Kumaraswamy Ponnambalam

Generation and Transmission Expansion Under Risk Using Stochastic Programming

In this paper, a new model for generation and transmission expansion is presented. This new model considers as random events the demand, the equivalent availability of the generating plants, and the transmission capacity factor of the transmission lines. In order to incorporate these random events into an optimization model, stochastic programming and probabilistic constraints are used. A risk factor is introduced in the objective function by means of the mean-variance Markowitz theory. The solved optimization problem is a mixed integer nonlinear program. The expected value of perfect information is obtained in order to show the cost of ignoring uncertainty. The proposed model is illustrated by a six- and a 21-node network using a dc approximation.

Inferring operating rules for reservoir operations using fuzzy regression and ANFIS

The methods of ordinary least-squares regression (OLSR), fuzzy regression (FR), and adaptive network-based fuzzy inference system (ANFIS) are compared in inferring operating rules for a reservoir operations optimization problem. Dynamic programming (DP) is used as an example optimization tool to provide the input-output data set to be used by OLSR, FR, and ANFIS models. The coefficients of an FR model are found by solving a linear programming (LP) problem. The objective function of the LP is to minimize the total fuzziness of the FR model, which is related to the width of fuzzy coefficients in the regression model. Before applying FR to the reservoir operations problem, two FR formulations and interval regression (IR) are first examined in a simple tutorial example. ANFIS is also used to derive the reservoir operating rules as fuzzy IF-THEN rules. The OLSR, FR, and ANFIS based rules are then simulated and compared based on their performance in simulation. The methods are applied to a long-term planning problem as well as to a medium-term implicit stochastic optimization model. The results indicate that FR is useful to derive operating rules for a long-term planning model, where imperfect and partial information is available. ANFIS is beneficial in medium-term implicit stochastic optimization as it is able to extract important features of the system from the generated input-output set and represent those features as general operating rules.

A fast algorithm for power system optimization problems using an interior point method

Variants of simplex-based methodologies are generally used to solve underlying linear programming (LP) problems. An implementation of the dual affine (DA) algorithm (a variant of N. Karmarkars (1984) interior point method) is described in detail and some computational results are presented. This algorithm is particularly suitable for problems with a large number of constraints, and is applicable to linear and nonlinear optimization problems. In contrast with the simplex method, the number of iterations required by the DA algorithm to solve large-scale problems is relatively small. The DA algorithm has been implemented considering the sparsity of the constraint matrix. The normal equation that is required to be solved in every iteration is solved using a preconditioned conjugate gradient method. An application of the technique to a hydro-scheduling is presented; the largest problem is solved over nine times faster than an efficient simplex (MINOS) code. A new heuristic basis recovery procedure is implemented to provide primal and dual optimal basic solutions which are not generally available if interior point methods are used. The tested examples indicate that this new approach requires less than 10% of the original iterations of the simplex method to find the optimal basis. >

A unified approach to statistical design centering of integrated circuits with correlated parameters

This paper presents a general method for statistical design centering of integrated circuits with correlated parameters. It unifies worst-case design, nominal design and tolerance design in a single framework by selecting appropriate norms to measure the distances from the nominal values. The method uses an advanced first-order second moment technique as an alternative to the simplicial algorithm. Yield estimation is calculated in the original space and no transformation to uncorrelated variables is required. The solution algorithms are based on the recently developed interior-point methods for semi-definite programming. One tutorial and one practical example explain the application.

Minimizing variance of reservoir systems operations benefits using soft computing tools

Soft computing based tools including fuzzy inference systems (FIS), artificial neural networks (ANN), and genetic algorithms (GA) are used here to tackle the minimization of variance of benefits from reservoir operation. Variance reduction is a very hard optimization problem and solvable only using implicit methods like simulation, especially if the problem is nonlinear. First, a recently developed stochastic optimization method develops the optimal release policy (which is simply the recommended release in each season) of the system whose objective function maximizes the expected benefits. The policy is then simulated for a long inflow series to provide the trajectories of optimal releases and storages of the reservoir. These trajectories are then used as input-output data to train an adaptive neuro fuzzy inference system (ANFIS) to obtain updated fuzzy operating rules. The ANFIS based fuzzy rules are simulated and compared with policies developed using a multiple regression analysis, a commonly used method in water resources optimization. As the ANFIS performed better, further, a parameterized T-norm operator is applied and its parameters (numbering only two) are optimized through GA but with the objective of variance reduction in the benefits achieved. Results compare the better performance of the ANFIS based policies with other methods such as stochastic dynamic programming and the original stochastic method to demonstrate the usefulness of GA optimized parameters of a T-norm fuzzy operator for variance reduction.

