**Applications of Simulated Annealing-Based Approaches to Electric Power Systems**

Yann-Chang Huang, Huo-Ching Sun and Kun-Yuan Huang

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50388

## **1. Introduction**

104 Simulated Annealing – Advances, Applications and Hybridizations

Nondestructive Evaluation 20: 1936-1943.

Wood Science and Technology 27: 373-380.

density, Wood Science and Technology 25: 341-349.

Technical Note, Forest product journal 39: 62-64.

Conference on Natural Computation 1459:1463.

Journal of Computer Vision 4: 321-331.

Logistics 23:28.

Processing 445-448.

220(4598): 671–680.

Physics, 90: 161-175.

Modelling, 18(11):29-57.

multiple hardwood species, Wood and Fiber Science 32: 287-300.

Industrial Tomography, Forest Product Journal 34: 42-46.

[14] Sarigul E, Abbott L, Schmoldt D (2001) Nondestructive rule-based defect detection and indentification system in CT images of hardwood logs, Review of Progress in

[15] Oja J, Grundberg S, Gronlund A (1998) Measuring the outer shape of pinus sylvestris

[16] Sandoz J (1993) Moisture content and temperature effect on ultrasound timber grading,

[17] Lindgren L (1991) Medical CAT-Scanning: x-ray CT-numbers and their relation to wood

[18] Schmoldt D, He J, Abbort A (2000) Automated Labeling of log feature in CT imagery of

[19] Taylor F, Wagner F, McMillin C, Morgan I, Hopkins F (1984) Locating Knots by

[20] Wagner F, Roder F (1989) Ultrafast CT scanning of an oak log for internal defect,

[21] Dawei Q, Zhang P, Xuejing J (2010), Detection of Wood Image, Sixth International

[22] Dawei Q, Hongbo M, Mingming Z, Lei Y (2008) Detection of Wood Defects From X-ray Image by ANN, Proceedings of the IEEE International Conference on Automation and

[23] Borianne P, Pernaudat R, Subsol G (2011) Automated Delineation of Tree-Rings in X-Ray Computed Tomography Images of Wood. IEEE International Conference on Image

[24] Kass M, Wikins K, Terzopoulos D (1988) Snake: Active Contour Model, International

[25] Kirkpatrick S, Gelatt C, Vecchi M (1983) Optimization by Simulated Annealing, Science

[26] Ingber L, (1993) Simulated annealing: Practice versus theory, Mathl. Comput.

[27] Dueck G, Scheuer T, (1990) Threshold accepting: A general purpose optimization algorithm appearing superior to simulated annealing, Journal of Computational

saw logs with an x-ray log scanner, Scandinavian Journal Forest 13: 340-347.

In the last decade, many heuristic methods have evolved for solving optimization problems that were previously difficult or impossible to solve. These methods include simulated annealing (SA), tabu search (TS), genetic algorithm (GA), differential evolution (DE), evolutionary programming (EP), evolutionary strategy (ES), ant colony optimization (ACO), and particle swarm optimization (PSO). This chapter reviews papers in international journals that present the SA-based methods for electric power system applications.

The numerical simulation of physical phenomena has become a major issue in the design phase or optimization of many systems. The complexity of the phenomena demands that multi-scale and multi-physical aspects are considered. In recent years, many researchers are concerned with the development, study, and implementation of efficient numerical methods that can be used in applications in engineering, natural and other sciences. They also focus on numerical methods for solving highly complex and CPU-time problems using highperformance computing. For the advancement of computing schemes that improve the efficiency and even optimize calculation processes, researchers must be able to generate novel systematic methods for solving multi-physical problems. The applications of SAbased methods for electric power systems are surveyed.

This chapter reviews various SA-based methods (including SA and other meta-heuristics methods) for the planning, operation, and optimization of power systems, including relevant recent and historical developments. Relevant publications in international journals that cover a broad range of applications of SA methods to solving power system problems are reviewed. As is well known among power engineers, many kinds of combinatorial optimization problems arise in the planning and operation of power systems. The generation and transmission expansion planning, generator maintenance scheduling and unit commitment, reactive power planning, load forecasting and economic dispatch, and distribution systems planning and operation are typical problems.

© 2012 Huang et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Since finding the global optimal solution to a problem is difficult because of the vast numbers of combinations of tentative solutions, many approximation algorithms have been developed to find acceptable near-optimal solutions over several decades. In the 1960's to 1970's, many mathematical programming methods were proposed and discussed. In the early 1980's, the expert system was introduced to solve such problems. However, it was not very effective in the most complex problems. In the early 1990's, the artificial neural network (ANN), or the simulation of the brains as a living entity, was revived in the field of artificial intelligence by the calculation speeds of contemporary computers. The ANN can solve combinatorial optimization problems and load forecasting problems extremely rapidly although the size of such problems that could be solved is limited. At that time, other methods for solving combinatorial optimization problems attracted the attention of many researchers. These methods skillfully simulate physical phenomena, natural evolution, and mathematical heuristics, and yield good results when applied to the combinatorial optimization problems. These methods are referred to as "modern heuristic approaches" or "meta-heuristic approaches".

Applications of Simulated Annealing-Based Approaches to Electric Power Systems 107

Minimize *f*(*x*) (1)

subject to *gi*(*x*) = 0, *i* = 1, 2, …, *I*

*hj*(*x*)0, *j* = 1, 2, …, *J*

where *xT* = [*x*1, *x*2, …, *xN*] and the elements of *x* are the decision variables that are to be determined. The constrained optimization problem that is described by (1) can be transformed to an unconstrained optimization problem using the penalty method. With the equality constraints represented as inequality constraints, the unconstrained optimization

Minimize *f*(*x*) + *p* ΣΘ[*hi*(*x*)] (2)

where *p* is the penalty coefficient and Θ is the penalty function. When *f*(*x*) is multi-modal, SA can be used to find the global optimal values of the decision variables *x* that are described in (2). Let *xd* be a dependent parameter in a constrained problem that is randomly selected at any stage in the optimization process. The constrained problem can thus be

Minimize *f*(*x*', *xd*) (3)

 *xd* = *g*'(*x*') (4)

This chapter reviews SA-based optimization methods for solving the constrained

Generation expansion planning (GEP) problems are important in planning activities to determine what generating units should be constructed and when such units should come on line over a long-term planning horizon. The major objectives of GEP are to minimize the total investment and operating costs of the generating units, while meeting capacity constraints, energy constraints, operating constraints, and the reliability criteria. The GEP problem is equivalent to finding a set of optimal decision vectors over a planning horizon that minimize the investment and operating costs under various constraints. Therefore, GEP

GEP for a power system is challenging because the large-scale, long-term, nonlinear and discrete nature of generation units. Many emerging techniques (including expert systems,

subject to *hj*(*x*)0, *j* = 1, 2, …, *J* where *x*' is *x* excluding *xd*. In minimizing the objective function in (3), the values of the nondependent parameters in *x*' are firstly determined in each iteration of the optimization process. When the values of the parameters in *x*' are known, *xd* can be determined by

applying the relationship between the dependent and non-dependent parameters:

optimization problem that is formulated as (3) and (4).

**3. Generation and transmission expansion planning** 

is a highly constrained, nonlinear, discrete optimization problem.

problem can be formulated as:

formulated as

Many applications of modern heuristic approaches to power systems have been proposed in the past 20 years, especially in generation and transmission expansion planning, generator maintenance scheduling and unit commitment, reactive power planning, load forecasting and economic dispatch, distribution systems planning and operation, and other applications. In this chapter, articles that have been published on the use of SA, TS, GA, DE, EP, ES, and combinations thereof in relation to power systems are systematically reviewed. These articles were published in international journals and cover a broad range of SA-based methods. A total of 81 journal papers are listed in the references.

This chapter provides a broad and relatively technical treatment of important topics at a level suitable for advanced students and for researchers with a background in electric power systems or in engineering optimization applications. It will review key areas of electric power system planning, operation, and optimization in a citation-rich format that is similar to that used by leading review journals.

The rest of this chapter is organized as follows.


## **2. Formulation of power system optimization problems**

Many optimization problems in power system planning, control, and operation can be formulated mathematically as follows:

$$\text{Minimize } f(\mathbf{x}) \tag{1}$$

$$\text{subject to } g(\mathbf{x}) = 0, \text{ } i = 1, 2, \dots, I$$

$$h(\mathbf{x}) \ge 0, \mathbf{j} = 1, 2, \dots, I$$

where *xT* = [*x*1, *x*2, …, *xN*] and the elements of *x* are the decision variables that are to be determined. The constrained optimization problem that is described by (1) can be transformed to an unconstrained optimization problem using the penalty method. With the equality constraints represented as inequality constraints, the unconstrained optimization problem can be formulated as:

106 Simulated Annealing – Advances, Applications and Hybridizations

"meta-heuristic approaches".

to that used by leading review journals.

Reactive Power Planning

Other Applications

Conclusion

The rest of this chapter is organized as follows.

 Load Forecasting and Economic Dispatch Distribution Systems Planning and Operation

formulated mathematically as follows:

 Formulation of Power System Optimization Problems Generation and Transmission Expansion Planning

Generator Maintenance Scheduling and Unit Commitment

**2. Formulation of power system optimization problems** 

Since finding the global optimal solution to a problem is difficult because of the vast numbers of combinations of tentative solutions, many approximation algorithms have been developed to find acceptable near-optimal solutions over several decades. In the 1960's to 1970's, many mathematical programming methods were proposed and discussed. In the early 1980's, the expert system was introduced to solve such problems. However, it was not very effective in the most complex problems. In the early 1990's, the artificial neural network (ANN), or the simulation of the brains as a living entity, was revived in the field of artificial intelligence by the calculation speeds of contemporary computers. The ANN can solve combinatorial optimization problems and load forecasting problems extremely rapidly although the size of such problems that could be solved is limited. At that time, other methods for solving combinatorial optimization problems attracted the attention of many researchers. These methods skillfully simulate physical phenomena, natural evolution, and mathematical heuristics, and yield good results when applied to the combinatorial optimization problems. These methods are referred to as "modern heuristic approaches" or

Many applications of modern heuristic approaches to power systems have been proposed in the past 20 years, especially in generation and transmission expansion planning, generator maintenance scheduling and unit commitment, reactive power planning, load forecasting and economic dispatch, distribution systems planning and operation, and other applications. In this chapter, articles that have been published on the use of SA, TS, GA, DE, EP, ES, and combinations thereof in relation to power systems are systematically reviewed. These articles were published in international journals and cover a broad range of SA-based

This chapter provides a broad and relatively technical treatment of important topics at a level suitable for advanced students and for researchers with a background in electric power systems or in engineering optimization applications. It will review key areas of electric power system planning, operation, and optimization in a citation-rich format that is similar

Many optimization problems in power system planning, control, and operation can be

methods. A total of 81 journal papers are listed in the references.

$$\text{Minimize } f(\mathbf{x}) \star p \, \Sigma \Theta[h(\mathbf{x})] \tag{2}$$

where *p* is the penalty coefficient and Θ is the penalty function. When *f*(*x*) is multi-modal, SA can be used to find the global optimal values of the decision variables *x* that are described in (2). Let *xd* be a dependent parameter in a constrained problem that is randomly selected at any stage in the optimization process. The constrained problem can thus be formulated as

$$\text{Minimize } f(\mathbf{x}', \mathbf{x}) \tag{3}$$

$$\text{subject to } h(\mathbf{x}) \ge 0, \mathbf{j} = 1, 2, \dots, \mathbf{j}$$

where *x*' is *x* excluding *xd*. In minimizing the objective function in (3), the values of the nondependent parameters in *x*' are firstly determined in each iteration of the optimization process. When the values of the parameters in *x*' are known, *xd* can be determined by applying the relationship between the dependent and non-dependent parameters:

$$\mathbf{x}d = \mathbf{g}'(\mathbf{x}') \tag{4}$$

This chapter reviews SA-based optimization methods for solving the constrained optimization problem that is formulated as (3) and (4).

#### **3. Generation and transmission expansion planning**

Generation expansion planning (GEP) problems are important in planning activities to determine what generating units should be constructed and when such units should come on line over a long-term planning horizon. The major objectives of GEP are to minimize the total investment and operating costs of the generating units, while meeting capacity constraints, energy constraints, operating constraints, and the reliability criteria. The GEP problem is equivalent to finding a set of optimal decision vectors over a planning horizon that minimize the investment and operating costs under various constraints. Therefore, GEP is a highly constrained, nonlinear, discrete optimization problem.

GEP for a power system is challenging because the large-scale, long-term, nonlinear and discrete nature of generation units. Many emerging techniques (including expert systems,

fuzzy logic, neural networks, analytic hierarchy process, network flow, decomposition methods, SA and GA) have been used in GEP. These emerging optimization techniques and their potential use in challenging GEP in the future competitive environment of the power industry have been reviewed, and some useful information and resources for future GEP were provided (Zhu & Chow, 1997).

Applications of Simulated Annealing-Based Approaches to Electric Power Systems 109

decomposition with an implicit zero-one enumeration procedure. Tests have been performed on three systems. Two smaller systems for which optimal solutions are known have been used to tune the main parameters of the SA process. The SA method has then been applied to a larger example system for which no optimal solutions are known. Therefore, an entire family of interesting solutions have been obtained a cost of

A parallel SA (PSA) algorithm for solving the long-term TEP problem was proposed (Gallego et al., 1997). A strategy that does not affect the basic convergence properties of the sequential SA have been implemented and tested. They studied the conditions under which the PSA is most efficient and tested the PSA on three example networks: a small 6-bus network and two complex real-life networks. Excellent results were reported in the test. In addition to reducing computing times, the proposed PSA algorithm greatly improved

Three families of non-convex optimization approaches for solving the TEP: SA, GA, and TS, were compared and an integrated view of these methodologies was proposed (Gallego et al., 1998a). Test results obtained using large-scale, real-life networks verified that the presented hybrid approach greatly outperformed those obtained using any individual approach.

An extended GA was proposed to solve the optimal TEP (Gallego et al., 1998b). They improved the GA in two main ways: the initial population was obtained by conventional optimization based methods, and the mutation approach was used in the SA. Test results revealed excellent performance for a difficult large-scale real-life problem, and demonstrated a substantial reduction in investment cost relative to earlier solutions that

A parallel TS was proposed to solve the TEP (Gallego et al., 2000). The presented method is a third-generation TS procedure with many advanced features. It exhibits the features of many other approaches, such as heuristic search, SA and GA. In all studied test cases, new generation and load sites can be connected to an existing main network, and these connections may require more than one line and the addition of a transformer, making the

The proper generator maintenance scheduling (GMS) in a power system is very important to its economic and reliable operation. To prevent premature aging and the failure of generators in a power system, which would cause unplanned and costly power outages, preventive maintenance must be performed at regular intervals. The generator maintenance in a power system involves scheduling and executing actual maintenance work. The GMS must be solved in the planning of the secure and reliable operation of a power system, primarily because other short- and long-term planning activities, including unit commitment, generation dispatch, import/export of power and GEP are directly influenced by related decisions. Modern power systems have become larger as the demand for

problem harder to solve in the sense that more combinations must be considered.

**4. Generator maintenance scheduling and unit commitment** 

approximately 7% less than those of the best solutions for the example system.

solution quality when applied to the largest of the test networks.

were obtained by conventional optimization methods and SA.

Meta-heuristic techniques, such as GA, DE, EP, ES, ACO, PSO, TS, SA, and hybrid approaches have been used in GEP and compared (Kannan et al., 2005). The original GEP problem has been modified using the virtual mapping procedure and the penalty factor method to improve the efficiency of these meta-heuristic techniques. Further, intelligent initial population generation has been introduced to reduce computational time. The GEP problem considered synthetic test systems for 6-year, 14-year, and 24-year planning horizons and five types of candidate units. The results obtained using these proposed techniques were compared and validated against conventional dynamic programming, and the effectiveness of each of the proposed methods was verified.

A novel GEP model was proposed (Kannan et al., 2007) for developing countries in a partially deregulated environment, in which both utilities and independent power producers (IPPs) participate in the generation market. In this model, the utility purchases electric power from the IPPs and sells it to the consumer. The utility maximizes its profit and ensures profits for all of the participating IPPs. Additionally, the utility checks under/over investment and considers system security, national security (fuel-mix ratio), social welfare and reliability simultaneously. The budget constraints of the utility are considered in the expansion plan. Meta-heuristic methods, such as GA, DE, EP, ES, PSO, TS, SA, and the hybrid approaches have been used to solve the restructured GEP problem, and the performances of each was evaluated and validated against the dynamic programming method for a synthetic test system with five types of candidate plant for the utility and three types of candidate plant for IPPs, with a 6-year planning horizon. The effectiveness of the proposed modifications and techniques has been demonstrated.

Transmission expansion planning (TEP) involves determining when and where new circuits must be installed, as well as the number that must be installed, to ensure that the power system will meet customer's demand with sufficient quality over a long planning horizon, while minimizing investment, operating, and interruption costs. TEP in a deregulated environment is highly complex. The possibility of generation planning and demand-side management as substitutes for transmission expansion must be considered when candidates for transmission expansions are generated. The deregulation of a power system produces new objectives in expansion planning such as the minimization of congestion cost. The optimized TEP considers such primary objectives as network investment cost, congestion cost, reliability, and environmental impact, subject to the relevant technical constraints.

The SA method was proposed for solving long-term TEP problems which are hard, largescale combinatorial problems (Romero et al., 1996). The proposed approach has been compared with a more conventional optimization technique that is based on mathematical decomposition with an implicit zero-one enumeration procedure. Tests have been performed on three systems. Two smaller systems for which optimal solutions are known have been used to tune the main parameters of the SA process. The SA method has then been applied to a larger example system for which no optimal solutions are known. Therefore, an entire family of interesting solutions have been obtained a cost of approximately 7% less than those of the best solutions for the example system.

