Piecewise Quadratic Cost Functions Research Articles

Passive Attack Detection for a Class of Stealthy Intermittent Integrity Attacks

This paper proposes a passive methodology for detecting a class of stealthy intermittent integrity attacks in cyber-physical systems subject to process disturbances and measurement noise. A stealthy intermittent integrity attack strategy is first proposed by modifying a zero-dynamics attack model. The stealthiness of the generated attacks is rigorously investigated under the condition that the adversary does not know precisely the system state values. In order to help detect such attacks, a backward-in-time detection residual is proposed based on an equivalent quantity of the system state change, due to the attack, at a time prior to the attack occurrence time. A key characteristic of this residual is that its magnitude increases every time a new attack occurs. To estimate this unknown residual, an optimal fixed-point smoother is proposed by minimizing a piece-wise linear quadratic cost function with a set of specifically designed weighting matrices. The smoother design guarantees robustness with respect to process disturbances and measurement noise, and is also able to maintain sensitivity as time progresses to intermittent integrity attack by resetting the covariance matrix based on the weighting matrices. The adaptive threshold is designed based on the estimated backward-in-time residual, and the attack detectability analysis is rigorously investigated to characterize quantitatively the class of attacks that can be detected by the proposed methodology. Finally, a simulation example is used to demonstrate the effectiveness of the developed methodology.

Gradient Descent for Resource Allocation with Packet Loss*

This paper studies the effect of packet loss during the application of the weighted gradient descent to solve a resource allocation problem with piecewise quadratic cost functions in a multi-agent system. We define two performance metrics that measure, respectively, the deviation from the constraint and the error on the expected cost function. We derive upper bounds on both metrics: both bounds are proportional to the difference between the initial cost function and the cost function evaluated at the minimizer. Then, we extend the analysis of the constraint violation to open multi-agent systems where agents are replaced: based on a preliminary result and simulations we show that the combination of replacements and losses makes the constraint violation error diverge with time.

Non-convex constrained economic power dispatch with prohibited operating zones and piecewise quadratic cost functions

This paper is focused on the solution of the non-convex economic power dispatch problem with piecewise quadratic cost functions and practical operation constraints of generation units. The constraints of the economic dispatch problem are power balance constraint, generation limits constraint, prohibited operating zones and transmission power losses. To solve this problem, a meta-heuristic optimization algorithm named crow search algorithm is proposed. A constraint handling technique is also implemented to satisfy the constraints effectively. For the verification of the effectiveness and the superiority of the proposed algorithm, it is tested on 6-unit, 10-unit and 15-unit test systems. The simulation results and statistical analysis show the efficiency of the proposed algorithm. Also, the results confirm the superiority and the high-quality solutions of the proposed algorithm when compared to the other reported algorithms.

Reverse Search Strategy Based Optimization Technique to Economic Dispatch Problems with Multiple Fuels

This research article constitutes an efficient way to extract the minimum fuel cost for the economic dispatch problem considering multiple fuel options along with valve point effects. In general one convex cost function for one generator is assumed. The piecewise quadratic cost function is represented by selecting the convex cost function more than one for a generator. A new approach referred as Power Search Algorithm (PSA) based on reverse search strategy is suggested along with various constraints in ED. The proposed algorithm has been evaluated on a ten unit multi-fuel system for various power demands with different fuel input options. The outcomes acquired from the simulation are of good quality and furthermore it reveals that the proposed algorithm performs superior than the various algorithms discussed.

Optimum Dispatch of Hybrid Solar Thermal (HSTP) Electric Power Plant Using Non-Smooth Cost Function and Emission Function for IEEE-30 Bus System

The basic objective of economic load dispatch (ELD) is to optimize the total fuel cost of hybrid solar thermal electric power plant (HSTP). In ELD problems the cost function for each generator has been approximated by a single quadratic cost equation. As cost of coal increases, it becomes even more important have a good model for the production cost of each generator for the solar thermal hybrid system. A more accurate formulation is obtained for the ELD problem by expressing the generation cost function as a piece wise quadratic cost function. However, the solution methods for ELD problem with piece wise quadratic cost function requires much complicated algorithms such as the hierarchical structure approach along with evolutionary computations (ECs). A test system comprising of 10 units with 29 different fuel [7] cost equations is considered in this paper. The applied genetic algorithm method will provide optimal solution for the given load demand.

