Combined heat and power dynamic economic emission dispatch (CHPDEED) problem is a complicated nonlinear constrained multiobjective optimization problem with nonconvex characteristics. CHPDEED determines the optimal heat and power schedule of committed generating units by minimizing both fuel cost and emission simultaneously under ramp rate constraints and other constraints. This paper proposes hybrid differential evolution (DE) and sequential quadratic programming (SQP) to solve the CHPDEED problem with nonsmooth and nonconvex cost function due to valve point effects. DE is used as a global optimizer, and SQP is used as a fine tuning to determine the optimal solution at the final. The proposed hybrid DESQP method has been tested and compared to demonstrate its effectiveness.
Recently, combined heat and power (CHP) units, known as cogeneration or distributed generation, have played an increasingly important role in the utility industry. CHP units can provide not only electrical power but also heat to the customers. While the efficiency of the normal power generation is between 50% and 60%, the power and heat cogeneration increases the efficiency to around 90% [
In order to utilize the integrated CHP system more CO_{2} economically, combined heat and power economic dispatch (CHPED) problem is applied. The objective of the CHPED problem is to determine both power generation and heat production from units by minimizing the fuel cost such that both heat and power demands are met, while the combined heat and power units are operated in a bounded heat versus power plane. For most CHP units the heat production capacities depend on the power generation. This mutual dependency of the CHP units introduces a complication to the problem [
In the literature, several optimization techniques have been used to solve the CHPED problem with complex objective functions or constraints such as Lagrangian relaxation (LR) [
In the CHPED formulation the ramp rate limits of the units are neglected. Plant operators, to avoid lifeshortening of the turbines and boilers, try to keep thermal stress on the equipments within the safe limits. This mechanical constraint is usually transformed into a limit on the rate of change of the electrical output of generators. Such ramp rate constraints link the generator operation in two consecutive time intervals. Combined heat and power dynamic economic dispatch (CHPDED) problem is an extension of CHPED problem where the ramp rate constraint is considered. The primary objective of the CHPDED problem is to determine the heat and power schedule of the committed units so as to meet the predicted heat and electricity load demands over a time horizon at minimum operating cost under ramp rate constraints and other constraints [
The traditional dynamic economic dispatch (DED) problem which considers only thermal units that provide only electric power has been studied by several authors (see the review paper [
Differential evolution algorithm (DE), which was proposed by Storn and Price [
The DE shares many similarities with evolutionary computation techniques such as genetic algorithms (GA) techniques. The system is initialized with a population of random solutions and searches for optima by updating generations. DE has evolution operators such as crossover and mutation. Although DE seem to be good methods to solve the CHPDEED problem with nonsmooth and nonconvex cost functions, solutions obtained are just near global optimum with long computation time. Therefore, hybrid methods such as DESQP can be effective in solving the CHPDEED problems with valve point effects.
The main contributions of the paper are as follows. (1) A multiobjective optimization problem is formulated using CHPDEED approach. The multiobjective optimization problem is converted into a singleobjective optimization using the weighting method. (2) Hybrid DESQP method is proposed and validated for solving the CHPDEED problem with nonsmooth and nonconvex objective function. DE is used as a base level search for global exploration and SQP is used as a local search to finetune the solution obtained from DE. (3) The effectiveness of the proposed method is shown for test systems.
In this section we formulate the CHPDEED problem. The system under consideration has three types of generating units, conventional thermal units (TU), CHP units, and heatonly units (H). The power is generated by conventional thermal units and CHP units, while the heat is generated by CHP units and heatonly units. The objective of the CHPDEED problem is to simultaneously minimize the system’s production cost and emission so as to meet the predicted heat and power load demands over a time horizon under ramp rate and other constraints. The following objectives and constraints are taken into account in the formulation of the CHPDEED problem.
In this section, we introduce the cost and emission functions of three types of generating units, conventional thermal units which produce power only, CHP units which produce both heat and power, and heatonly units which produce heat only.
Let
There are three kinds of constraints considered in the CHPDEED problem, that is, the equilibrium constraints of power and heat production, the capacity limits of each unit, and the ramp rate limits.
Aggregating the objectives and constraints, the CHPDEED problem can be mathematically formulated as a nonlinear constrained multiobjective optimization problem which can be converted into a singleobjective optimization using the weighting method as
DE is a simple yet powerful heuristic method for solving nonlinear, nonconvex, and nonsmooth optimization problems. DE algorithm is a population based algorithm using three operators; mutation, crossover, and selection to evolve from randomly generated initial population to final individual solution [
According to the target vector
The selection process determines which of the vectors will be chosen for the next generation by implementing onetoone competition between the offsprings and their corresponding parents. If
In this paper, we define the evaluation function for evaluating the fitness of each individual in the population in DE algorithm as follows:
SQP method can be considered as one of the best nonlinear programming methods for constrained optimization problems [
In this section we present two examples. The first example shows the efficiency of the proposed DESQP method for the DED problem. In the second example, the hybrid DESQP method is applied to the CHPDEED problem. In DESQP method, the control parameters are chosen as
This example consists of ten conventional thermal units to investigate the effectiveness of the proposed DESQP technique in solving the DED problem with valve point effects and transmission line losses. The technical data of the units as well as the demand for the 10unit system are taken from [
Hourly generation (MW) schedule obtained from DED using DESQP for 10unit system.
H 










Loss 

































































































































































































































































































Comparison results of 10thermalunit system (cost ×
Method  EP [ 
PSO [ 
AIS [ 
NSGAII [ 
IBFA [ 
DESQP 
 
cost ( 






This example is 11unit system (eight conventional thermal units, two CHP units, and one heatonly unit) for solving the CHPDED, CHPDEED, and CHPPDED problems using DESQP method. We shall solve the CHPDEED problem when
Data of the CHP units and heatonly unit system.
CHP units 






























 
Heatonly units 








 








Heat load demand of the threeunit system for 24 hours.
Time 
Demand (MWth) 

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Heatpower feasible operating region for CHP unit 1.
Heatpower feasible operating region for CHP unit 2.
The best solutions of the CHPDED, CHPDEED, and CHPPDED problems for DESQP algorithm are given in Tables
Hourly heat and power schedule obtained from CHPDED.
H 










Loss 












































































































































































































































































































































































Cost ($) =
Hourly heat and power schedule obtained from CHPDEED (
t 










Loss 












































































































































































































































































































































































Cost ($) =
Hourly heat and power schedule obtained from CHPPDED.
H 










Loss 












































































































































































































































































































































































Cost ($) =
This paper presents a hybrid method combining differential evolution (DE) and sequential quadratic programming (SQP) for solving dynamic dispatch (CHPDED, CHPDEED, and CHPPDED) problems with valvepoint effects including generator ramp rate limits. In this paper, DE is first applied to find the best solution. This best solution is given to SQP as an initial condition that fine tunes the optimal solution at the final. The feasibility and efficiency of the DESQP were illustrated by conducting case studies with system consisting of eight conventional thermal units, two CHP units, and one heatonly unit.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (130107D1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.