108 Simulated Annealing – Advances, Applications and Hybridizations

the effectiveness of each of the proposed methods was verified.

proposed modifications and techniques has been demonstrated.

relevant technical constraints.

were provided (Zhu & Chow, 1997).

fuzzy logic, neural networks, analytic hierarchy process, network flow, decomposition methods, SA and GA) have been used in GEP. These emerging optimization techniques and their potential use in challenging GEP in the future competitive environment of the power industry have been reviewed, and some useful information and resources for future GEP

Meta-heuristic techniques, such as GA, DE, EP, ES, ACO, PSO, TS, SA, and hybrid approaches have been used in GEP and compared (Kannan et al., 2005). The original GEP problem has been modified using the virtual mapping procedure and the penalty factor method to improve the efficiency of these meta-heuristic techniques. Further, intelligent initial population generation has been introduced to reduce computational time. The GEP problem considered synthetic test systems for 6-year, 14-year, and 24-year planning horizons and five types of candidate units. The results obtained using these proposed techniques were compared and validated against conventional dynamic programming, and

A novel GEP model was proposed (Kannan et al., 2007) for developing countries in a partially deregulated environment, in which both utilities and independent power producers (IPPs) participate in the generation market. In this model, the utility purchases electric power from the IPPs and sells it to the consumer. The utility maximizes its profit and ensures profits for all of the participating IPPs. Additionally, the utility checks under/over investment and considers system security, national security (fuel-mix ratio), social welfare and reliability simultaneously. The budget constraints of the utility are considered in the expansion plan. Meta-heuristic methods, such as GA, DE, EP, ES, PSO, TS, SA, and the hybrid approaches have been used to solve the restructured GEP problem, and the performances of each was evaluated and validated against the dynamic programming method for a synthetic test system with five types of candidate plant for the utility and three types of candidate plant for IPPs, with a 6-year planning horizon. The effectiveness of the

Transmission expansion planning (TEP) involves determining when and where new circuits must be installed, as well as the number that must be installed, to ensure that the power system will meet customer's demand with sufficient quality over a long planning horizon, while minimizing investment, operating, and interruption costs. TEP in a deregulated environment is highly complex. The possibility of generation planning and demand-side management as substitutes for transmission expansion must be considered when candidates for transmission expansions are generated. The deregulation of a power system produces new objectives in expansion planning such as the minimization of congestion cost. The optimized TEP considers such primary objectives as network investment cost, congestion cost, reliability, and environmental impact, subject to the

The SA method was proposed for solving long-term TEP problems which are hard, largescale combinatorial problems (Romero et al., 1996). The proposed approach has been compared with a more conventional optimization technique that is based on mathematical A parallel SA (PSA) algorithm for solving the long-term TEP problem was proposed (Gallego et al., 1997). A strategy that does not affect the basic convergence properties of the sequential SA have been implemented and tested. They studied the conditions under which the PSA is most efficient and tested the PSA on three example networks: a small 6-bus network and two complex real-life networks. Excellent results were reported in the test. In addition to reducing computing times, the proposed PSA algorithm greatly improved solution quality when applied to the largest of the test networks.

Three families of non-convex optimization approaches for solving the TEP: SA, GA, and TS, were compared and an integrated view of these methodologies was proposed (Gallego et al., 1998a). Test results obtained using large-scale, real-life networks verified that the presented hybrid approach greatly outperformed those obtained using any individual approach.

An extended GA was proposed to solve the optimal TEP (Gallego et al., 1998b). They improved the GA in two main ways: the initial population was obtained by conventional optimization based methods, and the mutation approach was used in the SA. Test results revealed excellent performance for a difficult large-scale real-life problem, and demonstrated a substantial reduction in investment cost relative to earlier solutions that were obtained by conventional optimization methods and SA.

A parallel TS was proposed to solve the TEP (Gallego et al., 2000). The presented method is a third-generation TS procedure with many advanced features. It exhibits the features of many other approaches, such as heuristic search, SA and GA. In all studied test cases, new generation and load sites can be connected to an existing main network, and these connections may require more than one line and the addition of a transformer, making the problem harder to solve in the sense that more combinations must be considered.

## **4. Generator maintenance scheduling and unit commitment**

The proper generator maintenance scheduling (GMS) in a power system is very important to its economic and reliable operation. To prevent premature aging and the failure of generators in a power system, which would cause unplanned and costly power outages, preventive maintenance must be performed at regular intervals. The generator maintenance in a power system involves scheduling and executing actual maintenance work. The GMS must be solved in the planning of the secure and reliable operation of a power system, primarily because other short- and long-term planning activities, including unit commitment, generation dispatch, import/export of power and GEP are directly influenced by related decisions. Modern power systems have become larger as the demand for

electricity has considerably increased, increasing the number of generators, reducing reserve margins, and making the GMS problem even more complex.

Applications of Simulated Annealing-Based Approaches to Electric Power Systems 111

generator and the thermal generator are fully considered. The relative operational capacities of the hydroplant and the thermal plant are also considered. A coarse-grained parallel SA algorithm was presented for short-term hydrothermal UC (Wong & Wong, 1994b). The design of the algorithm considers load balancing, processor synchronization reduction, communication overhead reduction and memory contention elimination. The test results were compared with those obtained using a sequential algorithm and the results revealed that the proposed method provides an almost linear reduction in computation time. Two parallel SA concepts, speculative computation and serial subset, were proposed to the UC (Annakkage et al., 1995). A combined scheme in which speculative computation is used in the initial phase and the serial subset is used in the final phase. The test results revealed that

the proposed parallel schemes greatly improved the computing performance of SA.

determining the UC.

system for four fuels.

A hybrid GA/SA method was developed to the UC (Wong & Wong, 1995). The proposed method can typically provide feasible schedules in the solution process. The hybrid method can handle the nonconvexity of the UC. The authors subsequently provided a new formulation for short-term UC with a take-or-pay fuel contract (Wong & Wong, 1996) and a used a fuzzy set approach to help to find schedules that yield, as closely as possible, the take-or-pay fuel consumption. They extended the formulation to cover the economic dispatch problem when fuel consumption exceeds the agreed amount in the take-or-pay contract, and the extended formulation was combined with the GA and SA algorithms for

A new formulation for short-term multiple-fuel-constrained UC was presented (Wong & Wong, 1997). In the formulation, the power balance constraint, operating limits of the generators, fuel availability factors of the generators, efficiency factors of the fuels and the supply limits of the fuels are fully considered. They combined the new formulation with GA, SA and hybrid GA/SA methods to establish new algorithms. They demonstrated the new algorithms by using them to determine the most economical generation schedule for 25 generators in a local power system and the schedule of the

An enhanced SA was adopted to solve the UC by applying mechanisms to ensure that the generated candidate solutions are feasible and satisfy all of the constraints (Wong, 1998). The performance of the enhanced SA was demonstrated and compared with that of conventional methods. The UC was divided into two subproblems: a combinatorial optimization problem and a nonlinear programming problem (Mantawy et al., 1998). They solved the former using the SA and the latter using a quadratic programming routine.

Numerical results revealed an improvement in the cost associated with the solutions.

those obtained using GA, TS and SA methods, and two exact algorithms.

A new algorithm based on integrating GA, TS and SA methods to solve the UC was presented (Mantawy et al., 1999). The core of the proposed algorithm is based on GA. TS is used to generate new population members in the reproduction phase of the GA. The SA is used to accelerate the convergence of the GA by applying the SA test for all the population members. Numerical results showed the superiority of the solutions thus obtained over

The objective of GMS is to plan preventive maintenance scheduling for generators based on load forecasts. The purpose is to minimize the operating cost, maximize the profit or minimize the power shortage annually. Many constraints must be considered, including the system load, the maintenance window of each generator, maintenance resources (including human resources, machinery, equipment, and others), and the estimated power output. Therefore, GMS is a mixed-integer programming problem, in which a combination of 0s and 1s specifies whether a generator unit is undergoing maintenance, subject to some equality and inequality constraints. Before deregulation of the electric power industry, system reliability and loss of load probability were two major objectives of the GMS problem. After deregulation, however, maximizing profit is the driving interest of an IPP. Hence, the profitability of power plants is the objective function for the GMS problem.

The thermal power plant GMS problem has been formulated as a mixed-integer programming problem and solved efficiently using the SA (Satoh & Nara, 1991). The GA, SA and TS methods have been used together to solve the large-scale, long-term GMS problem (Kim et al., 1997). This combined solution algorithm has the advantages of the individual algorithms and supports a reasonable combination of local and global searches. The method considers the maintenance class and many consecutive years scheduling. Several real-scale numerical examples demonstrate the effectiveness of the proposed method.

The application of meta-heuristic approaches, such as GA, SA and their hybrid for GMS in power systems was proposed (Dahal & Chakpitak, 2007). The presented paper mainly focuses on the application of GA/SA and GA/SA/heuristic hybrid approaches. The GA/SA hybrid used the probabilistic acceptance criterion of SA in the GA framework. The GA/SA/heuristic hybrid used heuristic methods with the GA/SA hybrid to seed the initial population. The authors formulated the GMS as an integer programming problem using a reliability-based objective function and typical problem constraints. They discussed the implementation and performance of the meta-heuristic methods and their hybrid in a test case. The obtained results are promising and reveal that the hybrid methods are less sensitive to variations in the parameters of the technique and are effective alternatives to other methods for performing GMS.

The main objective of unit commitment (UC) is how to schedule the on/off status of the generators to minimize the production cost of electricity. A typical UC problem is combinatorial and involves a large set of physical, operating and contractual constraints, making the problem difficult to solve. The SA method was originally proposed to solve the UC problem (Zhuang & Galiana, 1990). It is highly flexible in handling UC constraints, and numerical results on test systems of up to 100 units were reported.

A short-term hydrothermal UC based on the SA method was proposed (Wong & Wong, 1994a). In the algorithm, the power balance constraint, total water discharge constraint, reservoir volume limits and constraints on the operational limits of the hydrothermal generator and the thermal generator are fully considered. The relative operational capacities of the hydroplant and the thermal plant are also considered. A coarse-grained parallel SA algorithm was presented for short-term hydrothermal UC (Wong & Wong, 1994b). The design of the algorithm considers load balancing, processor synchronization reduction, communication overhead reduction and memory contention elimination. The test results were compared with those obtained using a sequential algorithm and the results revealed that the proposed method provides an almost linear reduction in computation time. Two parallel SA concepts, speculative computation and serial subset, were proposed to the UC (Annakkage et al., 1995). A combined scheme in which speculative computation is used in the initial phase and the serial subset is used in the final phase. The test results revealed that the proposed parallel schemes greatly improved the computing performance of SA.

110 Simulated Annealing – Advances, Applications and Hybridizations

method.

other methods for performing GMS.

margins, and making the GMS problem even more complex.

profitability of power plants is the objective function for the GMS problem.

electricity has considerably increased, increasing the number of generators, reducing reserve

The objective of GMS is to plan preventive maintenance scheduling for generators based on load forecasts. The purpose is to minimize the operating cost, maximize the profit or minimize the power shortage annually. Many constraints must be considered, including the system load, the maintenance window of each generator, maintenance resources (including human resources, machinery, equipment, and others), and the estimated power output. Therefore, GMS is a mixed-integer programming problem, in which a combination of 0s and 1s specifies whether a generator unit is undergoing maintenance, subject to some equality and inequality constraints. Before deregulation of the electric power industry, system reliability and loss of load probability were two major objectives of the GMS problem. After deregulation, however, maximizing profit is the driving interest of an IPP. Hence, the

The thermal power plant GMS problem has been formulated as a mixed-integer programming problem and solved efficiently using the SA (Satoh & Nara, 1991). The GA, SA and TS methods have been used together to solve the large-scale, long-term GMS problem (Kim et al., 1997). This combined solution algorithm has the advantages of the individual algorithms and supports a reasonable combination of local and global searches. The method considers the maintenance class and many consecutive years scheduling. Several real-scale numerical examples demonstrate the effectiveness of the proposed

The application of meta-heuristic approaches, such as GA, SA and their hybrid for GMS in power systems was proposed (Dahal & Chakpitak, 2007). The presented paper mainly focuses on the application of GA/SA and GA/SA/heuristic hybrid approaches. The GA/SA hybrid used the probabilistic acceptance criterion of SA in the GA framework. The GA/SA/heuristic hybrid used heuristic methods with the GA/SA hybrid to seed the initial population. The authors formulated the GMS as an integer programming problem using a reliability-based objective function and typical problem constraints. They discussed the implementation and performance of the meta-heuristic methods and their hybrid in a test case. The obtained results are promising and reveal that the hybrid methods are less sensitive to variations in the parameters of the technique and are effective alternatives to

The main objective of unit commitment (UC) is how to schedule the on/off status of the generators to minimize the production cost of electricity. A typical UC problem is combinatorial and involves a large set of physical, operating and contractual constraints, making the problem difficult to solve. The SA method was originally proposed to solve the UC problem (Zhuang & Galiana, 1990). It is highly flexible in handling UC constraints, and

A short-term hydrothermal UC based on the SA method was proposed (Wong & Wong, 1994a). In the algorithm, the power balance constraint, total water discharge constraint, reservoir volume limits and constraints on the operational limits of the hydrothermal

numerical results on test systems of up to 100 units were reported.

A hybrid GA/SA method was developed to the UC (Wong & Wong, 1995). The proposed method can typically provide feasible schedules in the solution process. The hybrid method can handle the nonconvexity of the UC. The authors subsequently provided a new formulation for short-term UC with a take-or-pay fuel contract (Wong & Wong, 1996) and a used a fuzzy set approach to help to find schedules that yield, as closely as possible, the take-or-pay fuel consumption. They extended the formulation to cover the economic dispatch problem when fuel consumption exceeds the agreed amount in the take-or-pay contract, and the extended formulation was combined with the GA and SA algorithms for determining the UC.

A new formulation for short-term multiple-fuel-constrained UC was presented (Wong & Wong, 1997). In the formulation, the power balance constraint, operating limits of the generators, fuel availability factors of the generators, efficiency factors of the fuels and the supply limits of the fuels are fully considered. They combined the new formulation with GA, SA and hybrid GA/SA methods to establish new algorithms. They demonstrated the new algorithms by using them to determine the most economical generation schedule for 25 generators in a local power system and the schedule of the system for four fuels.

An enhanced SA was adopted to solve the UC by applying mechanisms to ensure that the generated candidate solutions are feasible and satisfy all of the constraints (Wong, 1998). The performance of the enhanced SA was demonstrated and compared with that of conventional methods. The UC was divided into two subproblems: a combinatorial optimization problem and a nonlinear programming problem (Mantawy et al., 1998). They solved the former using the SA and the latter using a quadratic programming routine. Numerical results revealed an improvement in the cost associated with the solutions.

A new algorithm based on integrating GA, TS and SA methods to solve the UC was presented (Mantawy et al., 1999). The core of the proposed algorithm is based on GA. TS is used to generate new population members in the reproduction phase of the GA. The SA is used to accelerate the convergence of the GA by applying the SA test for all the population members. Numerical results showed the superiority of the solutions thus obtained over those obtained using GA, TS and SA methods, and two exact algorithms.

An extended mean field annealing neural network (NN) approach was presented to shortterm UC (Liang et al., 2000). The annealing NN provides the high solution quality of the SA with the rapid convergence of the NN. Test results confirmed that their approach was very effective in finding the optimum solution to the UC. An approach combining the feedforward NN and the SA method was presented to solve UC (Nayak & Sharma, 2000). The NN is used to determine the discrete variables that correspond to the state of each unit at various times. The SA is used to generate the continuous variables that correspond to the power output of each unit and the production cost. A set of load profiles as inputs and the corresponding UC schedules as outputs that satisfy the minimum up–down times, spinning reserve and crew constraints were used to train the NN. The experimental results demonstrate that the proposed approach can solve the UC in a reduced computational time.

Applications of Simulated Annealing-Based Approaches to Electric Power Systems 113

An absolutely stochastic SA method was proposed to UC (Saber et al., 2007) and fuzzy UC using the absolutely stochastic SA was presented (Saber et al., 2006). In both papers, all of the solutions that involved high and low costs, are associated with acceptance probabilities and an early jump from one local minimum to another, enabling more local minima to be found and compared in a particular time or number of iterations. The number of bits to be flipped is determined by the appropriate control of parameter. Excess units with a systemdependent probability distribution handle constraints efficiently. The sensitivity of the distribution parameters is satisfactory. To reduce the number of required economic load dispatch calculations, a sign bit vector was introduced. Numerical results indicate an improvement in the cost and time required to find a solution relative to those using the proposed algorithms. Besides, An EP based SA was proposed to the short-term UC (Christober Asir Rajan & Mohan, 2007) and the UC using SA embedded EP approach was presented to hydro-thermal UC (Christober Asir Rajan, 2011). Numerical results are used to compare the costs of the solutions and computation times obtained when the proposed

approaches and conventional methods are used to determine the optimal UC.

Reactive power planning, or Var planning seeks to optimize the allocation of reactive power sources in a power system, based on location and size. The objectives of Var planning are to minimize the cost of new reactive power supplies. The many variants of this objective involve the cost of real power losses or the cost of fuel. Furthermore, such technical indices as deviation from a given voltage schedule or the security margin may be used as objectives for optimization. The installation of reactive power sources also releases system capacity

The Var planning greatly influences the secure and economic operation of electric power systems. SA was used to contingency-constrained optimal Var planning in large-scale power systems (Hsiao et al., 1993). Their problem formulation considered practical aspects of Var sources, load constraints, and the operating constraints at different load levels. The proposed SA determines the locations of installed Var sources, the types and sizes of Var sources to be installed, and the settings of Var sources at different loading conditions. Test results confirm that the proposed approach is suitable for large-scale power systems

SA based computer package for multi-objective, Var planning in large-scale power systems was proposed (Hsiao et al., 1994). Optimal Var planning is reformulated as a constrained, multi-objective, nondifferentiable optimization problem. The new formulation considers four objective functions that re related to investment in the system, its operating efficiency, its security and the system service quality. Their new formulation also considered load, operating constraints and contingency constraints. The problem formulation allows both the objective functions and the equality and inequality constraints to be nondifferentiable, making the problem formulation more realistic. The package uses a two-stage solution algorithm that is based on an extended SA and the ε-constraint method. The first-stage of

**5. Reactive power planning** 

and improves the voltage level.

applications.

A combined SA and TS approach was used to solve the UC (Purushothama et al., 2003). In their stochastic extended neighborhood algorithm, SA is the main stochastic algorithm, and TS is used to perform an extended neighborhood search and thus locally improve the solution obtained by SA. The neighborhood search uses local domain-knowledge, resulting in rapid convergence of the SA. The results obtained for many example systems illustrate the potential of the hybrid approach. The SA with local search hybrid algorithm was proposed to solve the UC (Purushothama & Jenkins, 2003). The hybrid algorithm is robust and provides faster convergence than earlier algorithms. The results verified its potential for solving the UC.

A scheduling method for representing the thermal stress of turbine shafts as ramp rate constraints in the UC (Li & Shahidehpour, 2003). In the UC, thermal stress over the elastic limit is used to calculate the ramping cost. Determination of the contribution of the thermal stress to the generation cost requires that a set of solution that includes thermal stress at the end of each time step be calculated; this requirement establishes a complex problem that cannot be solved using an ordinary optimization method. An improved SA was used to determine the optimal trajectory of each generating unit, and they elucidated the economics of frequently ramping up/down of low-cost generating units in relation to the cost of replacing their turbine rotors with a shorter life span. The results demonstrated the effectiveness of the proposed method.