Preferred Economic Dispatch of Thermal Power Units

Economic Dispatch (ED) problem is one of the main concerns of the power generation operations which are basically solved to generate optimal amount of power from the generating units in the system by minimizing the fuel cost and by satisfying the system constraints. The accuracy of ED solutions is highly influenced by the fuel cost parameters of the generating units. Generally, the parameters are subjected to transform due to aging process and other external issues. Further the parameters associated with the transmission line modelling also change due to aforementioned issues. The loss coefficients which are the functions of transmission line parameters get altered from the original value over a time. Hence, the periodical estimation of these coefficients is highly essential in power system problems for obtaining ideal solutions for ED problem. Estimating the ideal parameters of the ED problem may be the best solution for this issue. This paper presents the Teaching Learning Based Optimization (TLBO) algorithm to estimate the parameters associated with ED problem. The estimation problem is formulated as an error minimization problem. This work provides a frame work for the computation of coefficients for quadratic function, piecewise quadratic cost function, emission function, transmission line parameters and loss coefficients. The effectiveness of TLBO is tested with 2 standard test systems and an Indian utility system.

Optimal power flow using artificial bee colony algorithm with global and local neighborhoods

Optimal power flow (OPF) is one of the most requisite tools for power system operation analysis. This problem has a complex mathematical formulation which is relatively hard to solve. This paper presents a swarm intelligence-based approach to solve the OPF problem. The proposed approach describes the use of a modified artificial bee colony (ABC) algorithm called ABC with global and local neighborhoods (ABCGLN) to determine the optimal settings of OPF control variables. ABCGLN is a recent modified version of basic ABC algorithm that can handle non-differentiable, non-linear, and multi modal objective functions. The ABCGLN approach is tested here on the standard IEEE 30-bus test system with three different objective functions for minimizing quadratic fuel cost function, piecewise quadratic cost function and quadratic cost function with valve point effects. The simulation results demonstrate the potential of ABCGLN algorithm of finding effective and robust quality solutions to solve OPF problem with various objective functions for the considered system as compared to those available in the literature.

Improved parameters for economic dispatch problems by teaching learning optimization

Abstract The solution of Economic Dispatch (ED) problems mainly depends on the modelling of thermal generators. The physical variations such as aging and ambient temperature affect the modelling parameters and are unavoidable. As these parameters are the backbone of ED solution, the periodical estimation of these characteristics coefficients is necessary for accurate dispatch. The process is formulated as an error minimization problem and a nature inspired algorithm namely Teaching Learning Based Optimization (TLBO) is proposed as an estimator. This work provides a frame work for the computation of coefficients for quadratic and cubic cost functions, valve point loading, piece-wise quadratic cost and emission functions. The effectiveness of TLBO is demonstrated on 5 standard test systems and a practical Indian utility system, involving varying degree of complexity. TLBO yields better results than benchmark Least Error Square (LES) method and other evolutionary algorithms. The economic deviation is also tested with existing systems.

An efficient equivalent thermal cost function model for nonlinear mid-term hydrothermal generation planning

This paper presents an efficient model to represent nonlinear thermal generation costs in the mid-term hydrothermal generation planning problem. The proposed approach comprises two procedures: the first one consists in obtaining an exact piecewise quadratic equivalent thermal cost function for total thermal generation cost based on quadratic cost functions for each individual unit. The second procedure consists in using a dynamic piecewise linear model to represent such function in the optimization problem to be solved. The combination of both procedures yield a linearized model for the equivalent thermal generation cost curve for each system area and time step, which are represented in the problem as a set of constraints. Numerical results for some instances of the multi-period, stochastic nonlinear hydrothermal planning problem show a remarkable CPU time reduction and an improved accuracy in the final solution of the problem, as compared to an individual static piecewise linear model, which is the usual approach adopted in the literature.

