C:\Documents and Settings\shiri\Desktop\20.jpg = repair times of components
C:\Documents and Settings\shiri\Desktop\19.jpg = outage rate of component C:\Documents and Settings\shiri\Desktop\18.jpg during period j
C:\Documents and Settings\shiri\Desktop\16.jpg = component age
C:\Documents and Settings\shiri\Desktop\22.jpg = NHPP parameter
Dynamic and Integer programming model:
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.
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. A dynamic programming formulation is developed to obtain replacement policies for individual components (e.g., transformers) over a finite horizon.
Dynamic Programming algorithm 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. Algorithm is presented to solve the finite horizon equipment replacement problem with general costs.
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, where 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 if 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.
Genetic Algorithms (GAs) were invented by John Holland and developed by him and his students and colleagues. Holland's book "Adaption in Natural and Artificial Systems" published in 1975.
Genetic algorithms are a part of evolutionary computing, which is a rapidly growing area of artificial intelligence. Inspired by Darwin's theory of evolution - Genetic Algorithms (GAs) are computer programs which create an environment where populations of data can compete and
Only the fittest survive, sort of evolution on a computer.
Genetic Algorithms are a search method that can be used for both solving problems and modelling evolutionary systems. Since it is heuristic (it estimates a solution) you won't know if the solution is exact. Most real- life problems are like that: you estimate a solution, you don't calculate it exactly.
Most problems don't have any formula for solving the problem because it is either too complex or it takes too long to calculate the solution exactly. One example could be space optimization - the best way to put objects of varying size into a room so they take as little space as possible. A
heuristic method is a more feasible approach.
GAs differ from other heuristic methods in several ways. One important difference is that it works on a population of possible solutions, which other heuristic methods use a single solution in their iterations (Astar). Another difference is that GAs are probabilistic and not deterministic.
Use of generic algorithm:
Optimization is the art of selecting the best alternative among a given set of options. In any optimization problem there is an objective function or objective that depends on a set of variables. To reach an optimum' does not necessarily mean " maximum". It means the best value for the function.
GAs are excellent for all tasks requiring optimization and are highly effective in any situation where many inputs (variables) interact to produce a large number of possible outputs (solutions). Some example situations are:
Optimization such as data fiting, clustering, trend spotting, path finding, ordering.
Management: Distribution, scheduling, project management, Courier routing, container packing, task assignment, time tables.
Financial: Portfolio balancing, budgeting, forecasting, investment analysis and payment scheduling.
Engineering: Structural design (eg beam sizes), electrical design (eg circuit boards), mechanical design (eg optimize weight, size & cost), process control, network design (eg computer networks).
R & D: Curve and surface fitting, neural network connection matrices, function optimization, fuzzy logic, population modeling, molecular modeling and drug design.
Advantages and Disadvantages:
A GA has a number of advantages. It can quickly scan a vast solution set. Bad proposals do not affect the end solution negatively as they are simply discarded. The inductive nature of the GA means that it doesn't have to know any rules of the problem - it works by its own internal rules. This is very useful for complex or loosely defined problems.
GAs has drawbacks too, of course. While the great advantage of Gas is the fact that they find a solution through evolution, this is also the biggest disadvantage. Evolution is inductive; in nature life does not evolve towards a good solution - it evolves away from bad circumstances
This can cause a species to evolve into an evolutionary dead end. Likewise, GAs risk finding a suboptimal solution.
Constraint handling techniques:
There are several approaches proposed in GAs to handle constrained Optimization problems.
Methods based on penalty functions
- Death Penalty
- Static Penalties
- Dynamic Penalties
- Annealing Penalties
- Adaptive Penalties
This simple and popular method just rejects unfeasible solutions from the population: P(x) = +âˆž , x âˆˆS âˆ’ F . In this case, there will never be any unfeasible solutions in the population. If a feasible search space is convex or a reasonable part of the whole search space, it can be expected that this method can work well. However, when the problem is highly constrained, the algorithm will spend a lot of time to find too few feasible solutions. Also, considering only the points that are in feasible region of the search space prevents to find better solutions.
In the methods of this group, penalty parameters don't depend on the current generation number and a constant penalty is applied to unfeasible solutions. Homaifar et al proposed a static penalty approach in which users describe some levels of violation.
The disadvantage of this method is the large number of parameters that must be set.
In the methods of this group, penalty parameters are usually dependent on the current generation number. This dynamic method increases the penalty as generation grows.
There are some methods based on annealing algorithm in this group. Michalewicz and Attia developed a method (GENECOP II) based on the idea of simulated annealing.
GENOCOP II distinguishes between linear and nonlinear constraints. In the algorithm only active constraints are under consideration at every iteration. While the temperature Ï„ decrease, selective pressure on unfeasible solutions increases. As an interesting point of the method, there is no diversity of the initial generation that consists of multiple copies of a solution satisfied all linear constraints. At each generation the temperature Ï„ decreases and the best solution found at the previous iteration is used as a starting point of next iteration. The algorithm is terminated at a previously determined freezing point Ï„f .
In the methods of this group, penalty parameters are updated for every generation according to information gathered from the population. The purpose of this method is to protect the diversity of the population and to protect the unfeasible solutions, which form the big part of the population in the early generations, from high penalties. Gen and Cheng claims that their approach is independent from the problem. However, this approach is only used in an optimization problem so their claim can be appreciated as suspicious.
Le Riche et al.  developed a segregated GA (sometimes it is called as yet another method) that uses two penalty parameters (say, p1 and p2) in two different populations. The aim is to overcome the problem of too high and too low penalties. If a small value is selected for p1 and a large value for p2, a simultaneous convergence from feasible and unfeasible sides can be achieved