A new SA combined with a dynamic economic dispatch method was designed to solve the short-term UC (Simopoulos et al., 2006a). SA was used to schedule the generating units, while a dynamic economic dispatch method, incorporating the ramp rate constraints, was used to solve the UC. The ramp rates are considered by performing either a backward or a forward sequence of conventional economic dispatches with modified limits on the generating units. The proposed algorithm is relatively fast and provides feasible nearoptimal solutions. A new method for the incorporation of the unit unavailability and the uncertainty of the load forecast in the solution of the short-term UC solved by the SA was presented in (Simopoulos et al., 2006b). The required spinning reserve capacity was conducted by imposing reliability constraints, based on the expected unserved energy and the loss of load probability indices. Numerical simulations demonstrated the efficiency of the proposed method.

An absolutely stochastic SA method was proposed to UC (Saber et al., 2007) and fuzzy UC using the absolutely stochastic SA was presented (Saber et al., 2006). In both papers, all of the solutions that involved high and low costs, are associated with acceptance probabilities and an early jump from one local minimum to another, enabling more local minima to be found and compared in a particular time or number of iterations. The number of bits to be flipped is determined by the appropriate control of parameter. Excess units with a systemdependent probability distribution handle constraints efficiently. The sensitivity of the distribution parameters is satisfactory. To reduce the number of required economic load dispatch calculations, a sign bit vector was introduced. Numerical results indicate an improvement in the cost and time required to find a solution relative to those using the proposed algorithms. Besides, An EP based SA was proposed to the short-term UC (Christober Asir Rajan & Mohan, 2007) and the UC using SA embedded EP approach was presented to hydro-thermal UC (Christober Asir Rajan, 2011). Numerical results are used to compare the costs of the solutions and computation times obtained when the proposed approaches and conventional methods are used to determine the optimal UC.

## **5. Reactive power planning**

112 Simulated Annealing – Advances, Applications and Hybridizations

solving the UC.

effectiveness of the proposed method.

the proposed method.

An extended mean field annealing neural network (NN) approach was presented to shortterm UC (Liang et al., 2000). The annealing NN provides the high solution quality of the SA with the rapid convergence of the NN. Test results confirmed that their approach was very effective in finding the optimum solution to the UC. An approach combining the feedforward NN and the SA method was presented to solve UC (Nayak & Sharma, 2000). The NN is used to determine the discrete variables that correspond to the state of each unit at various times. The SA is used to generate the continuous variables that correspond to the power output of each unit and the production cost. A set of load profiles as inputs and the corresponding UC schedules as outputs that satisfy the minimum up–down times, spinning reserve and crew constraints were used to train the NN. The experimental results demonstrate that the proposed approach can solve the UC in a reduced computational time. A combined SA and TS approach was used to solve the UC (Purushothama et al., 2003). In their stochastic extended neighborhood algorithm, SA is the main stochastic algorithm, and TS is used to perform an extended neighborhood search and thus locally improve the solution obtained by SA. The neighborhood search uses local domain-knowledge, resulting in rapid convergence of the SA. The results obtained for many example systems illustrate the potential of the hybrid approach. The SA with local search hybrid algorithm was proposed to solve the UC (Purushothama & Jenkins, 2003). The hybrid algorithm is robust and provides faster convergence than earlier algorithms. The results verified its potential for

A scheduling method for representing the thermal stress of turbine shafts as ramp rate constraints in the UC (Li & Shahidehpour, 2003). In the UC, thermal stress over the elastic limit is used to calculate the ramping cost. Determination of the contribution of the thermal stress to the generation cost requires that a set of solution that includes thermal stress at the end of each time step be calculated; this requirement establishes a complex problem that cannot be solved using an ordinary optimization method. An improved SA was used to determine the optimal trajectory of each generating unit, and they elucidated the economics of frequently ramping up/down of low-cost generating units in relation to the cost of replacing their turbine rotors with a shorter life span. The results demonstrated the

A new SA combined with a dynamic economic dispatch method was designed to solve the short-term UC (Simopoulos et al., 2006a). SA was used to schedule the generating units, while a dynamic economic dispatch method, incorporating the ramp rate constraints, was used to solve the UC. The ramp rates are considered by performing either a backward or a forward sequence of conventional economic dispatches with modified limits on the generating units. The proposed algorithm is relatively fast and provides feasible nearoptimal solutions. A new method for the incorporation of the unit unavailability and the uncertainty of the load forecast in the solution of the short-term UC solved by the SA was presented in (Simopoulos et al., 2006b). The required spinning reserve capacity was conducted by imposing reliability constraints, based on the expected unserved energy and the loss of load probability indices. Numerical simulations demonstrated the efficiency of Reactive power planning, or Var planning seeks to optimize the allocation of reactive power sources in a power system, based on location and size. The objectives of Var planning are to minimize the cost of new reactive power supplies. The many variants of this objective involve the cost of real power losses or the cost of fuel. Furthermore, such technical indices as deviation from a given voltage schedule or the security margin may be used as objectives for optimization. The installation of reactive power sources also releases system capacity and improves the voltage level.

The Var planning greatly influences the secure and economic operation of electric power systems. SA was used to contingency-constrained optimal Var planning in large-scale power systems (Hsiao et al., 1993). Their problem formulation considered practical aspects of Var sources, load constraints, and the operating constraints at different load levels. The proposed SA determines the locations of installed Var sources, the types and sizes of Var sources to be installed, and the settings of Var sources at different loading conditions. Test results confirm that the proposed approach is suitable for large-scale power systems applications.

SA based computer package for multi-objective, Var planning in large-scale power systems was proposed (Hsiao et al., 1994). Optimal Var planning is reformulated as a constrained, multi-objective, nondifferentiable optimization problem. The new formulation considers four objective functions that re related to investment in the system, its operating efficiency, its security and the system service quality. Their new formulation also considered load, operating constraints and contingency constraints. The problem formulation allows both the objective functions and the equality and inequality constraints to be nondifferentiable, making the problem formulation more realistic. The package uses a two-stage solution algorithm that is based on an extended SA and the ε-constraint method. The first-stage of

the solution algorithm uses an extended SA to find a global, noninferior solution. The primary objective function and the trade-off tolerances are then used to transform the constrained multi-objective optimization problem into a single-objective optimization problem with more constraints by applying the ε-constraint method. The second-stage uses the SA to obtain the global optimal solution.

Applications of Simulated Annealing-Based Approaches to Electric Power Systems 115

Three GA/SA/TS combined algorithms for Var planning were proposed (Liu et al., 2002). Trying reasonably to combine local and global search, they adopt the acceptance probability of SA to improve the convergence of the simple GA, and to apply TS to find more accurate solutions. Test results confirmed that the proposed method is effective to find better solutions than those of existing methods within reasonable time. A projection-based twolayer SA was used to solve the multi-objective optimization problems (Chen & Ke, 2004). The SA yielded a desirable, globally efficient solution to such problems, even when the

A new approach was presented to model and solve Var planning under the static voltage stability constraint (Wang et al., 2011). First, the fuzzy clustering method was used to select new candidate Var source locations. Then, modified Gray code was applied to represent a series of non-uniform Var capacity intervals at different candidate buses. Under the new ordering of the Var capacity intervals, a simplified piecewise linear function that relates the total transfer capability with new Var capacity was derived and applied as a static voltage stability constraint in Var planning. Finally, the optimization problem was solved by an enhanced SA using modified Gray code. The proposed SA adopted a modified definition of neighborhood selection and a novel approach to generating new random solutions. Test results demonstrated that the proposed method is a simple and effective approach for

A multi-objective SA was proposed to provide decision support in reactive power compensation (Antunes et al., 2011). Their method computed a set of well-distributed and diversified solutions underlying distinct trade-offs, even for a challenging network. The characteristics of the non-dominated front are relevant information that helps planning engineers select satisfactory compromise solutions that improve the operating conditions of

Accurate forecasting of electricity load has been one of the most important issues in the planning, operation, and control of electric power systems. Recently, following power system privatization and deregulation, the accurate forecasting of electricity load has received increasing attention. An optimal fuzzy inference method for short-term load forecasting was presented (Mori & Kobayashi, 1996). Their proposed method constructs an optimal structure of the simplified fuzzy inference that minimizes model errors and the number of membership functions to capture the nonlinear behavior of short-term loads in

the power system. The model was identified by SA and the steepest descent method.

integrated moving average model and the general regression NN model.

Support vector machines (SVMs) have been successfully used to solve nonlinear regression and time series problems. The SVM with SA approach was proposed to forecast electricity load (Pai & Hong, 2005). They used SA to select the parameters of the SVM model. Empirical results indicate that the proposed model outperforms the autoregressive

solution space is nonconvex and the objective functions are nondifferentiable.

voltage stability-constrained Var planning with contingency considered.

**6. Load forecasting and economic dispatch** 

the network.

A constrained, multi-objective and non-differentiable optimisation for Var planning problem was proposed (Chen & Liu, 1994). The objectives were minimization of active power loss cost, minimization of the cost of investment in Var sources, robustness of the system security margin and minimization of the voltage deviation of the system. The operating constraints, load constraints and expansion constraints of the system were considered. The goal-attainment method, based on SA, for solving general multi-objective optimization problems by assuming that the decision-maker has goals for each of the objective functions was used to solve the problem. The solution methodology involved finding a desirable, global noninferior solution to the problem, even when the objective space is nonconvex.

An interactive satisfying method, a two-level structure, was proposed to solve the multiobjective power system Var planning (Chen & Liu, 1995). The analysis level involved calculation of a possible or set of possible solutions to the multi-objective problem, and the decision level generate noninferior solutions that meet the preferences of the decision makers. In the analysis level, the ε-constraint method that is based on SA is used to find a global noninferior solution. The proposed method guarantees the solution to be a desirable, global noninferior solution for a general multiobjective Var planning, according to the preferences of the decision makers.

Var planning was presented as a multi-objective optimization problem in terms of maximum system security and minimum operation cost (Jwo et al., 1995). An effective algorithm based on hybrid expert system and SA was proposed to obtain the global optimal solution considering both quality and speed. A weak bus-oriented criterion for identifying candidate buses for Var planning was presented in (Chen, 1996). A voltage collapse proximity indicator was first used to identify weak buses. Then appropriate Var planning for those weak buses increased the system security margin to prevent voltage collapse. The goal attainment method, based on the SA, was applied to solving the multi-objective problem by assuming that the decision-maker has goals for each of the objective functions. The presented method both provides a good final solution and reduces the solution space.

An innovative fast global optimization technique, hybrid partial gradient descent/SA (HPGDSA), for optimal Var planning was presented (Liu et al., 1997). The basic concept of the HPGDSA is that partial gradient descent and SA alternate with each other to reduce the CPU time below that of the conventional SA while the ability to find the global optimal of the SA is retained. A hybrid SA/GA approach was proposed to solve the Var planning (Jwo et al., 1999). That approach found the near-global optimal solution in a finite time. Moreover, the solution time was much less than that of the conventional SA. The proposed method yielded promising results, relative to those obtained using SA, GA and HPGDSA.

Three GA/SA/TS combined algorithms for Var planning were proposed (Liu et al., 2002). Trying reasonably to combine local and global search, they adopt the acceptance probability of SA to improve the convergence of the simple GA, and to apply TS to find more accurate solutions. Test results confirmed that the proposed method is effective to find better solutions than those of existing methods within reasonable time. A projection-based twolayer SA was used to solve the multi-objective optimization problems (Chen & Ke, 2004). The SA yielded a desirable, globally efficient solution to such problems, even when the solution space is nonconvex and the objective functions are nondifferentiable.

A new approach was presented to model and solve Var planning under the static voltage stability constraint (Wang et al., 2011). First, the fuzzy clustering method was used to select new candidate Var source locations. Then, modified Gray code was applied to represent a series of non-uniform Var capacity intervals at different candidate buses. Under the new ordering of the Var capacity intervals, a simplified piecewise linear function that relates the total transfer capability with new Var capacity was derived and applied as a static voltage stability constraint in Var planning. Finally, the optimization problem was solved by an enhanced SA using modified Gray code. The proposed SA adopted a modified definition of neighborhood selection and a novel approach to generating new random solutions. Test results demonstrated that the proposed method is a simple and effective approach for voltage stability-constrained Var planning with contingency considered.

A multi-objective SA was proposed to provide decision support in reactive power compensation (Antunes et al., 2011). Their method computed a set of well-distributed and diversified solutions underlying distinct trade-offs, even for a challenging network. The characteristics of the non-dominated front are relevant information that helps planning engineers select satisfactory compromise solutions that improve the operating conditions of the network.

## **6. Load forecasting and economic dispatch**

114 Simulated Annealing – Advances, Applications and Hybridizations

the SA to obtain the global optimal solution.

space is nonconvex.

preferences of the decision makers.

the solution algorithm uses an extended SA to find a global, noninferior solution. The primary objective function and the trade-off tolerances are then used to transform the constrained multi-objective optimization problem into a single-objective optimization problem with more constraints by applying the ε-constraint method. The second-stage uses

A constrained, multi-objective and non-differentiable optimisation for Var planning problem was proposed (Chen & Liu, 1994). The objectives were minimization of active power loss cost, minimization of the cost of investment in Var sources, robustness of the system security margin and minimization of the voltage deviation of the system. The operating constraints, load constraints and expansion constraints of the system were considered. The goal-attainment method, based on SA, for solving general multi-objective optimization problems by assuming that the decision-maker has goals for each of the objective functions was used to solve the problem. The solution methodology involved finding a desirable, global noninferior solution to the problem, even when the objective

An interactive satisfying method, a two-level structure, was proposed to solve the multiobjective power system Var planning (Chen & Liu, 1995). The analysis level involved calculation of a possible or set of possible solutions to the multi-objective problem, and the decision level generate noninferior solutions that meet the preferences of the decision makers. In the analysis level, the ε-constraint method that is based on SA is used to find a global noninferior solution. The proposed method guarantees the solution to be a desirable, global noninferior solution for a general multiobjective Var planning, according to the

Var planning was presented as a multi-objective optimization problem in terms of maximum system security and minimum operation cost (Jwo et al., 1995). An effective algorithm based on hybrid expert system and SA was proposed to obtain the global optimal solution considering both quality and speed. A weak bus-oriented criterion for identifying candidate buses for Var planning was presented in (Chen, 1996). A voltage collapse proximity indicator was first used to identify weak buses. Then appropriate Var planning for those weak buses increased the system security margin to prevent voltage collapse. The goal attainment method, based on the SA, was applied to solving the multi-objective problem by assuming that the decision-maker has goals for each of the objective functions. The presented method both provides a good final solution and reduces the solution space.

An innovative fast global optimization technique, hybrid partial gradient descent/SA (HPGDSA), for optimal Var planning was presented (Liu et al., 1997). The basic concept of the HPGDSA is that partial gradient descent and SA alternate with each other to reduce the CPU time below that of the conventional SA while the ability to find the global optimal of the SA is retained. A hybrid SA/GA approach was proposed to solve the Var planning (Jwo et al., 1999). That approach found the near-global optimal solution in a finite time. Moreover, the solution time was much less than that of the conventional SA. The proposed method yielded promising results, relative to those obtained using SA, GA and HPGDSA.

Accurate forecasting of electricity load has been one of the most important issues in the planning, operation, and control of electric power systems. Recently, following power system privatization and deregulation, the accurate forecasting of electricity load has received increasing attention. An optimal fuzzy inference method for short-term load forecasting was presented (Mori & Kobayashi, 1996). Their proposed method constructs an optimal structure of the simplified fuzzy inference that minimizes model errors and the number of membership functions to capture the nonlinear behavior of short-term loads in the power system. The model was identified by SA and the steepest descent method.

Support vector machines (SVMs) have been successfully used to solve nonlinear regression and time series problems. The SVM with SA approach was proposed to forecast electricity load (Pai & Hong, 2005). They used SA to select the parameters of the SVM model. Empirical results indicate that the proposed model outperforms the autoregressive integrated moving average model and the general regression NN model.

A fuzzy NN combined with a chaos-search GA (CGA) and SA, applied to short-term powersystem load forecasting was presented (Liao & Tsao, 2006). They used a fuzzy hyperrectangular composite NN for forecasting the initial load. An integrated CGA and SA method is then used to find the optimal NN parameters. The CGA is effective in global search but ineffective in local search, while the SA is effective in local optimal search. The paper combined the two methods to exploit the advantages of both and to eliminate the known downside of the traditional NN. Their test results demonstrated superior a forecasting accuracy than other commonly used forecasting methods.

Applications of Simulated Annealing-Based Approaches to Electric Power Systems 117

A parallel TS for determining ramp rate constrained ED for generating units with nonmonotonically and monotonically increasing incremental cost functions was proposed (Ongsakul et al., 2004). To parallelize TS efficiently, neighborhood decomposition was performed to balance the computing load, and competitive selection was used to update the best solution reached among subneighborhoods. The proposed approach optimizes the compromises between the experimental speedup and the solution quality for the best performance with different subneighborhood sizes. The proposed approach is potentially viable for online ED because of it provides substantial generator fuel cost savings and the

A novel multiobjective optimization method for economic emission load dispatch of fixed head hydro plants and thermal plants with nonsmooth fuel cost and emission level functions was presented (Basu, 2005). In this problem, economic and emission objectives were competing. Based on the assumption that the decision-maker has goals for each of the objective functions, the multiobjective problem is converted into a single-objective optimization problem by the goal-attainment method, which is then handled by the SA. The solution method yields a global or near-global noninferior solution that will be close to meeting the decision-maker's requirements. Test results confirmed the applicability and

Dynamic ED determines the optimal operation of units with predicted load demands over a certain period with the objective of minimizing total production cost while the system is operating within its ramp rate limits. SA was proposed to obtain the global or near global optimum dynamic ED (Panigrahi et al., 2006). They incorporated load balance constraints, operating limits, valve point loading, ramp constraints, and network losses into the dynamic ED. Numerical results revealed the performance and applicability of the proposed method. Since generators have quadratic fuel cost functions, classical techniques ignore or flatten out the portions of the incremental fuel cost curves and so may have difficulty in determining the global optimum solution for non-differentiable fuel cost functions. EP based SA approach was presented to ED in a large-scale power system (Christober Asir Rajan, 2010). The proposed techniques can offer global or near-global optimum dispatch solutions. Test results demonstrate that the proposed integrated approach can provide accurate solutions

Network reconfiguration problem was formulated as a constrained, multi-objective and non-differential optimization problem for both loss reduction and load balancing that considers load constraints and operating constraints (Chiang & Jean-Jumeau, 1990a, 1990b). The number of switch-on/switch-off operations that are involved in network reconfiguration was included as a constraint. Then, a two-stage solution method that is based on a modified SA and the ε-constraint method for general multi-objective optimization were presented. The proposed approach allows designers to obtain a desirable, global noninferior solution in a reasonable computation time. Given a target number of switch-on/switch-off operations

high upper bound on speedup.

validity of the proposed method.

within reasonable times for any fuel cost functions.