Augmented Lagrange Hopfield network initialized by quadratic programming for economic dispatch with piecewise quadratic cost functions and prohibited zones

This paper proposes a method based on quadratic programming (QP) and augmented Lagrange Hopfield network (ALHN) for solving economic dispatch (ED) problem with piecewise quadratic cost functions and prohibited zones. The ALHN method is a continuous Hopfield neural network with its energy function based on augmented Lagrange function which can properly deal with constrained optimization problems. In the proposed method, the QP method is firstly used to determine the fuel cost curve for each unit and initialize for the ALHN method, then a heuristic search is used for repairing prohibited zone violations, and the ALHN method is finally applied for solving the problem if any violations found. The proposed method has been tested on different systems and the obtained results are compared to those from many other methods in the literature. The result comparison has indicated that the proposed method has obtained better solution quality than many other methods. Therefore, the proposed QP-ALHN method could be a favorable method for solving the ED problem with piecewise quadratic cost functions and prohibited zones.

Economic dispatch with multiple fuel types by enhanced augmented Lagrange Hopfield network

This paper proposes an enhanced augmented Lagrange Hopfield network (EALHN) for solving economic dispatch (ED) with piecewise quadratic cost functions. The EALHN is an augmented Lagrange Hopfield neural network (ALHN), a continuous Hopfield neural network with its energy function based on augmented Lagrangian function, enhanced by a heuristic search for determination of fuel type. The proposed EALHN solves the ED problem in two phases. In the first phase, a heuristic search based on the average production cost of generating units is used to determine the most suitable fuel type for each unit so that total maximum power generation from all units is sufficient for supplying to load demand. In the last phase, the ALHN is applied to find optimal solution corresponding to the chosen fuel types. The proposed method is tested on several systems with various load demands and the obtained test results are compared to those from many other methods in the literature. Test results have indicated that the proposed method is efficient and fast for the ED problems with multiple fuel types represented by quadratic cost functions.

Convex parametric piecewise quadratic optimization: Theory and algorithms

In this paper we study the problem of parametric minimization of convex piecewise quadratic functions. Our study provides a unifying framework for convex parametric quadratic and linear programs. Furthermore, it extends parametric optimization algorithms to problems with piecewise quadratic cost functions, paving the way for new applications of parametric optimization in explicit dynamic programming and optimal control with quadratic stage cost.

Reinforcement Learning approaches to Economic Dispatch problem

This paper presents Reinforcement Learning (RL) approaches to Economic Dispatch problem. In this paper, formulation of Economic Dispatch as a multi stage decision making problem is carried out, then two variants of RL algorithms are presented. A third algorithm which takes into consideration the transmission losses is also explained. Efficiency and flexibility of the proposed algorithms are demonstrated through different representative systems: a three generator system with given generation cost table, IEEE 30 bus system with quadratic cost functions, 10 generator system having piecewise quadratic cost functions and a 20 generator system considering transmission losses. A comparison of the computation times of different algorithms is also carried out.

Optimal power flow using differential evolution algorithm

This paper presents an evolutionary-based approach to solve the optimal power flow (OPF) problem. The proposed approach employs differential evolution (DE) algorithm for optimal settings of OPF control variables. The proposed approach is examined and tested on the standard IEEE 30-bus test system with different objective functions that reflect fuel cost minimization, voltage profile improvement, and voltage stability enhancement. In addition, non-smooth piecewise quadratic cost function has been considered. The simulation results of the proposed approach are compared to those reported in the literature. The results demonstrate the potential of the proposed approach and show its effectiveness and robustness to solve the OPF problem for the systems considered.