Probabilistic design of systems with general distributions of parameters

This paper presents a new method for finding optimal solutions of systems with design parameters which are random variables distributed with various general and possibly non-symmetrical distributions. A double-bounded density function is used to approximate the distributions. Specifications may require tracking constraints in time domain and stability conditions in frequency domain. Using sensitivity information, the proposed method first finds a linearized feasible region. Afterwards it attempts to place a tolerance box of the design parameters such that the region with higher yield lies in the feasible region. The yield is estimated by the joint cumulative density function over a portion of the tolerance box contained in the feasible region. Optimal designs are found for a fourth-order servomechanism and actual yields are evaluated by Monte-Carlo simulation. Copyright

Maximization of Manufacturing Yield of Systems with Arbitrary Distributions of Component Values

This paper presents a general method for maximizing manufacturing yield when the realizations of system components are independent random variables with arbitrary distributions. Design specifications define a feasible region which, in the nonlinear case, is linearized using a first-order approximation. The method attempts to place the given tolerance hypercube of the uncertain parameters such that the area with higher yield lies in the feasible region. The yield is estimated by using the joint cumulative density function over the portion of the tolerance hypercube that is contained in the feasible region. A double-bounded density function is used to approximate various bounded distributions for which optimal designs are demonstrated on a tutorial example. Monte Carlo simulation is used to evaluate the actual yields of optimal designs.

Constrained state formulation for the stochastic control of multireservoir systems

A new formulation is presented for the analysis of multireservoir systems, based on the development of first and second moment expressions for the stochastic storage state variables. These expressions give explicit consideration to the maximum and minimum storage bounds in the reservoir system, a feature not incorporated in some existing formulations based on traditional control theory. Using this analysis, expected values of the storage states, variances of storage, release policies, reliability levels, and failure probabilities (useful information in the context of reservoir operations and design) can be obtained from a nonlinear programming solution. The approach does not involve any discretization of the system variables. The results for the means and standard deviations of the storage states for a multireservoir system compare favorably with those obtained from simulation.

Stochastic partial differential equations in groundwater hydrology

Part I of this series of two papers (Unny, 1989) dealt with the theoretical derivation of the moment equations for the stochastic partial differential equation in the water table depth forced by stochastic rainfall input. Part I also developed a maximum likelihood estimation procedure for parameter determination. The primary aim of the present manuscript is the application of the parameter estimation procedure to the Borden aquifer, an aquifer designated as an experimental site, where extensive field measurements have been carried out. Estimates of hydraulic conductivity and transmissivity for the Borden aquifer, derived from the maximum likelihood algorithm, have been compared with estimates obtained by “traditional” procedures. The paper also presents the simulated solution of the governing differential equation in the one dimensional problem applied to the Borden aquifer.

An application of Karmarkar's interior-point linear programming algorithm for multi-reservoir operations optimization

Optimization of multi-reservoir systems operations is typically a very large scale optimization problem. The following are the three types of optimization problems solved using linear programming (LP): (i) deterministic optimization for multiple periods involving fine stage intervals, for example, from an hour to a week (ii) implicit stochastic optimization using multiple years of inflow data, and (iii) explicit stochastic optimization using probability distributions of inflow data. Until recently, the revised simplex method has been the most efficient solution method available for solving large scale LP problems. In this paper, we show that an implementation of the Karmarkars interior-point LP algorithm with a newly developed stopping criterion solves optimization problems of large multi-reservoir operations more efficiently than the simplex method. For example, using a Micro VAX II minicomputer, a 40 year, monthly stage, two-reservoir system optimization problem is solved 7.8 times faster than the advanced simplex code in MINOS 5.0. The advantage of this method is expected to be greater as the size of the problem grows from two reservoirs to multiples of reservoirs. This paper presents the details of the implementation and testing and in addition, some other features of the Karmarkars algorithm which makes it a valuable optimization tool are illuminated.