**7. Distribution systems planning and operation** 

Economic dispatch (ED) is an important daily optimization task in the operation of a power system. Most calculus-based industrial algorithms for solving the ED problem, such as the Lagrangian multiplier method, require the incremental cost curves to be monotonically increasing or piece-wise linear. When the generating units have non-monotonically increasing or non-linear incremental cost curves, the conventional procedure either ignores or flattens out the portions of the incremental cost curves that are not continuous or monotonically increasing. Inaccurate dispatch results are thus obtained. In these cases, accurate dispatch results can only be obtained using more general approaches without restrictions on the shape of fuel cost functions.

SA-based ED was presented to obtain a global or near-global optimum dispatch solution (Wong & Fung, 1993). In the algorithm, the load balance constraint and the operating limit constraints of the generators are fully accounted for. Transmission losses were firstly discounted and subsequently incorporated in the algorithm using the B-matrix loss formula. Test results are obtained by the proposed approach more economically than using the dynamic programming method with a zoom feature.

A combination of the incremental GA and SA was presented to obtain the global or nearglobal optimum solution for the ED and developed to minimize the memory requirement (Wong & Wong 1994c). They proposed a method for solving the problem of discretization in the encoding of generator loadings. The algorithms include a method for ensuring that the dispatch solutions that are generated by the solution process are feasible and valid. The effects of valve-point loading and ramping characteristics of the generators are considered. The developed algorithms were demonstrated by applying them to a power system, and they were shown to be general and are computationally faster than the earlier SA-based method.

A new multi-objective stochastic search technique for ED was proposed (Das & Patvardhan, 1998). Their heuristic combined a real coded GA and SA. The proposed approach provides the values of various parameters that optimize different objectives, and the best compromise between them in a single run. The test results were compared with those obtained using other methods and the proposed heuristic was found to converge rapidly to better solutions. Additionally, perturbation analysis demonstrated that the solutions that were obtained by the proposed algorithm were truly pareto-optimal, meaning that no objective could be further improved without degrading the others. SA approach was used to solve optimal power flow (OPF) problem that involved both the load flow and the ED (Roa-Sepulveda & Pavez-Lazo, 2003). Test results confirmed the effectiveness of the solution technique.

A parallel TS for determining ramp rate constrained ED for generating units with nonmonotonically and monotonically increasing incremental cost functions was proposed (Ongsakul et al., 2004). To parallelize TS efficiently, neighborhood decomposition was performed to balance the computing load, and competitive selection was used to update the best solution reached among subneighborhoods. The proposed approach optimizes the compromises between the experimental speedup and the solution quality for the best performance with different subneighborhood sizes. The proposed approach is potentially viable for online ED because of it provides substantial generator fuel cost savings and the high upper bound on speedup.

116 Simulated Annealing – Advances, Applications and Hybridizations

restrictions on the shape of fuel cost functions.

dynamic programming method with a zoom feature.

method.

forecasting accuracy than other commonly used forecasting methods.

A fuzzy NN combined with a chaos-search GA (CGA) and SA, applied to short-term powersystem load forecasting was presented (Liao & Tsao, 2006). They used a fuzzy hyperrectangular composite NN for forecasting the initial load. An integrated CGA and SA method is then used to find the optimal NN parameters. The CGA is effective in global search but ineffective in local search, while the SA is effective in local optimal search. The paper combined the two methods to exploit the advantages of both and to eliminate the known downside of the traditional NN. Their test results demonstrated superior a

Economic dispatch (ED) is an important daily optimization task in the operation of a power system. Most calculus-based industrial algorithms for solving the ED problem, such as the Lagrangian multiplier method, require the incremental cost curves to be monotonically increasing or piece-wise linear. When the generating units have non-monotonically increasing or non-linear incremental cost curves, the conventional procedure either ignores or flattens out the portions of the incremental cost curves that are not continuous or monotonically increasing. Inaccurate dispatch results are thus obtained. In these cases, accurate dispatch results can only be obtained using more general approaches without

SA-based ED was presented to obtain a global or near-global optimum dispatch solution (Wong & Fung, 1993). In the algorithm, the load balance constraint and the operating limit constraints of the generators are fully accounted for. Transmission losses were firstly discounted and subsequently incorporated in the algorithm using the B-matrix loss formula. Test results are obtained by the proposed approach more economically than using the

A combination of the incremental GA and SA was presented to obtain the global or nearglobal optimum solution for the ED and developed to minimize the memory requirement (Wong & Wong 1994c). They proposed a method for solving the problem of discretization in the encoding of generator loadings. The algorithms include a method for ensuring that the dispatch solutions that are generated by the solution process are feasible and valid. The effects of valve-point loading and ramping characteristics of the generators are considered. The developed algorithms were demonstrated by applying them to a power system, and they were shown to be general and are computationally faster than the earlier SA-based

A new multi-objective stochastic search technique for ED was proposed (Das & Patvardhan, 1998). Their heuristic combined a real coded GA and SA. The proposed approach provides the values of various parameters that optimize different objectives, and the best compromise between them in a single run. The test results were compared with those obtained using other methods and the proposed heuristic was found to converge rapidly to better solutions. Additionally, perturbation analysis demonstrated that the solutions that were obtained by the proposed algorithm were truly pareto-optimal, meaning that no objective could be further improved without degrading the others. SA approach was used to solve optimal power flow (OPF) problem that involved both the load flow and the ED (Roa-Sepulveda &

Pavez-Lazo, 2003). Test results confirmed the effectiveness of the solution technique.

A novel multiobjective optimization method for economic emission load dispatch of fixed head hydro plants and thermal plants with nonsmooth fuel cost and emission level functions was presented (Basu, 2005). In this problem, economic and emission objectives were competing. Based on the assumption that the decision-maker has goals for each of the objective functions, the multiobjective problem is converted into a single-objective optimization problem by the goal-attainment method, which is then handled by the SA. The solution method yields a global or near-global noninferior solution that will be close to meeting the decision-maker's requirements. Test results confirmed the applicability and validity of the proposed method.

Dynamic ED determines the optimal operation of units with predicted load demands over a certain period with the objective of minimizing total production cost while the system is operating within its ramp rate limits. SA was proposed to obtain the global or near global optimum dynamic ED (Panigrahi et al., 2006). They incorporated load balance constraints, operating limits, valve point loading, ramp constraints, and network losses into the dynamic ED. Numerical results revealed the performance and applicability of the proposed method.

Since generators have quadratic fuel cost functions, classical techniques ignore or flatten out the portions of the incremental fuel cost curves and so may have difficulty in determining the global optimum solution for non-differentiable fuel cost functions. EP based SA approach was presented to ED in a large-scale power system (Christober Asir Rajan, 2010). The proposed techniques can offer global or near-global optimum dispatch solutions. Test results demonstrate that the proposed integrated approach can provide accurate solutions within reasonable times for any fuel cost functions.

## **7. Distribution systems planning and operation**

Network reconfiguration problem was formulated as a constrained, multi-objective and non-differential optimization problem for both loss reduction and load balancing that considers load constraints and operating constraints (Chiang & Jean-Jumeau, 1990a, 1990b). The number of switch-on/switch-off operations that are involved in network reconfiguration was included as a constraint. Then, a two-stage solution method that is based on a modified SA and the ε-constraint method for general multi-objective optimization were presented. The proposed approach allows designers to obtain a desirable, global noninferior solution in a reasonable computation time. Given a target number of switch-on/switch-off operations

involved in the network configuration, the solution algorithm can identify the most effective operations. To reduce the required computing time, the researchers studied the idea of approximate calculations and incorporated them into the solution algorithm, in which two efficient load-flow methods were used: one for high temperatures and the other for low temperatures.

Applications of Simulated Annealing-Based Approaches to Electric Power Systems 119

A single comprehensive algorithm for distribution system switch reconfiguration and capacitor control was proposed (Jiang & Baldick, 1996). They used SA to optimize the switch configuration of the distribution system, and a discrete optimization algorithm to find the optimal capacitor control. They evaluated the benefits of the optimal switch configuration and capacitor control, in terms of both reduced loss and decreased voltage bandwidth. A nonlinear constrained, non-differentiable approach for optimal network routing in distribution system planning was presented (Jonnavithula & Billinton, 1996). The main objective was to minimize the total cost, which is the summation of reliability costs, the cost of feeder resistive loss, investment costs and maintenance costs. They used SA to find a

SA-based approach for loss minimization by using an automatic switching operation in large-scale distribution systems was presented (Jeon et al., 2002). SA is particularly well suited for a large combinatorial optimization problem since it can avoid local minima by accepting improvements in cost. However, it commonly requires a meaningful cooling schedule and a special strategy, which makes use of the property of distribution systems in finding the optimal solution. This paper expands the cost function by adding the operating conditions of a distribution system, improving the perturbation mechanism with system topology, and using the polynomial-time cooling schedule, which is based on the statistical calculation in the search. Test results validated and confirmed the effectiveness of the

Hybrid SA and TS approach was applied to minimize real power loss in distribution systems (Jeon et al., 2004). SA is very suitable for large combinational optimization problems, but the SA requires excessive computing time. TS attempts to determine a better solution in the manner of a greatest-descent algorithm, but it cannot guarantee convergence. The hybrid SA and TS algorithm was applied to improve the computing time and convergence. Numerical examples validated and established the effectiveness of their

A method for optimal planning of radial distribution networks was solved by a combination of the steepest descent and the SA (Nahman & Peric, 2008). Their objective was to find the complete network of available routes and the optimization goal was to obtain the routes that provide the minimal total annual cost. The solution with minimum capital cost, obtained using the steepest descent approach, was used as the initial solution in SA to obtain the solution with minimum total cost. The costs associated with capital recovery, energy loss

SA for optimal tearing of networks was presented to divide a power system network model into a number of sub-networks to optimize the use of parallel computer systems for network analysis (Irving & Sterling, 1990). Test results were compared with those obtained using the iterative improvement method, and the proposed SA yielded significantly better solutions.

global optimum solution to the problem.

and undelivered energy costs were considered.

**8. The other applications** 

proposed approach.

hybrid approach.

A comprehensive approach to strategic planning of Var compensators in a nonsinusoidal distribution system was presented (Chu et al., 1994). The problem was formulated as a nondifferentiable, combinatorial optimization problem to minimize the system costs while meeting various operating constraints and harmonic limits. SA was used to determine the optimal locations, types and sizes, and settings of these Var compensators. Their proposed approach could handle discrete rather than continuous Var compensator values and determine whether the Var compensators were fixed or switchable for different load levels.

A modified SA was presented for network reconfiguration for loss reduction in distribution systems and the switching limitation was considered (Chang & Kuo, 1994). They proposed a set of simplified line flow equations for approximate loss calculation. They then used an efficient perturbation scheme and an initialization procedure to determine a better starting temperature for the SA. The computing time of the SA was greatly reduced without loss of quality of the solution. Additionally, the proposed SA could rapidly provide a global optimal or near-optimal solution to the problem, and numerical results confirmed the effectiveness of the proposed method.

Optimal capacitor placement, replacement and control in large-scale unbalanced, radial or loop distribution networks were formulated in a combinatorial optimization problem with a non-differentiable objective function (Chiang et al., 1995a, 1995b). They solved this problem using SA and the greedy search technique to obtain high-quality of solutions at a high computing speed. The challenge is to determine the optimal locations in which to install (or replace, or remove) capacitors, the types and sizes of the capacitors to be installed (or replaced) and, for each load level, the control schemes for each capacitor in the nodes of a general three-phase unbalanced distribution system, such that a desired objective function is minimized while the load constraints, network constraints and operating constraints at various load levels are satisfied. The objective function incorporated both the cost of energy loss and costs related to capacitor purchase, capacitor installation, capacitor replacement and capacitor removal. Analysis of the computational complexity of the solution algorithm reveals that the algorithm is also effective for large-scale distribution systems as the computational efforts is reasonable.

A new formulation for power system sectionalizing device placement considering outage, maintenance and investments costs was proposed (Billinton & Jonnavithula, 1996). They formulated the problem as a combinatorial constrained optimization problem with a nonlinear, nondifferentiable objective function. The SA was used to determine the number of sectionalizing switches and the locations of the switches. Test results revealed that the proposed approach yielded a global optimal solution to the sectionalizing device placement problem, considering reliability, investment and maintenance costs.

A single comprehensive algorithm for distribution system switch reconfiguration and capacitor control was proposed (Jiang & Baldick, 1996). They used SA to optimize the switch configuration of the distribution system, and a discrete optimization algorithm to find the optimal capacitor control. They evaluated the benefits of the optimal switch configuration and capacitor control, in terms of both reduced loss and decreased voltage bandwidth. A nonlinear constrained, non-differentiable approach for optimal network routing in distribution system planning was presented (Jonnavithula & Billinton, 1996). The main objective was to minimize the total cost, which is the summation of reliability costs, the cost of feeder resistive loss, investment costs and maintenance costs. They used SA to find a global optimum solution to the problem.

SA-based approach for loss minimization by using an automatic switching operation in large-scale distribution systems was presented (Jeon et al., 2002). SA is particularly well suited for a large combinatorial optimization problem since it can avoid local minima by accepting improvements in cost. However, it commonly requires a meaningful cooling schedule and a special strategy, which makes use of the property of distribution systems in finding the optimal solution. This paper expands the cost function by adding the operating conditions of a distribution system, improving the perturbation mechanism with system topology, and using the polynomial-time cooling schedule, which is based on the statistical calculation in the search. Test results validated and confirmed the effectiveness of the proposed approach.

Hybrid SA and TS approach was applied to minimize real power loss in distribution systems (Jeon et al., 2004). SA is very suitable for large combinational optimization problems, but the SA requires excessive computing time. TS attempts to determine a better solution in the manner of a greatest-descent algorithm, but it cannot guarantee convergence. The hybrid SA and TS algorithm was applied to improve the computing time and convergence. Numerical examples validated and established the effectiveness of their hybrid approach.

A method for optimal planning of radial distribution networks was solved by a combination of the steepest descent and the SA (Nahman & Peric, 2008). Their objective was to find the complete network of available routes and the optimization goal was to obtain the routes that provide the minimal total annual cost. The solution with minimum capital cost, obtained using the steepest descent approach, was used as the initial solution in SA to obtain the solution with minimum total cost. The costs associated with capital recovery, energy loss and undelivered energy costs were considered.

## **8. The other applications**

118 Simulated Annealing – Advances, Applications and Hybridizations

effectiveness of the proposed method.

computational efforts is reasonable.

temperatures.

involved in the network configuration, the solution algorithm can identify the most effective operations. To reduce the required computing time, the researchers studied the idea of approximate calculations and incorporated them into the solution algorithm, in which two efficient load-flow methods were used: one for high temperatures and the other for low

A comprehensive approach to strategic planning of Var compensators in a nonsinusoidal distribution system was presented (Chu et al., 1994). The problem was formulated as a nondifferentiable, combinatorial optimization problem to minimize the system costs while meeting various operating constraints and harmonic limits. SA was used to determine the optimal locations, types and sizes, and settings of these Var compensators. Their proposed approach could handle discrete rather than continuous Var compensator values and determine whether the Var compensators were fixed or switchable for different load levels. A modified SA was presented for network reconfiguration for loss reduction in distribution systems and the switching limitation was considered (Chang & Kuo, 1994). They proposed a set of simplified line flow equations for approximate loss calculation. They then used an efficient perturbation scheme and an initialization procedure to determine a better starting temperature for the SA. The computing time of the SA was greatly reduced without loss of quality of the solution. Additionally, the proposed SA could rapidly provide a global optimal or near-optimal solution to the problem, and numerical results confirmed the

Optimal capacitor placement, replacement and control in large-scale unbalanced, radial or loop distribution networks were formulated in a combinatorial optimization problem with a non-differentiable objective function (Chiang et al., 1995a, 1995b). They solved this problem using SA and the greedy search technique to obtain high-quality of solutions at a high computing speed. The challenge is to determine the optimal locations in which to install (or replace, or remove) capacitors, the types and sizes of the capacitors to be installed (or replaced) and, for each load level, the control schemes for each capacitor in the nodes of a general three-phase unbalanced distribution system, such that a desired objective function is minimized while the load constraints, network constraints and operating constraints at various load levels are satisfied. The objective function incorporated both the cost of energy loss and costs related to capacitor purchase, capacitor installation, capacitor replacement and capacitor removal. Analysis of the computational complexity of the solution algorithm reveals that the algorithm is also effective for large-scale distribution systems as the

A new formulation for power system sectionalizing device placement considering outage, maintenance and investments costs was proposed (Billinton & Jonnavithula, 1996). They formulated the problem as a combinatorial constrained optimization problem with a nonlinear, nondifferentiable objective function. The SA was used to determine the number of sectionalizing switches and the locations of the switches. Test results revealed that the proposed approach yielded a global optimal solution to the sectionalizing device placement

problem, considering reliability, investment and maintenance costs.

SA for optimal tearing of networks was presented to divide a power system network model into a number of sub-networks to optimize the use of parallel computer systems for network analysis (Irving & Sterling, 1990). Test results were compared with those obtained using the iterative improvement method, and the proposed SA yielded significantly better solutions.

The placement of a minimal set of phasor measurement units (PMUs) to make the system measurement model observable was presented (Baldwin et al., 1993). A PMU at a bus measures the voltage and all of the current phasors at that bus, requiring extension of the topological observability theory. The minimal PMU set is found using a dual search algorithm, which uses both a modified bisecting search and SA. The former fixes the number of PMUs while the latter seeks a placement set that yields an observable network for a fixed number of PMUs. Test results verified the effectiveness of the proposed approach.