A Simplified Approach for Economic Dispatch with Piecewise Quadratic Cost Functions

This paper presents a simple methodology for solving economic dispatch problems with multiple fuel options. The problem has been solved in two stages. In the first stage, the most economic fuel of each unit is identified and in the second stage Economic Load Dispatch (ELD) for the selected fuels is performed. The Composite Cost Function (CCF) is used to identify the most economical fuel and the feasible operating region of each unit. With these details, Sequential Approach with Matrix Framework (SAMF) is performed to determine the optimal dispatch of each unit. The effectiveness of the proposed methodology is tested with a standard ten unit system. Numerical simulation results indicate that the proposed methodology has close agreement with recent reports.

Exact Cost Performance Analysis of Piecewise Affine Systems

A method for the exact cost performance analysis of autonomous discrete-time piecewise affine systems is presented. Cost performance refers to the average trajectory cost over the entire region of attraction of the origin. The trajectory cost is based on the infinite-horizon cost. First, a reverse reachability algorithm which constructs the explicit piecewise quadratic trajectory cost function over the entire region of attraction of the origin is presented. Then, an explicit expression for the integral of a quadratic function over a simplex of arbitrary dimension is derived. Thus, the piecewise quadratic trajectory cost function can be integrated exactly and efficiently in order to determine the cost performance of the system as a whole. This alleviates the need to perform a large number of simulations.

Application of Adaptive Evolutionary Algorithm to Economic Load Dispatch with Nonconvex Cost Functions

This paper presents a new adaptive evolutionary algorithm (AEA) by integrating genetic algorithm (GA) and evolution strategy (ES) to solve the economic dispatch with piecewise quadratic cost functions and prohibited operating zones. AEA consists of GA for a global search capability and ES for a local search in an adaptive manner when the present generation evolves into the next generation. AEA is algorithm that ratio of population of GA and ES is adaptively modified in reproduction according to the fitness. Numerical results show that the proposed method is more effective than other previously developed methods and it can be applied successfully in problems with nonconvex functions.

Performance and stability analysis of discontinuous PWA systems by piecing together PWQ functions

An algorithm for evaluating the cost performance of discontinuous autonomous discrete-time piecewise affine systems is presented. The algorithm performs reverse reachability analysis and constructs a piecewise quadratic trajectory cost function over the entire region of attraction of the origin while explicitly taking into account the exact spatial evolution of the trajectories and the exact switching structure of the system as a whole. Available explicitly, this cost function can be integrated in order to evaluate the cost performance of the entire system. The reverse reachability algorithm is applied to the problem of constructing Lyapunov functions. The resulting Lyapunov functions are less conservative than other forms of Lyapunov function commonly used for stability analysis of autonomous discrete-time piecewise affine systems.

Fast computation evolutionary programming algorithm for the economic dispatch problem

AbstractThis paper essentially aims to propose a new EP based algorithm for solving the ED problem. The ED problem is solved using EP with system lambda as decision variable and power mismatch as fitness function. The algorithm is made fast through judicious modifications in initialization of the parent population, offspring generation and selection of the normal distribution curve. The proposed modifications reduce the search region progressively and generate only effective offsprings. The proposed algorithm is tested on a number of sample systems with quadratic cost function and also on a 10‐unit system with piecewise quadratic cost function. The computational results reveal that the proposed algorithm has an excellent convergence characteristic and is superior to other EP based methods in many respects. Copyright © 2005 John Wiley & Sons, Ltd.

Evolutionary programming techniques for different kinds of economic dispatch problems

This paper is aimed at exploring the performance of the different evolutionary programming (EP) techniques for all kinds of economic dispatch (ED) problems. The three EP techniques considered here differ in the kind of mutation they use: Gaussian, Cauchy and combined Gaussian–Cauchy mutation. The kinds of problems tested are: ED of generators with prohibited operating zones (POZ), ED of generators with piecewise quadratic cost functions (PQCF), combined economic–environmental dispatch (CEED) and multi-area economic dispatch (MAED). The three EP techniques have been developed in Matlab 6.5 and numerical examples typical to each kind of ED have been tested. The simulation results are compared and discussed to show the relative performances of the different EP techniques.