Applications of Simulated Annealing-Based Approaches to Electric Power Systems 121

loading conditions and system configurations to the left in the s-plane. The incorporation of SA as a derivative-free optimization approach in PSS design greatly reduces the computational burden. Test results demonstrated the effectiveness of the proposed approach under various disturbances and loading conditions for two multimachine power

SA-based approach to PSS and flexible alternating current transmission systems (FACTS) based stabilizer tuning was presented (Abido, 2000b). The problem of designing PSS and FACTS-based stabilizers was formulated as an optimization problem. An eigenvalue-based objective function to increase system damping was proposed, and the SA was used to search for optimal stabilizer parameters. Different control schemes have been proposed and tested on a weakly connected power system under different disturbances, loading conditions, and parameter variations. Nonlinear simulation results indicate the potential usefulness of the SA in the problem of tuning PSS and FACTS-based stabilizer. The effectiveness and robustness of the proposed control schemes under a wide range of loading conditions and

A pole placement technique for PSS and thyristor controlled series capacitor (TCSC) based stabilizer using SA was proposed (Abido, 2000c). The design problem is formulated as an optimization problem where SA was used to search for the optimal setting of the design parameters, and considered a pole placement-based objective function to shift the dominant eigenvalues to the left in the s-plane. Eigenvalue analysis and nonlinear simulation results confirmed the effectiveness and the robustness of the proposed stabilizers and their ability

SA was applied to evaluate harmonics and frequency for power system quality analysis and frequency relaying (Soliman et al., 2004). The sum of the squares of errors is the objective function to be minimized for evaluating the amplitude and phase angle of each harmonic component as well as the fundamental frequency of the voltage signal. The proposed algorithm applied digitized samples of the voltage signal where the power quality is to be measured and the frequency relaying is to be implemented. The proposed SA had an adaptive cooling schedule and used a variable discretization to accelerate the convergence of the original SA. Numerical results revealed that the proposed approach can identify the

Calculation of the optimum installation angle for the fixed solar-cell panels based on GA and SA was presented (Chen et al., 2005). The incident angle of sunlight strongly affects the output power of a solar-cell panel, and its efficiency can be improved if the solar-cell panel is properly installed at the optimum angle. Both GA and SA with climatic data are utilized to calculate the optimum installation angle of the solar-cell panel at various locations in Taiwan. Experimental results reveal that the best monthly installation angles are very close

Identifying placement sites for PMUs in a power system based on incomplete observability was presented (Nuqui & Phadke, 2005). They introduced the novel concept of depth of unobservability and explained its effect on the number of PMU placements. They extended

system parameter variations have been demonstrated.

to provide efficient damping of low frequency oscillations.

to those determined by the computer simulation results.

harmonic spectrum in the signal.

systems.

A parallel SA was presented to decompose power systems into subsystems that were with equal numbers of nodes and control variables (Mori & Takeda, 1994). The decomposition of a power system is a difficult discrete combinatorial problem. The researchers' numerical results revealed that the parallel SA yielded better solutions than the conventional SA.

SA was applied to multi-partition an observable power system state estimation network into two or more observable sub-networks (Habiballah & Irving, 1995). The proposed SA was theoretically based on combinatorial optimization, rather than a heuristic derivation. Numerical results demonstrated the effectiveness of the proposed SA.

A novel method for designing power system damping controllers was presented (Chen et al., 1996). They used SA to optimize the controller parameters in the nonlinear optimization problem, and considered the all of the design criteria of the controllers simultaneously. Their proposed method can also be used to design controllers that are robust under a specified set of operating conditions.

Design of output feedback controllers for thyristor controlled series compensators in a meshed power system was proposed (Chen et al., 1998). They used SA to optimize the output feedback gains for the controllers. Conflicting design objectives, such as improvement in the damping of the critical modes, any deterioration of the damping of the noncritical modes and the saturation of the controller actuators, were simultaneously considered. Numerical results verified that the SA can be applied to design robust controllers that satisfy the required performance criteria under many operating conditions.

Feeder imbalance describes a situation in which the magnitudes of the voltages of a threephase voltage source are not equal, or the phase differences between them are not 120 electrical degrees, or both. Phase balancing make the voltages balanced at each load point of the feeder. Phase balancing optimization is currently attracting more attention in the power industry, especially following deregulation. Nonlinear effects, such as voltage drops and energy losses, make the problem difficult to solve. SA was used as an effective method to solve a power distribution phase balancing problem with its nonlinear effects (Zhu et al., 1999). Test results verified the effectiveness of the proposed approach.

Robust design of multi-machine power system stabilizers (PSS) using SA was proposed (Abido, 2000a). The SA can obtain optimal parameter settings of an extensively used conventional fixed-structure lead-lag PSS. The parameters of the proposed SA-based PSS were optimized to shift simultaneously the system electromechanical modes under different loading conditions and system configurations to the left in the s-plane. The incorporation of SA as a derivative-free optimization approach in PSS design greatly reduces the computational burden. Test results demonstrated the effectiveness of the proposed approach under various disturbances and loading conditions for two multimachine power systems.

120 Simulated Annealing – Advances, Applications and Hybridizations

approach.

The placement of a minimal set of phasor measurement units (PMUs) to make the system measurement model observable was presented (Baldwin et al., 1993). A PMU at a bus measures the voltage and all of the current phasors at that bus, requiring extension of the topological observability theory. The minimal PMU set is found using a dual search algorithm, which uses both a modified bisecting search and SA. The former fixes the number of PMUs while the latter seeks a placement set that yields an observable network for a fixed number of PMUs. Test results verified the effectiveness of the proposed

A parallel SA was presented to decompose power systems into subsystems that were with equal numbers of nodes and control variables (Mori & Takeda, 1994). The decomposition of a power system is a difficult discrete combinatorial problem. The researchers' numerical results revealed that the parallel SA yielded better solutions than the conventional SA.

SA was applied to multi-partition an observable power system state estimation network into two or more observable sub-networks (Habiballah & Irving, 1995). The proposed SA was theoretically based on combinatorial optimization, rather than a heuristic derivation.

A novel method for designing power system damping controllers was presented (Chen et al., 1996). They used SA to optimize the controller parameters in the nonlinear optimization problem, and considered the all of the design criteria of the controllers simultaneously. Their proposed method can also be used to design controllers that are robust under a

Design of output feedback controllers for thyristor controlled series compensators in a meshed power system was proposed (Chen et al., 1998). They used SA to optimize the output feedback gains for the controllers. Conflicting design objectives, such as improvement in the damping of the critical modes, any deterioration of the damping of the noncritical modes and the saturation of the controller actuators, were simultaneously considered. Numerical results verified that the SA can be applied to design robust controllers that satisfy the required performance criteria under many operating conditions.

Feeder imbalance describes a situation in which the magnitudes of the voltages of a threephase voltage source are not equal, or the phase differences between them are not 120 electrical degrees, or both. Phase balancing make the voltages balanced at each load point of the feeder. Phase balancing optimization is currently attracting more attention in the power industry, especially following deregulation. Nonlinear effects, such as voltage drops and energy losses, make the problem difficult to solve. SA was used as an effective method to solve a power distribution phase balancing problem with its nonlinear effects (Zhu et al.,

Robust design of multi-machine power system stabilizers (PSS) using SA was proposed (Abido, 2000a). The SA can obtain optimal parameter settings of an extensively used conventional fixed-structure lead-lag PSS. The parameters of the proposed SA-based PSS were optimized to shift simultaneously the system electromechanical modes under different

Numerical results demonstrated the effectiveness of the proposed SA.

1999). Test results verified the effectiveness of the proposed approach.

specified set of operating conditions.

SA-based approach to PSS and flexible alternating current transmission systems (FACTS) based stabilizer tuning was presented (Abido, 2000b). The problem of designing PSS and FACTS-based stabilizers was formulated as an optimization problem. An eigenvalue-based objective function to increase system damping was proposed, and the SA was used to search for optimal stabilizer parameters. Different control schemes have been proposed and tested on a weakly connected power system under different disturbances, loading conditions, and parameter variations. Nonlinear simulation results indicate the potential usefulness of the SA in the problem of tuning PSS and FACTS-based stabilizer. The effectiveness and robustness of the proposed control schemes under a wide range of loading conditions and system parameter variations have been demonstrated.

A pole placement technique for PSS and thyristor controlled series capacitor (TCSC) based stabilizer using SA was proposed (Abido, 2000c). The design problem is formulated as an optimization problem where SA was used to search for the optimal setting of the design parameters, and considered a pole placement-based objective function to shift the dominant eigenvalues to the left in the s-plane. Eigenvalue analysis and nonlinear simulation results confirmed the effectiveness and the robustness of the proposed stabilizers and their ability to provide efficient damping of low frequency oscillations.

SA was applied to evaluate harmonics and frequency for power system quality analysis and frequency relaying (Soliman et al., 2004). The sum of the squares of errors is the objective function to be minimized for evaluating the amplitude and phase angle of each harmonic component as well as the fundamental frequency of the voltage signal. The proposed algorithm applied digitized samples of the voltage signal where the power quality is to be measured and the frequency relaying is to be implemented. The proposed SA had an adaptive cooling schedule and used a variable discretization to accelerate the convergence of the original SA. Numerical results revealed that the proposed approach can identify the harmonic spectrum in the signal.

Calculation of the optimum installation angle for the fixed solar-cell panels based on GA and SA was presented (Chen et al., 2005). The incident angle of sunlight strongly affects the output power of a solar-cell panel, and its efficiency can be improved if the solar-cell panel is properly installed at the optimum angle. Both GA and SA with climatic data are utilized to calculate the optimum installation angle of the solar-cell panel at various locations in Taiwan. Experimental results reveal that the best monthly installation angles are very close to those determined by the computer simulation results.

Identifying placement sites for PMUs in a power system based on incomplete observability was presented (Nuqui & Phadke, 2005). They introduced the novel concept of depth of unobservability and explained its effect on the number of PMU placements. They extended

their model to recognize limitations in the availability of communication facilities around the network and thus formulated the constrained placement problem. The SA was further used to solve the pragmatic phased installation of PMUs. Results demonstrated that the proposed SA provides utilities with systematic approach for incrementally placing PMUs, to help manage the impact of their cost.

Applications of Simulated Annealing-Based Approaches to Electric Power Systems 123

Abido, M. A. (2000b). Simulated annealing based approach to PSS and FACTS based stabilizer tuning. *International Journal of Electrical Power & Energy Systems*, Vol. 22, No. 4,

Abido, M. A. (2000c). Pole placement technique for PSS and TCSC-based stabilizer design using simulated annealing. *International Journal of Electrical Power & Energy Systems*, Vol.

Annakkage, U. D.; Numnonda, T. & Pahalawaththa, N. C. (1995). Unit commitment by parallel simulated annealing. *IEE Proceedings-Generation, Transmission and Distribution*,

Antunes, C. H.; Lima, P.; Oliveira, E. & Pires, D. F. (2011). A multi-objective simulated annealing approach to reactive power compensation. *Engineering Optimization*, Vol. 43,

Baldwin, T. L.; Mili, L.; Boisen, M. B., Jr. & Adapa, R. (1993). Power system observability with minimal phasor measurement placement. *IEEE Transactions on Power Systems*, Vol.

Basu, M. (2005). A simulated annealing-based goal-attainment method for economic emission load dispatch of fixed head hydrothermal power systems. *International Journal* 

Chang, H. C. & Kuo, C. C. (1994). Network reconfiguration in distribution systems using simulated annealing. *Electric Power Systems Research*, Vol. 29, No. 3, (May 1994), pp. 227-

Chen, X. R.; Pahalawaththa, N. C.; Annakkage, U. D. & Kumble, C. S. (1996). Simulated annealing for the design of power system damping controllers. *Electric Power Systems* 

Chen, X. R.; Pahalawaththa, N. C.; Annakkage, U. D. & Kumble, C. S. (1998). Design of decentralised output feedback TCSC damping controllers by using simulated annealing. *IEE Proceedings-Generation, Transmission and Distribution*, Vol. 145, No. 5,

Chen, Y. L. & Liu, C. C. (1994). Multiobjective Var planning using the goal-attainment method. *IEE Proceedings-Generation, Transmission and Distribution*, Vol. 141, No. 3, (May

Chen, Y. L. & Liu, C. C. (1995). Optimal multi-objective Var planning using an interactive satisfying method. *IEEE Transactions on Power Systems*, Vol. 10, No. 2, (May 1995), pp.

Chen, Y. L. (1996). Weak bus-oriented optimal multi-objective Var planning. *IEEE Transactions on Power Systems*, Vol. 11, No. 4, (November 1996), pp. 1885-1890 Chen, Y. M.; Lee, C. H. & Wu, H. C. (2005). Calculation of the optimum installation angle for fixed solar-cell panels based on the genetic algorithm and the simulated-annealing method. *IEEE Transactions on Energy Conversion*, Vol. 20, No. 2, (June 2005), pp. 467- 473

*of Electrical Power & Energy Systems*, Vol. 27, No. 2, (February 2005), pp. 147-153 Billinton, R. & Jonnavithula, S. (1996). Optimal switching device placement in radial distribution systems. *IEEE Transactions on Power Delivery*, Vol. 11, No. 3, (July 1996), pp.

(May 2000), pp. 247-258

22, No. 8, (November 2000), pp. 543-554

No. 10, (October 2011), pp. 1063-1077

8, No. 2, (May 1993), pp. 707-715

(September 1998), pp. 553-558

1994), pp. 227-232

664-670

1646-1651

238

Vol. 142, No. 6, (November 1995), pp. 595-600

*Research*, Vol. 39, No. 1, (October 1996), pp. 67-72

SA was applied to optimize size of a PV/wind integrated hybrid energy system with battery storage (Ekren & Ekren, 2010). Their SA used a stochastic gradient search to find the global optimization solution that minimized the total cost of the hybrid energy system. The decision variables included PV size, area swept by the wind turbine rotor, and battery capacity. Test results confirmed that SA yielded better results than those obtained from response surface method.

## **9. Conclusions**

This chapter reviewed journal papers that have presented the applications of SA-based approaches to electric power systems, especially in generation and transmission expansion planning, generator maintenance scheduling and unit commitment, reactive power planning, load forecasting and economic dispatch, distribution systems planning and operation, and the other applications. The SA-based approaches have the following advantages. They may find a global optimum; they can produce a number of alternative solutions; no mathematical restrictions on the problem formulation exist, and they are relatively easy to program and numerically robust. The purpose of the review of papers and example applications in this chapter is to illustrate the potential application of the SA-based approaches in the optimization of electric power systems, and the advantages of such methods. Recently, these new heuristic tools have been combined with each other and with knowledge to solve extremely challenging problems. Hybrid approaches typically seem both to combine complementary strengths and to overcome the drawbacks of single methods by embedding in them one or more steps that involve different techniques. Developing solutions using such tools provides two major advantages: development time is much shorter than when more traditional approaches are used, and the solutions are very robust.

## **Author details**

Yann-Chang Huang, Huo-Ching Sun, and Kun-Yuan Huang *Department of Electrical Engineering, Cheng Shiu University, Kaohsiung, Taiwan* 

## **10. References**

Abido, M. A. (2000a). Robust design of multimachine power system stabilizers using simulated annealing. *IEEE Transactions on Energy Conversion*, Vol. 15, No. 3, (September 2000), pp. 297-304

Abido, M. A. (2000b). Simulated annealing based approach to PSS and FACTS based stabilizer tuning. *International Journal of Electrical Power & Energy Systems*, Vol. 22, No. 4, (May 2000), pp. 247-258

122 Simulated Annealing – Advances, Applications and Hybridizations

help manage the impact of their cost.

response surface method.

**9. Conclusions** 

robust.

**Author details** 

**10. References** 

2000), pp. 297-304

Yann-Chang Huang, Huo-Ching Sun, and Kun-Yuan Huang

*Department of Electrical Engineering, Cheng Shiu University, Kaohsiung, Taiwan* 

Abido, M. A. (2000a). Robust design of multimachine power system stabilizers using simulated annealing. *IEEE Transactions on Energy Conversion*, Vol. 15, No. 3, (September

their model to recognize limitations in the availability of communication facilities around the network and thus formulated the constrained placement problem. The SA was further used to solve the pragmatic phased installation of PMUs. Results demonstrated that the proposed SA provides utilities with systematic approach for incrementally placing PMUs, to

SA was applied to optimize size of a PV/wind integrated hybrid energy system with battery storage (Ekren & Ekren, 2010). Their SA used a stochastic gradient search to find the global optimization solution that minimized the total cost of the hybrid energy system. The decision variables included PV size, area swept by the wind turbine rotor, and battery capacity. Test results confirmed that SA yielded better results than those obtained from

This chapter reviewed journal papers that have presented the applications of SA-based approaches to electric power systems, especially in generation and transmission expansion planning, generator maintenance scheduling and unit commitment, reactive power planning, load forecasting and economic dispatch, distribution systems planning and operation, and the other applications. The SA-based approaches have the following advantages. They may find a global optimum; they can produce a number of alternative solutions; no mathematical restrictions on the problem formulation exist, and they are relatively easy to program and numerically robust. The purpose of the review of papers and example applications in this chapter is to illustrate the potential application of the SA-based approaches in the optimization of electric power systems, and the advantages of such methods. Recently, these new heuristic tools have been combined with each other and with knowledge to solve extremely challenging problems. Hybrid approaches typically seem both to combine complementary strengths and to overcome the drawbacks of single methods by embedding in them one or more steps that involve different techniques. Developing solutions using such tools provides two major advantages: development time is much shorter than when more traditional approaches are used, and the solutions are very


Chen, Y. L. & Ke, Y. L. (2004). Multi-objective Var planning for large-scale power systems using projection-based two-layer simulated annealing algorithms. *IEE Proceedings-Generation, Transmission and Distribution*, Vol. 151, No. 4, (July 2004), pp. 555-560

Applications of Simulated Annealing-Based Approaches to Electric Power Systems 125

Gallego, R. A.; Monticelli, A. & Romero, R. (1998a). Comparative studies on nonconvex optimization methods for transmission network expansion planning. *IEEE Transactions* 

Gallego, R. A.; Monticelli, A. & Romero, R. (1998b). Transmission system expansion planning by an extended genetic algorithm. *IEE Proceedings-Generation, Transmission and* 

Habiballah, I. O. & Irving, M. R. (1995). Multipartitioning of power system state estimation networks using simulated annealing. *Electric Power Systems Research*, Vol. 34, No. 2,

Hsiao, Y. T.; Chiang, H. D.; Liu, C. C. & Chen, Y. L. (1994). A computer package for optimal multi-objective Var planning in large scale power systems. *IEEE Transactions on Power* 

Hsiao, Y. T.; Liu, C. C.; Chiang, H. D. & Chen, Y. L. (1993). A new approach for optimal Var sources planning in large scale electric power systems. *IEEE Transactions on Power* 

Irving, M. R. & Sterling, M. J. H. (1990). Optimal network tearing using simulated annealing. *IEE Proceedings-Generation, Transmission and Distribution*, Vol. 137, No. 1, (January 1990),

Jeon, Y. J.; Kim, J. C.; Kim, J. O; Shin, J. R. & Lee, K. Y. (2002). An efficient simulated annealing algorithm for network reconfiguration in large-scale distribution systems.

Jiang, D. & Baldick, R. (1996). Optimal electric distribution system switch reconfiguration and capacitor control. *IEEE Transactions on Power Systems*, Vol. 11, No. 2, (May 1996),

Jonnavithula, S. & Billinton, R. (1996). Minimum cost analysis of feeder routing in distribution system planning. *IEEE Transactions on Power Delivery*, Vol. 11, No. 4,

Jwo, W. S.; Liu, C. W. & Liu, C. C. (1999). Large-scale optimal Var planning by hybrid simulated annealing/genetic algorithm. *International Journal of Electrical Power & Energy* 

Jwo, W. S.; Liu, C. W.; Liu, C. C. & Hsiao, Y. T. (1995). Hybrid expert system and simulated annealing approach to optimal reactive power planning. *IEE Proceedings-Generation,* 

Kannan, S.; Slochanal, S. M. R. & Padhy, N. P. (2005). Application and comparison of metaheuristic techniques to generation expansion planning problem. *IEEE Transactions* 

Kannan, S.; Slochanal, S. M. R.; Baskar, S.; Murugan, P. (2007). Application and comparison of metaheuristic techniques to generation expansion planning in the partially deregulated environment. *IET Generation, Transmission & Distribution*, Vol. 1, No. 1,

*Transmission and Distribution*, Vol. 142, No. 4, (July 1995), pp. 381-385

*on Power Systems*, Vol. 20, No. 1, (February 2005), pp. 466-475

*IEEE Transactions on Power Delivery*, Vol. 17, No. 4, (October 2002), pp. 1070-1078 Jeon,Y. J. & Kim, J. C. (2004). Application of simulated annealing and tabu search for loss minimization in distribution systems. *International Journal of Electrical Power &* 

*on Power Systems*, Vol. 13, No. 3, (August 1998), pp. 822-828

*Distribution*, Vol. 145, No. 3, (May 1998), pp. 329-335

*Systems*, Vol. 9, No. 2, (May 1994), pp. 668-676

*Systems*, Vol. 8, No. 3, (August 1993), pp. 988-996

*Energy Systems*, Vol. 26, No. 1, (January 2004), pp. 9-18

*Systems*, Vol. 21, No. 1, (January 1999), pp. 39-44

(August 1995), pp. 117-120

pp. 69-72

pp. 890-897

(October 1996), pp. 1935-1940

(January 2007), pp. 111-118


Gallego, R. A.; Monticelli, A. & Romero, R. (1998a). Comparative studies on nonconvex optimization methods for transmission network expansion planning. *IEEE Transactions on Power Systems*, Vol. 13, No. 3, (August 1998), pp. 822-828

124 Simulated Annealing – Advances, Applications and Hybridizations

Chen, Y. L. & Ke, Y. L. (2004). Multi-objective Var planning for large-scale power systems using projection-based two-layer simulated annealing algorithms. *IEE Proceedings-*

Chiang, H. D. & Jean-Jumeau, R. (1990b). Optimal network reconfigurations in distribution systems: part II: solution algorithms and numerical results. *IEEE Transactions on Power* 

Chiang, H. D.; Wang, J. C.; Tong, J. & Darling, G. (1995a). Optimal capacitor placement, replacement and control in large-scale unbalanced distribution systems: modeling and a new formulation. *IEEE Transactions on Power Systems*, Vol. 10, No. 1, (February 1995),

Chiang, H. D.; Wang, J. C.; Tong, J. & Darling, G. (1995b). Optimal capacitor placement, replacement and control in large-scale unbalanced distribution systems: system solution algorithms and numerical studies. *IEEE Transactions on Power Systems*, Vol. 10, No. 1,

Christober Asir Rajan, C. & Mohan, M. R. (2007). An evolutionary programming based simulated annealing method for solving the unit commitment problem. *International Journal of Electrical Power & Energy Systems*, Vol. 29, No. 7, (September 2007), pp. 540-550 Christober Asir Rajan, C. (2010). A solution to the economic dispatch using EP based SA algorithm on large scale power system. *International Journal of Electrical Power & Energy* 

Christober Asir Rajan, C. (2011). Hydro-thermal unit commitment problem using simulated annealing embedded evolutionary programming approach. *International Journal of* 

Chu, R. F.; Wang, J. C. & Chiang H. D. (1994). Strategic planning of LC compensators in nonsinusoidal distribution systems. *IEEE Transactions on Power Delivery*, Vol. 9, No. 3,

Dahal, K. P. & Chakpitak, N. (2007). Generator maintenance scheduling in power systems using metaheuristic-based hybrid approaches. *Electric Power Systems Research*, Vol. 77,

Das, D. B. & Patvardhan, C.; (1998). New multi-objective stochastic search technique for economic load dispatch. *IEE Proceedings-Generation, Transmission and Distribution*, Vol.

Ekren, O. & Ekren, B. Y. (2010). Size optimization of a PV/wind hybrid energy conversion system with battery storage using simulated annealing. *Applied Energy*, Vol. 87, No. 2,

Gallego, R. A.; Romero, R. & Monticelli, A. J. (2000). Tabu search algorithm for network synthesis. *IEEE Transactions on Power Systems*, Vol. 15, No. 2, (May 2000), pp. 490-495 Gallego, R. A.; Alves, A. B.; Monticelli, A. & Romero, R. (1997). Parallel simulated annealing applied to long term transmission network expansion planning. *IEEE Transactions on* 

*Electrical Power & Energy Systems*, Vol. 33, No. 4, (May 2011), pp. 939-946

*Generation, Transmission and Distribution*, Vol. 151, No. 4, (July 2004), pp. 555-560 Chiang, H. D. & Jean-Jumeau, R. (1990a). Optimal network reconfigurations in distribution systems: part I: a new formulation and a solution methodology. *IEEE Transactions on* 

*Power Delivery*, Vol. 5, No. 4, (October 1990), pp. 1902-1909

*Delivery*, Vol. 5, No. 3, (July 1990), pp. 1568-1574

*Systems*, Vol. 32, No. 6, (July 2010), pp. 583-591

pp. 356-362

(February 1995), pp. 363-369

(July 1994), pp. 1558-1563

No. 7, (May 2007), pp. 771-779

(February 2010), pp. 592-598

145, No. 6, (November 1998), pp. 747-752

*Power Systems*, Vol. 12, No. 1, (February 1997), pp. 181-188

	- Kim, H.; Hayashi, Y. & Nara, K. (1997). An algorithm for thermal unit maintenance scheduling through combined use of GA, SA and TS. *IEEE Transactions on Power Systems*, Vol. 12, No. 1, (February 1997), pp. 329-335

Applications of Simulated Annealing-Based Approaches to Electric Power Systems 127

Panigrahi, C. K.; Chattopadhyay, P. K.; Chakrabarti, R. N. & Basu, M. (2006). Simulated annealing technique for dynamic economic dispatch. *Electric Power Components and* 

Purushothama, G. K.; Jenkins, L. (2003). Simulated annealing with local search-a hybrid algorithm for unit commitment. *IEEE Transactions on Power Systems*, Vol. 18, No. 1,

Purushothama, G. K.; Narendranath, U. A.; Jenkins, L. (2003). Unit commitment using a stochastic extended neighbourhood search. *IEE Proceedings-Generation, Transmission and* 

Roa-Sepulveda, C. A. & Pavez-Lazo, B. J. (2003). A solution to the optimal power flow using simulated annealing. *International Journal of Electrical Power & Energy Systems*, Vol. 25,

Romero, R.; Gallego, R. A.; Monticelli, A. (1996). Transmission system expansion planning by simulated annealing. *IEEE Transactions on Power Systems*, Vol. 11, No. 1, (February

Saber, A. Y.; Senjyu, T.; Miyagi, T.; Urasaki, N. & Funabashi, T. (2007). Unit commitment by heuristics and absolutely stochastic simulated annealing. *IET Generation, Transmission &* 

Saber, A. Y.; Senjyu, T.; Miyagi, T.; Urasaki, N. & Funabashi, T. (2006). Fuzzy unit commitment scheduling using absolutely stochastic simulated annealing. *IEEE* 

Satoh, T. & Nara, K. (1991). Maintenance scheduling by using simulated annealing method.

Simopoulos, D. N.; Kavatza, S. D. & Vournas, C. D. (2006b). Reliability constrained unit commitment using simulated annealing. *IEEE Transactions on Power Systems*, Vol. 21,

Simopoulos, D. N.; Kavatza, S. D. & Vournas, C. D. (2006a). Unit commitment by an enhanced simulated annealing algorithm. *IEEE Transactions on Power Systems*, Vol. 21,

Soliman, S. A.; Mantaway, A. H. & El-Hawary, M. E. (2004). Simulated annealing optimization algorithm for power systems quality analysis. International Journal of

Wang, Y.; Li, F.; Wan, Q. & Chen, H. (2011). Reactive power planning based on fuzzy clustering, gray code, and simulated annealing. *IEEE Transactions on Power Systems*, Vol.

Wong, K. P. & Fung, C. C. (1993). Simulated annealing based economic dispatch algorithm. *IEE Proceedings-Generation, Transmission and Distribution*, Vol. 140, No. 6, (November

Wong, K. P. & Wong, Y. W. (1994a). Short-term hydrothermal scheduling. Part I: simulated annealing approach. *IEE Proceedings-Generation, Transmission and Distribution*, Vol. 141,

Electrical Power & Energy Systems, Vol. 26, No. 1, (January 2004), pp. 31-36

*Transactions on Power Systems*, Vol. 21, No. 2, (May 2006), pp. 955-964

*IEEE Transactions on Power Systems*, Vol. 6, No. 2, (May 1991), pp. 850-857

*Systems*, Vol. 34, No. 5, (May 2006), pp. 577-586

*Distribution*, Vol. 150, No. 1, (January 2003), pp. 67-72

*Distribution*, Vol. 1, No. 2, (March 2007), pp. 234-243

No. 4, (November 2006), pp. 1699-1706

26, No. 4, (November 2011), pp. 2246-2255

No. 5, (September 1994), pp. 497-501

No. 1, (February 2006), pp. 68-76

1993), pp. 509-515

(February 2003), pp. 273-278

No. 1, (January 2003), pp. 47-57

1996), pp. 364-369


Panigrahi, C. K.; Chattopadhyay, P. K.; Chakrabarti, R. N. & Basu, M. (2006). Simulated annealing technique for dynamic economic dispatch. *Electric Power Components and Systems*, Vol. 34, No. 5, (May 2006), pp. 577-586

126 Simulated Annealing – Advances, Applications and Hybridizations

*Systems*, Vol. 12, No. 1, (February 1997), pp. 329-335

*Distribution*, Vol. 147, No. 3, (May 2000), pp. 164-170

*Systems*, Vol. 12, No. 1, (February 1997), pp. 437-443

(November 2002), pp. 765-769

1998), pp. 197-204

2008), pp. 790-795

No. 6, (August 2000), pp. 461-477

4, (October. 2005), pp. 2381-2388

No. 17, (October 2005), pp. 2669-2688

*Distribution*, Vol. 151, No. 2, (March 2004), pp. 157-166

Kim, H.; Hayashi, Y. & Nara, K. (1997). An algorithm for thermal unit maintenance scheduling through combined use of GA, SA and TS. *IEEE Transactions on Power* 

Li, Z. & Shahidehpour, M. (2003). Generation scheduling with thermal stress constraints. *IEEE Transactions on Power Systems*, Vol. 18, No. 4, (November 2003), pp. 1402-1409 Liang, R. H. & Kang, F. C. (2000). Thermal generating unit commitment using an extended mean field annealing neural network. *IEE Proceedings-Generation, Transmission and* 

Liao, G. C. & Tsao, T. P. (2006). Application of a fuzzy neural network combined with a chaos genetic algorithm and simulated annealing to short-term load forecasting. *IEEE* 

Liu, Y.; Ma, L. & Zhang, J. (2002). Reactive power optimization by GA/SA/TS combined algorithms. *International Journal of Electrical Power & Energy Systems*, Vol. 24, No. 9,

Mantawy, A. H.; Abdel-Magid, Y. L. & Selim, S. Z. (1998). A simulated annealing algorithm for unit commitment. *IEEE Transactions on Power Systems*, Vol. 13, No. 1, (February

Mantawy, A. H.; Abdel-Magid, Y. L. & Selim, S. Z. (1999). Integrating genetic algorithms, tabu search, and simulated annealing for the unit commitment problem. *IEEE* 

Mori, H. & Kobayashi, H. (1996). Optimal fuzzy inference for short-term load forecasting. *IEEE Transactions on Power Systems*, Vol. 11, No. 1, (February 1996), pp. 390-396 Mori, H. & Takeda, K. (1994). Parallel simulated annealing for power system decomposition.

Nahman, J. M. & Peric, D. M. (2008). Optimal planning of radial distribution networks by simulated annealing technique. *IEEE Transactions on Power Systems*, Vol. 23, No. 2, (May

Nayak, R. & Sharma, J. D. (2000). A hybrid neural network and simulated annealing approach to the unit commitment problem. *Computers & Electrical Engineering*, Vol. 26,

Nuqui, R. F. & Phadke, A. G. (2005). Phasor measurement unit placement techniques for complete and incomplete observability. *IEEE Transactions on Power Delivery*, Vol. 20, No.

Ongsakul, W.; Dechanupaprittha, S. & Ngamroo, I. (2004). Parallel tabu search algorithm for constrained economic dispatch. *IEE Proceedings-Generation, Transmission and* 

Pai, P. F. & Hong, W. C. (2005). Support vector machines with simulated annealing algorithms in electricity load forecasting. *Energy Conversion and Management*, Vol. 46,

*Transactions on Power Systems*, Vol. 14, No. 3, (August 1999), pp.829-836

*IEEE Transactions on Power Systems*, Vol. 9, No. 2, (May 1994), pp. 789-795

*Transactions on Evolutionary Computation*, Vol. 10, No. 3, (June 2006), pp. 330-340 Liu, C. W.; Jwo, W. S.; Liu, C. C. & Hsiao, Y. T. (1997). A fast global optimization approach to Var planning for the large scale electric power systems. *IEEE Transactions on Power* 

	- Wong, K. P. & Wong, Y. W. (1994b). Short-term hydrothermal scheduling. Part II: parallel simulated annealing approach. *IEE Proceedings-Generation, Transmission and Distribution*, Vol. 141, No. 5, (September 1994), pp. 502-506

**Chapter 7** 

© 2012 Kokubugata et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Improvements in Simulated Quenching** 

**Method for Vehicle Routing Problem** 

**Search History and Devising Means** 

**for Reducing the Number of Vehicles** 

Hisafumi Kokubugata, Yuji Shimazaki, Shuichi Matsumoto,

In many countries, rationalization of freight transportation is recognized to be an important problem. For example in Japan surrounded by sea, about 60% of freight transportation is carried out by road traffic. The proportion of freight road transportation to total road transportation is close to half. Although development of information technology accelerates electronic communication, physical distribution of goods is left behind. On the contrary, because electronic commerce has enhanced door-to-door delivery services, delivery distribution of goods has increased in urban areas. The demands for high-quality delivery services such as small-amount high frequency deliveries with time windows have been

From the aspect of freight carrier, decease of fuel consumption makes big profit, since the proportion of fuel to total cost is large. The rationalization in terms of increasing the loading rate and decreasing the total travel time is aimed not only for reducing operational costs in each freight carrier but also for relieving traffic congestion, saving energy and reducing exhaust gas. Effective distribution of goods should be realized by sophisticated delivery

A typical delivery problem is modelled mathematically in Vehicle Routing Problem (VRP). In VRP, scattered clients are serviced only once by exactly one of plural vehicles with load

and reproduction in any medium, provided the original work is properly cited.

**with Time Windows by Using** 

Hironao Kawashima and Tatsuru Daimon

Additional information is available at the end of the chapter

made by many clients (including companies and individuals).

http://dx.doi.org/10.5772/47854

**1. Introduction** 

planning.


**Improvements in Simulated Quenching Method for Vehicle Routing Problem with Time Windows by Using Search History and Devising Means for Reducing the Number of Vehicles** 

Hisafumi Kokubugata, Yuji Shimazaki, Shuichi Matsumoto, Hironao Kawashima and Tatsuru Daimon

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/47854

## **1. Introduction**

128 Simulated Annealing – Advances, Applications and Hybridizations

Vol. 141, No. 5, (September 1994), pp. 502-506

*Distribution*, Vol. 142, No. 4, (July 1995), pp. 372-380

*Systems*, Vol. 12, No. 2, (May 1997), pp. 776-784

359-368

1728

No. 5, (September 1994), pp. 507-513

Wong, K. P. & Wong, Y. W. (1994b). Short-term hydrothermal scheduling. Part II: parallel simulated annealing approach. *IEE Proceedings-Generation, Transmission and Distribution*,

Wong, K. P. & Wong, Y. W. (1994c). Genetic and genetic/simulated-annealing approaches to economic dispatch. *IEE Proceedings-Generation, Transmission and Distribution*, Vol. 141,

Wong, K. P. & Wong, Y. W. (1995). Thermal generator scheduling using hybrid genetic/simulated-annealing approach. *IEE Proceedings-Generation, Transmission and* 

Wong, K. P. & Wong, Y. W. (1996). Combined genetic algorithm/simulated annealing/fuzzy set approach to short-term generation scheduling with take-or-pay fuel contract. *IEEE* 

Wong, K. P. & Wong, Y. W. (1997). Hybrid genetic/simulated annealing approach to shortterm multiple-fuel-constrained generation scheduling. *IEEE Transactions on Power* 

Wong, Y. W. (1998). An enhanced simulated annealing approach to unit commitment. *International Journal of Electrical Power & Energy Systems*, Vol. 20, No. 5, (June 1998), pp.

Zhu, J. & Chow, M. Y. (1997). A review of emerging techniques on generation expansion planning. *IEEE Transactions on Power Systems*, Vol. 12, No. 4, (November 1997), pp. 1722-

Zhu, J.; Bilbro, G. & Chow, M. Y. (1999). Phase balancing using simulated annealing. *IEEE Transactions on Power Systems*, Vol. 14, No. 4, (November 1999), pp. 1508-1513 Zhuang, F.; Galiana, F. D. (1990). Unit commitment by simulated annealing. *IEEE* 

*Transactions on Power Systems*, Vol. 5, No. 1, (February 1990), pp. 311-318

*Transactions on Power Systems*, Vol. 11, No. 1, (February 1996), pp. 128-136

In many countries, rationalization of freight transportation is recognized to be an important problem. For example in Japan surrounded by sea, about 60% of freight transportation is carried out by road traffic. The proportion of freight road transportation to total road transportation is close to half. Although development of information technology accelerates electronic communication, physical distribution of goods is left behind. On the contrary, because electronic commerce has enhanced door-to-door delivery services, delivery distribution of goods has increased in urban areas. The demands for high-quality delivery services such as small-amount high frequency deliveries with time windows have been made by many clients (including companies and individuals).

From the aspect of freight carrier, decease of fuel consumption makes big profit, since the proportion of fuel to total cost is large. The rationalization in terms of increasing the loading rate and decreasing the total travel time is aimed not only for reducing operational costs in each freight carrier but also for relieving traffic congestion, saving energy and reducing exhaust gas. Effective distribution of goods should be realized by sophisticated delivery planning.

A typical delivery problem is modelled mathematically in Vehicle Routing Problem (VRP). In VRP, scattered clients are serviced only once by exactly one of plural vehicles with load

© 2012 Kokubugata et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

capacity which depart from a depot and return to it after touring the assigned clients. As mentioned above, clients often impose the earliest delivery time and the latest delivery time. A variation of VRP in which delivery time windows are included is called Vehicle Routing Problem with Time Windows (VRPTW). VRPTW is also applied to pick up operations such as cargo collection and garbage collection.

Improvements in Simulated Quenching Method for Vehicle Routing Problem with Time Windows by Using Search History and Devising Means for Reducing the Number of Vehicles 131

*Node with Demand* 

*Depot* 

**Figure 1.** Vehicle Routing Problem (VRP).

*A* 

*B* 

*Vehicle* 1

constraint.

including Simulated Annealing.

They are further classified into two types.

**2.2. Vehicle Routing Problem with Time Windows (VRPTW))** 

*Vehicle* 2

**0**

*C D* 

*G* 

**3. Precedent studies on heuristics for VRPTW** 

**3.1. Traditional heuristics approaches for VRPTW** 

In actual delivery operations, delivery time windows are often imposed by clients. Time window at node *i* is described as [*ei*, *li*], where *ei* is the earliest service starting time, *li* is the latest service starting time at node *i*. Vehicle routing problem taking account of time windows is called Vehicle Routing Problem with Time Windows (VRPTW). The first characteristic of VRPTW is the strictness of restriction on solutions. It often imposes increasing the number of operating vehicles. The second characteristic is the existence of waiting time. If the vehicle arrives before *ei*, it must wait until *ei* and then starts unloading service. Because VRP belongs to *NP*-hard problems, VRPTW belongs to them, too. Moreover, time windows make sequential delivery order restrictive. Hence, although both of VRP and VRPTW belong to same *NP*-hard problems in computational complexity theory, from a point of view with making practical algorithms, VRPTW is more difficult than VRP because of its tight

*E*

*F*

*Vehicle* 3

Because VRPTW belongs to *NP*-hard problems, exact methods are not fit for large problems. Therefore, heuristics have been important in the application of the VRPTW. Before the proposed method is explained, precedent studies on heuristics for VRPTW are introduced briefly. The heuristics for solving routing problems are classified into two major classes. The one is the family of traditional heuristics and the other is the family of metaheuristics

Comprehensive survey on traditional heuristics for VRPTW is presented in [3] by Bräysy & Gendreau. In this section, an outline of it is sketched. The traditional heuristics have been specially invented for solving VRPTW. They utilize the proper characteristics of VRPTW.

At the beginning of this chapter, VRP and VRPTW are introduced and followed by the explanation of precedent solution methods for VRPTW. And then, a practical solution method is proposed. It is composed by a data model, transformation rules of a solution on the data model and an overall search algorithm based on the refined Simulated Quenching (SQ) for VRPTW. The refined SQ procedures are derived from incorporating information of good solutions found in search history into basic SQ scheme. In the last section, the evaluation of the proposed method is conducted by comparisons on computational experiments with basic SQ.

## **2. Vehicle Routing Problem with Time Windows**

Typical routing problems are abstracted from actual logistics operations in urban areas and formalized as mathematical programming problems. They are categorized as the combinatorial optimization problems.

## **2.1. Vehicle Routing Problem (VRP)**

The Vehicle Routing Problem (VRP) is the most popular problem in routing problems. It involves the design of a set of minimum cost vehicle trips, originating and ending at a depot, for a fleet of vehicles with loading capacity that services a set of client spots with required demands. The problems studied in this chapter can be described in the style used by Crescenzi & Kann in [1] for their compendium of *NP* optimization problems. Although VRP is not listed in the compendium, it is given by Prins & Bouchenoua in [2] as follows.


Although the VRP in a narrow sense is defined above, the VRP in a broader sense includes the more comprehensive class of routing problems related to various conditions in which demands are located on nodes. It includes VRP with time windows imposed by clients, VRP with multiple depots, periodic VRP and etc. In this case, the simplest VRP defined above is called capacitated VRP (CVRP).

**Figure 1.** Vehicle Routing Problem (VRP).

**2. Vehicle Routing Problem with Time Windows** 

total demand processed by any trip cannot exceed *W*.

as cargo collection and garbage collection.

experiments with basic SQ.

natural numbers.

its traversed edges.

called capacitated VRP (CVRP).

combinatorial optimization problems.

**2.1. Vehicle Routing Problem (VRP)** 

capacity which depart from a depot and return to it after touring the assigned clients. As mentioned above, clients often impose the earliest delivery time and the latest delivery time. A variation of VRP in which delivery time windows are included is called Vehicle Routing Problem with Time Windows (VRPTW). VRPTW is also applied to pick up operations such

At the beginning of this chapter, VRP and VRPTW are introduced and followed by the explanation of precedent solution methods for VRPTW. And then, a practical solution method is proposed. It is composed by a data model, transformation rules of a solution on the data model and an overall search algorithm based on the refined Simulated Quenching (SQ) for VRPTW. The refined SQ procedures are derived from incorporating information of good solutions found in search history into basic SQ scheme. In the last section, the evaluation of the proposed method is conducted by comparisons on computational

Typical routing problems are abstracted from actual logistics operations in urban areas and formalized as mathematical programming problems. They are categorized as the

The Vehicle Routing Problem (VRP) is the most popular problem in routing problems. It involves the design of a set of minimum cost vehicle trips, originating and ending at a depot, for a fleet of vehicles with loading capacity that services a set of client spots with required demands. The problems studied in this chapter can be described in the style used by Crescenzi & Kann in [1] for their compendium of *NP* optimization problems. Although VRP is not listed in the compendium, it is given by Prins & Bouchenoua in [2] as follows.

 INSTANCE: Complete undirected graph *G* = (*V,E*), initial node *s V*, vehicle capacity *W N*, length *c*(*e*) *N* for each *e E*, demand *q*(*i*)  *N* for each *i V* , where *N* is the set of

 SOLUTION: A set of cycles (trips), each containing the initial node 0, that collectively traverses every node at least once. A node must be serviced by one single trip and the

MEASURE: The total cost of the trips, to be minimized. The cost of a trip is the sum of

Although the VRP in a narrow sense is defined above, the VRP in a broader sense includes the more comprehensive class of routing problems related to various conditions in which demands are located on nodes. It includes VRP with time windows imposed by clients, VRP with multiple depots, periodic VRP and etc. In this case, the simplest VRP defined above is

## **2.2. Vehicle Routing Problem with Time Windows (VRPTW))**

In actual delivery operations, delivery time windows are often imposed by clients. Time window at node *i* is described as [*ei*, *li*], where *ei* is the earliest service starting time, *li* is the latest service starting time at node *i*. Vehicle routing problem taking account of time windows is called Vehicle Routing Problem with Time Windows (VRPTW). The first characteristic of VRPTW is the strictness of restriction on solutions. It often imposes increasing the number of operating vehicles. The second characteristic is the existence of waiting time. If the vehicle arrives before *ei*, it must wait until *ei* and then starts unloading service. Because VRP belongs to *NP*-hard problems, VRPTW belongs to them, too. Moreover, time windows make sequential delivery order restrictive. Hence, although both of VRP and VRPTW belong to same *NP*-hard problems in computational complexity theory, from a point of view with making practical algorithms, VRPTW is more difficult than VRP because of its tight constraint.

## **3. Precedent studies on heuristics for VRPTW**

Because VRPTW belongs to *NP*-hard problems, exact methods are not fit for large problems. Therefore, heuristics have been important in the application of the VRPTW. Before the proposed method is explained, precedent studies on heuristics for VRPTW are introduced briefly. The heuristics for solving routing problems are classified into two major classes. The one is the family of traditional heuristics and the other is the family of metaheuristics including Simulated Annealing.

## **3.1. Traditional heuristics approaches for VRPTW**

Comprehensive survey on traditional heuristics for VRPTW is presented in [3] by Bräysy & Gendreau. In this section, an outline of it is sketched. The traditional heuristics have been specially invented for solving VRPTW. They utilize the proper characteristics of VRPTW. They are further classified into two types.

The first one is the type of constructive heuristics that produce vehicle routes by merging existing routes or inserting nodes into existing routes. Ioannou et al. proposed an efficient constructive heuristic in [4]. They use the generic sequential insertion framework proposed in [5] by Solomon.

Improvements in Simulated Quenching Method for Vehicle Routing Problem with Time Windows by Using Search History and Devising Means for Reducing the Number of Vehicles 133

*Node with Demand* 

*Depot*

procedures incorporated into them are rather complicated. In practical application of VRPTW algorithms to real-world problems, ease of implementation and flexibility are very important as well as quality of solution, running time and robustness. Hence, the authors of this chapter have proposed a simpler data model and a one-phase algorithm to solve VRPTW in [16]

The model to express a state of solution of VRPTW is realized as a sequence of integers, i.e., a string. In the string, the position of an integer, which is a symbol of the node with demand, implies not only which vehicle tours the node but also the routing order of it. An example of the string model is illustrated in Figure 2. The special number '0' should be interpreted not only as the depot but also as the delimiter which partitions the trips. If the number of vehicles is denoted by *m*, (*m*−1) '0's are provided in the string. If there is no integer between

This data model is coincidentally similar to that invented for the solution based on a kind of GA. It was introduced by Gendreau et al. in [17] as the original idea was given by Van Breedam in [18]. However, the proposed transformation rules in this chapter based on the data model are quite different from those of precedent methods. They will be described in

In a repetition in the proposed procedure, a new state of solution is generated from the present state by one of the following three types of transformation rules for generating neighbours. The first rule is to exchange an integer with another one in the string. The

*Vehicle* 2

*E*

which is not the two-phase algorithm composed by construction and improvement.

**4.1. Data modelling for VRPTW** 

'0' and '0', the relevant vehicle is not in use.

**Figure 2.** Proposed data model for VRPTW.

**4.2. Transformation rules for generating neighbours** 

*Vehicle* 1 *Vehicle* 2 *Vehicle* 3

**0**

*G A <sup>F</sup>*

*D* 

*Vehicle* 3

*C* 

*A B C* **0** *D E* **0** *F G*

the following section.

*Vehicle* 1

*B* 

The second one is the type of improvement heuristics which make changes in one vehicle route or between several vehicle routes. Bräysy proposed several efficient local search heuristics in [6] using a three-phase approach. In the first phase, several initial solutions are created using the route construction heuristics with different combinations of parameter values. In the second phase, an effort is put to reduce the number of routes. In the third phase, classical Or-opt exchanges, which replace three edges in the original tour by three new ones without modifying the orientation of the route, are used to minimize total travelled distance.

## **3.2. Metaheuristics for VRPTW**

Metaheuristics have been introduced into the solutions for VRPTW in the last two decades. Because metaheuristics are generally recognized to fit combinatorial optimizations, Simulated Annealing (SA), Tabu Search (TS), Genetic Algorithm (GA) and Ant Colony Optimization (ACO) have been tried to apply to VRPTW. Traditional heuristics explained in Sec. 3.1 are often embedded into these metaheuristics.

Cordeau et al. presented an efficient TS heuristic in [7]. Among the methods incorporating GA, the methods proposed by Homberger & Gehring in [8] and Berger et al. in [9] are reported to get good results. With respect to ACO, although not so many works on VRPTW are appeared in the literature, Gambardella et al. use an ACO approach with a hierarchy of two cooperative artificial ant colonies in [10]. Chiang & Russell developed a SA approach for VRPTW in [11]. They combined the SA process with the parallel construction approach that incorporates improvement procedures during the construction process.

In a comprehensive survey on metaheuristics for VRPTW given by Bräysy & Gendreau in [12], it is described that some hybrid methods are very effective and competitive with two good GA algorithms listed above. They are briefly introduced as follows. Bent & Van Hentenryck present a two-stage hybrid metaheuristic in [13], where in the first stage is a basic SA used to minimize the number of routes, and the second stage focuses on distance minimization using the large neighbourhood search. Bräysy presents a four-phase deterministic metaheuristic algorithm in [14] which is based on a modification of the variable neighbourhood search. Ibaraki et al. propose three metaheuristics in [15] to improve randomly generated initial solutions.

## **4. Data model and method of generating neighbours in searching process of simulated quenching for VRPTW**

Although some precedent methods based on metaheuristics mentioned above show good performance, their procedures are considerably complex. In particular, the local search procedures incorporated into them are rather complicated. In practical application of VRPTW algorithms to real-world problems, ease of implementation and flexibility are very important as well as quality of solution, running time and robustness. Hence, the authors of this chapter have proposed a simpler data model and a one-phase algorithm to solve VRPTW in [16] which is not the two-phase algorithm composed by construction and improvement.

## **4.1. Data modelling for VRPTW**

132 Simulated Annealing – Advances, Applications and Hybridizations

in [5] by Solomon.

travelled distance.

**3.2. Metaheuristics for VRPTW** 

Sec. 3.1 are often embedded into these metaheuristics.

improve randomly generated initial solutions.

**of simulated quenching for VRPTW** 

incorporates improvement procedures during the construction process.

The first one is the type of constructive heuristics that produce vehicle routes by merging existing routes or inserting nodes into existing routes. Ioannou et al. proposed an efficient constructive heuristic in [4]. They use the generic sequential insertion framework proposed

The second one is the type of improvement heuristics which make changes in one vehicle route or between several vehicle routes. Bräysy proposed several efficient local search heuristics in [6] using a three-phase approach. In the first phase, several initial solutions are created using the route construction heuristics with different combinations of parameter values. In the second phase, an effort is put to reduce the number of routes. In the third phase, classical Or-opt exchanges, which replace three edges in the original tour by three new ones without modifying the orientation of the route, are used to minimize total

Metaheuristics have been introduced into the solutions for VRPTW in the last two decades. Because metaheuristics are generally recognized to fit combinatorial optimizations, Simulated Annealing (SA), Tabu Search (TS), Genetic Algorithm (GA) and Ant Colony Optimization (ACO) have been tried to apply to VRPTW. Traditional heuristics explained in

Cordeau et al. presented an efficient TS heuristic in [7]. Among the methods incorporating GA, the methods proposed by Homberger & Gehring in [8] and Berger et al. in [9] are reported to get good results. With respect to ACO, although not so many works on VRPTW are appeared in the literature, Gambardella et al. use an ACO approach with a hierarchy of two cooperative artificial ant colonies in [10]. Chiang & Russell developed a SA approach for VRPTW in [11]. They combined the SA process with the parallel construction approach that

In a comprehensive survey on metaheuristics for VRPTW given by Bräysy & Gendreau in [12], it is described that some hybrid methods are very effective and competitive with two good GA algorithms listed above. They are briefly introduced as follows. Bent & Van Hentenryck present a two-stage hybrid metaheuristic in [13], where in the first stage is a basic SA used to minimize the number of routes, and the second stage focuses on distance minimization using the large neighbourhood search. Bräysy presents a four-phase deterministic metaheuristic algorithm in [14] which is based on a modification of the variable neighbourhood search. Ibaraki et al. propose three metaheuristics in [15] to

**4. Data model and method of generating neighbours in searching process** 

Although some precedent methods based on metaheuristics mentioned above show good performance, their procedures are considerably complex. In particular, the local search The model to express a state of solution of VRPTW is realized as a sequence of integers, i.e., a string. In the string, the position of an integer, which is a symbol of the node with demand, implies not only which vehicle tours the node but also the routing order of it. An example of the string model is illustrated in Figure 2. The special number '0' should be interpreted not only as the depot but also as the delimiter which partitions the trips. If the number of vehicles is denoted by *m*, (*m*−1) '0's are provided in the string. If there is no integer between '0' and '0', the relevant vehicle is not in use.

This data model is coincidentally similar to that invented for the solution based on a kind of GA. It was introduced by Gendreau et al. in [17] as the original idea was given by Van Breedam in [18]. However, the proposed transformation rules in this chapter based on the data model are quite different from those of precedent methods. They will be described in the following section.

**Figure 2.** Proposed data model for VRPTW.

## **4.2. Transformation rules for generating neighbours**

In a repetition in the proposed procedure, a new state of solution is generated from the present state by one of the following three types of transformation rules for generating neighbours. The first rule is to exchange an integer with another one in the string. The

second rule is to delete an arbitrary integer and then insert it to another position in the string. The third rule is that after a substring is taken out temporally, the direction of the substring is reversed, and then embedded in the place where the substring is taken out. These three transformation rules are illustrated in Figure 3.

Improvements in Simulated Quenching Method for Vehicle Routing Problem with Time Windows by Using Search History and Devising Means for Reducing the Number of Vehicles 135

> *Vehicle* <sup>1</sup> *Vehicle*<sup>3</sup> *<sup>F</sup> <sup>A</sup>*

> > *E*

*E* 

*Vehicle* 2

**0** *Vehicle* 2

*C D*

*<sup>F</sup> <sup>A</sup>*

**0**

*C D*

*One to One Exchange*

*A B C* **0** *D E* **0** *F A C B* **0** *D E* **0** *F* 

*Vehicle* 3

*E*

*Vehicle* 2

*Vehicle* 3

*E*

*Vehicle* 2

*B*

**Figure 4.** A result of 'one-to-one exchange' within a route.

*C D*

**0**

*C D*

*<sup>F</sup> <sup>A</sup>*

**0**

*<sup>F</sup> <sup>A</sup>*

*B* 

*Vehicle* 1

**4.3. Objective function** 

*B* 

*Vehicle* 1

**Figure 5.** A result of 'one-to-one exchange' between two non-zero integers striding over '0'.

The objective function of the VRPTW is formulated as follows.

The objective of the VRPTW is the minimization of total cost which is subject to constraints including the loading capacity of each vehicle and the time windows imposed by clients.

*One to One Exchange*

*A B C* **0** *D E* **0** *F A B F* **0** *D E* **0** *C*

1 0 ( ) *i ii n n*

*i i Es c d* 

where *s* = (*s*1, *s*2, · · · , *sn*) is a string that consists of the nodes with demands and a depot; *s*<sup>0</sup> and *sn*+1 are the implicit expressions of the depot omitted in the string *s*; *ck* is the servicing

*s ss*

<sup>1</sup> ,

*Vehicle* 3

*B*

*Vehicle* 1

(1)

Note that the rules are also applied to the special number '0' in the string illustrated in Figure 2. In other words, '0' is treated impartially with other integers. If 'one-to-one exchange' is executed within a substring partitioned by two '0's, only a route is changed. An example of the case is illustrated in Figure 4. If 'one-to-one exchange' is executed between two non-zero integers striding over '0', two nodes are exchanged between two routes. An example of this case is illustrated in Figure 5. If 'one-to-one exchange' is executed between a non-zero integer and '0', two routes are merged, while another route is divided into two routes. An example is illustrated in Figure 6.

When the second transformations rule 'delete and insert' is applied, several different cases also arise. If a non-zero integer is deleted and inserted at '0', a node is moved to another vehicle route. An example is illustrated in Figure 7.

When the third transformations rule 'partial reversal' is applied, several different cases also arise. If a substring including '0' is reversed, the relevant plural routes are changed. An example is illustrated in Figure 8. These three transformation rules were originally invented for VRP in [19] by the authors of this chapter.

**Figure 3.** Transformation rules for generating neighbours.

Improvements in Simulated Quenching Method for Vehicle Routing Problem with Time Windows by Using Search History and Devising Means for Reducing the Number of Vehicles 135

**Figure 4.** A result of 'one-to-one exchange' within a route.

**Figure 5.** A result of 'one-to-one exchange' between two non-zero integers striding over '0'.

#### **4.3. Objective function**

134 Simulated Annealing – Advances, Applications and Hybridizations

routes. An example is illustrated in Figure 6.

for VRP in [19] by the authors of this chapter.

*A*

**2. Delete & Insert** 

vehicle route. An example is illustrated in Figure 7.

*B*

**Figure 3.** Transformation rules for generating neighbours.

*A*

*B*

**0**

*A B C* **0** *<sup>A</sup> <sup>C</sup> <sup>B</sup>* **0**

**Present State 1. One to One** 

**0**

*C A B* **0** *C B A* **0**

*C*

*C*

These three transformation rules are illustrated in Figure 3.

second rule is to delete an arbitrary integer and then insert it to another position in the string. The third rule is that after a substring is taken out temporally, the direction of the substring is reversed, and then embedded in the place where the substring is taken out.

Note that the rules are also applied to the special number '0' in the string illustrated in Figure 2. In other words, '0' is treated impartially with other integers. If 'one-to-one exchange' is executed within a substring partitioned by two '0's, only a route is changed. An example of the case is illustrated in Figure 4. If 'one-to-one exchange' is executed between two non-zero integers striding over '0', two nodes are exchanged between two routes. An example of this case is illustrated in Figure 5. If 'one-to-one exchange' is executed between a non-zero integer and '0', two routes are merged, while another route is divided into two

When the second transformations rule 'delete and insert' is applied, several different cases also arise. If a non-zero integer is deleted and inserted at '0', a node is moved to another

When the third transformations rule 'partial reversal' is applied, several different cases also arise. If a substring including '0' is reversed, the relevant plural routes are changed. An example is illustrated in Figure 8. These three transformation rules were originally invented

**0** 

**0** 

*C*

*C*

*A*

**Exchange** 

**3. Partial Reversal** 

*A*

*B*

*B*

The objective of the VRPTW is the minimization of total cost which is subject to constraints including the loading capacity of each vehicle and the time windows imposed by clients. The objective function of the VRPTW is formulated as follows.

$$E(s) = \sum\_{i=1}^{n} c\_{s\_i} + \sum\_{i=0}^{n} d\_{s\_i, s\_{i+1}} \tag{1}$$

where *s* = (*s*1, *s*2, · · · , *sn*) is a string that consists of the nodes with demands and a depot; *s*<sup>0</sup> and *sn*+1 are the implicit expressions of the depot omitted in the string *s*; *ck* is the servicing

Improvements in Simulated Quenching Method for Vehicle Routing Problem with Time Windows by Using Search History and Devising Means for Reducing the Number of Vehicles 137

1

*C D*

*<sup>F</sup> <sup>A</sup>*

 

*E* 

**0** *Vehicle* **2** 

*k*

*l* is the latest service starting time at node *si*; *m* is the

*T t kT t k* ( 1) : ( ) (0 1) (3)

*ws* is the amount of demand at node *si*; *zk* is the position of *k*th '0'

*wW m*

(2)

*Vehicle* **1**

*Vehicle* **3**

*B*

**Figure 8.** A result of 'partial reversal' striding over '0'.

*C D*

*Es c d*

number of vehicles in use; *<sup>i</sup>*

SA.

where *is <sup>a</sup>* is arriving time at node *si*; *is*

**0**

*B* 

*Vehicle* **1** 

*<sup>F</sup> <sup>A</sup>*

vehicle *k*. *α*, *β* and *γ* are weight parameters.

,

<sup>1</sup>

*a l*

 

*i i i k iz*

*nn n m z*

10 1 1 1

*i ii i i i*

*Partial Reversal*

*A B C* **0** *D E* **0** *F A F* **0** *E D* **0** *C B* 

in the string *s* = (*s*1, *s*2, · · · , *sn*) (provided that *z*0 = 0; *zm* = *n*+1) and *Wk* is the loading capacity of

Simulated Quenching is adopted as the optimization technique for the proposed method since it is characterized by simple stochastic procedures and by global searching scope. In the original Simulated Annealing (SA), starting with a random initial state, it is expected to approach an equilibrium point. In order to obtain global optimum, cooling schedule should be logarithmic. However, it spends too much time to implement it. Hence, in practical

In the proposed method, it is adopted too. According to the strict theory of Simulated Annealing, the optimization technique using exponential cooling schedule (3) belongs to Simulated Qeunching (SQ) as described in [20]. SQ is considerd to be a practical variant of

In the proposed method, the three transformation rules described in Sec. 4.2 are applied randomly to the string model. The entire algorithm for the VRP is described as follows.

*s ss s s s k*

1

( ) max 0, *<sup>k</sup>*

*Vehicle* **3**

*E*

*Vehicle* **2**

**4.4. Optimization algorithm using Simulated Quenching** 

applications, exponential cooling schedule (3) is often adopted.

**Figure 6.** A result of 'one-to-one exchange' between non-zero integer and '0'.

cost at the node *k* (if *k* = 0,then *ck* = 0); *dk*,*<sup>l</sup>* is the minimal traversing cost from the node *k* to the node *l*. Each value of *dk*,*<sup>l</sup>* might be given by input data; or calculated as the Euclidean distances between a pair of coordinates of nodes; or calculated by the shortest path search algorithm (Warshall-Floyd's algorithm) when road network is given and vehicles must follow the roads in the network.

In order to impose solutions of VRPTW to satisfy time window constraints and load capacities and to reduce the number of vehicles in use, three penalty terms are added to the objective function (1) as follows:

**Figure 7.** A result of deleting non-zero and inserting it at '0'.

**Figure 8.** A result of 'partial reversal' striding over '0'.

*<sup>F</sup> <sup>A</sup>*

**Figure 6.** A result of 'one-to-one exchange' between non-zero integer and '0'.

*Vehicle* **3**

*E*

*Vehicle* **2**

*Vehicle* **3**

*E*

*Vehicle* **2**

follow the roads in the network.

**0**

*C D*

*B* 

*Vehicle* **1** 

objective function (1) as follows:

**0**

*C D*

*B* 

*Vehicle* **1** 

*<sup>F</sup> <sup>A</sup>*

**Figure 7.** A result of deleting non-zero and inserting it at '0'.

cost at the node *k* (if *k* = 0,then *ck* = 0); *dk*,*<sup>l</sup>* is the minimal traversing cost from the node *k* to the node *l*. Each value of *dk*,*<sup>l</sup>* might be given by input data; or calculated as the Euclidean distances between a pair of coordinates of nodes; or calculated by the shortest path search algorithm (Warshall-Floyd's algorithm) when road network is given and vehicles must

*One to One Exchange* 

*A B C* **0** *D E* **0** *F A* **0** *C* **0** *D E B F* 

*B*

*Vehicle* **1**

**0**

*C D* 

*Vehicle* **2**

*Vehicle* **<sup>1</sup>** *Vehicle***<sup>3</sup>** *<sup>F</sup> <sup>A</sup>*

*E* 

**0** *Vehicle* **2** 

*C D*

*<sup>F</sup> <sup>A</sup>*

*Vehicle* **3** 

*E* 

In order to impose solutions of VRPTW to satisfy time window constraints and load capacities and to reduce the number of vehicles in use, three penalty terms are added to the

*Delete and Insert*

*A B C* **0** *D E* **0** *F A C* **0** *D E* **0** *B F* 

*B*

$$E'(\mathbf{s}) = \left(\sum\_{i=1}^{n} c\_{s\_i} + \sum\_{i=0}^{n} d\_{s\_i, s\_{i+1}}\right) + \alpha \left(\sum\_{i=1}^{n+1} \max\left(0, a\_{s\_i} - l\_{s\_i}\right)\right) + \beta \left(\sum\_{k=1}^{m} \left(\sum\_{i=z\_{k-1}+1}^{z\_k} w\_{s\_i} - W\_k\right)\right) + \gamma \, m \tag{2}$$

where *is <sup>a</sup>* is arriving time at node *si*; *is l* is the latest service starting time at node *si*; *m* is the number of vehicles in use; *<sup>i</sup> ws* is the amount of demand at node *si*; *zk* is the position of *k*th '0' in the string *s* = (*s*1, *s*2, · · · , *sn*) (provided that *z*0 = 0; *zm* = *n*+1) and *Wk* is the loading capacity of vehicle *k*. *α*, *β* and *γ* are weight parameters.

#### **4.4. Optimization algorithm using Simulated Quenching**

Simulated Quenching is adopted as the optimization technique for the proposed method since it is characterized by simple stochastic procedures and by global searching scope. In the original Simulated Annealing (SA), starting with a random initial state, it is expected to approach an equilibrium point. In order to obtain global optimum, cooling schedule should be logarithmic. However, it spends too much time to implement it. Hence, in practical applications, exponential cooling schedule (3) is often adopted.

$$T(t+1) \coloneqq kT(t) \tag{3}$$

In the proposed method, it is adopted too. According to the strict theory of Simulated Annealing, the optimization technique using exponential cooling schedule (3) belongs to Simulated Qeunching (SQ) as described in [20]. SQ is considerd to be a practical variant of SA.

In the proposed method, the three transformation rules described in Sec. 4.2 are applied randomly to the string model. The entire algorithm for the VRP is described as follows.

```
{I. Preparation}
```
 *Read input data*;

 *If the link cost are not given from the input data, calculate the minimum path cost di*,*<sup>j</sup> between all pair of clients i, j including the depot* 0;

{II. *Initialization*} *Generate an initial feasible solution s*0 *by assigning nodes to vehicles in ascending order of the specified earliest arriving time*; *s* := *s*0; *s*\* := *s*;  *T* := *INITTEMP; Set N as the averaged neighbourhood size*;

{III. *Optimization by SQ*} *Minimize E by repetition of applying randomly one of the three transformation rules to the string model corresponding to x in the framework of SQ*;

{IV. *Output*} *Output the best solution s*\**.*

Step III, that is the main part of this algorithm, is detailed as follows.

*Repeat* 

 *trials* := 0*; changes* := 0;  *Repeat* 

 *trials* := *trials +* 1;

 *Generate a new state s*´ *from the current state s by applying randomly one of the three transformation rules to the string model of s*;

 *Calculate* Δ*E' = E'* (*s*´) *– E'* (*s*);  *If* Δ*E'* < *0 Then s*´ *is accepted as a new state*;  *If If* (*E'* (s´) < *E'* (*s*\* ) *and s*´ *is feasible*) *Then s*\* := *s*´  *Else s*´ *is accepted with probability* exp(*−*Δ*E'* /*T* )  *If s*´ *is accepted Then changes* := *changes +* 1; *s* := *s*´  *Until trials* ≥ *SIZEFACTOR* · *N or changes* ≥ *CUTOFF* · *N*;  *T* := *T* · *TEMPFACTOR Until T* ≤ *INITTEMP / FINDIVISOR*

The words noted by capital letters are parameters used in SA and SQ and values of them are specified in Sec 6.2. As descibed in Sec. 4.2, the transformation procedure of a solution of the proposed method is carried out randomly to all over the string data model. Hence, the transformation might derive changes in a vehicle route on one occation, it might derive changes over several vehicle routes on other occation. This method is applied to VRP in [19], VRP with backhaouls (VRPB) in [21], Pick up and Delivery Problem (PDP) and VRPTW in [16] by the autors of this chaper. It is also applied to other types of routing problems including Capacitated Arc Routing Problem (CARP) in [22] and a general routing problem with nodes, edges, and arcs (NEARP) in [23]. A precise analysis of this method is presented in [24].

## **5. Improvement of optimization algorithm based on SQ by adaptation of devices inspired by ACO**

Most of metaheuristics belong to stochastic local search (SLS) which starts at some position in search space and iteratively moves to neighbour, using randomised choices in generating and modifying candidates. In application of metaheuristics, both intensification and

(5)

(4)

on SQ.

as follows.

*Initialise pheromone trails*;

probability:

reached.

*While termination criterion is not satisfied* 

VRP, details are specified as follows.

that are not contained in

the weights *vs.* the heuristic values.

 *Perform subsidiary local search on sp*;  *Update pheromone trails based on sp*

 *Generate population sp of candidate solutions using subsidiary randomised constructive search*

*dij* is used, where *di j* is traversing cost of edge (*i*, *j*). 2. In the beginning, all weights are initialised to small value *τ*0.

iteratively extends partial round trip

**5.1. Ant Colony Optimization** 

Improvements in Simulated Quenching Method for Vehicle Routing Problem with Time Windows by Using Search History and Devising Means for Reducing the Number of Vehicles 139

(6)

with

diversification are important. For sufficient convergence of solution, intensification of search scope is necessary. On the other hand, in order to avoid stagnation in local but not global minimum area, diversification of search scope is also necessary. As compared to 'stupid fly', search process in pure SA and SQ is completely random. Making use of history records during search processes might be possible to improvement of optimization algorithm based

ACO was introduced by Dorigo et al. in [25]. It was inspired by foraging behaviour of ants. Ants often communicate via chemicals known as pheromones, which are deposited on the ground in the form of trails. With time, paths leading to the more interesting food sources become more frequented and are marked with larger amounts of pheromone. Pheromone trails provide the basis for stochastic trail-following behaviour underlying, for example, the collective ability to find shortest paths between a food source and the nest. ACO is described

In applying ACO to TSP (Travelling Salesman Problem) which is single vehicle version of

1. Pheromone trail *τij* is associated with each edge (*i*, *j*) in *G*, while heuristic values *ηij* = 1 /

3. In constructive search, each artificial ant starts with randomly chosen node and

by selecting node not contained in

, (7)

. *a* and *b* are parameters which control the relative impact of

[ ][ ]

 

*lNi il il*

4. After the constructive search, subsidiary local search which is iterative improvement based on standard 2-exchange neighbourhood is operated until local minimum is

 

*a b ij ij*

[ ][ ]

where *N'*(*i*) is the feasible neighbourhood of node *i*, that is, the set of all neighbours of *i*

*a b*

  diversification are important. For sufficient convergence of solution, intensification of search scope is necessary. On the other hand, in order to avoid stagnation in local but not global minimum area, diversification of search scope is also necessary. As compared to 'stupid fly', search process in pure SA and SQ is completely random. Making use of history records during search processes might be possible to improvement of optimization algorithm based on SQ.

### **5.1. Ant Colony Optimization**

(4)

(5)

138 Simulated Annealing – Advances, Applications and Hybridizations

 *between all pair of clients i, j including the depot* 0;

*one of the three transformation rules to the string model of s*;

(*s*´) *– E'*

(*s*);

 *= E'*

 *Else s*´ *is accepted with probability* exp(*−*Δ*E'*

 *If s*´ *is accepted Then changes* := *changes +* 1; *s* := *s*´

 *Until trials* ≥ *SIZEFACTOR* · *N or changes* ≥ *CUTOFF* · *N*;

 *s*´ *is accepted as a new state*;

 *T* := *T* · *TEMPFACTOR* 

**devices inspired by ACO** 

(s´) < *E'* (*s*\*

 *Until T* ≤ *INITTEMP / FINDIVISOR*

 *If the link cost are not given from the input data, calculate the minimum path cost di*,*<sup>j</sup>*

{III. *Optimization by SQ*} *Minimize E by repetition of applying randomly one of the three transformation rules to the string model corresponding to x in the framework of SQ*;

 *Generate a new state s*´ *from the current state s by applying randomly*

/*T* )

The words noted by capital letters are parameters used in SA and SQ and values of them are specified in Sec 6.2. As descibed in Sec. 4.2, the transformation procedure of a solution of the proposed method is carried out randomly to all over the string data model. Hence, the transformation might derive changes in a vehicle route on one occation, it might derive changes over several vehicle routes on other occation. This method is applied to VRP in [19], VRP with backhaouls (VRPB) in [21], Pick up and Delivery Problem (PDP) and VRPTW in [16] by the autors of this chaper. It is also applied to other types of routing problems including Capacitated Arc Routing Problem (CARP) in [22] and a general routing problem with nodes,

edges, and arcs (NEARP) in [23]. A precise analysis of this method is presented in [24].

**5. Improvement of optimization algorithm based on SQ by adaptation of** 

Most of metaheuristics belong to stochastic local search (SLS) which starts at some position in search space and iteratively moves to neighbour, using randomised choices in generating and modifying candidates. In application of metaheuristics, both intensification and

) *and s*´ *is feasible*) *Then s*\* := *s*´

{II. *Initialization*} *Generate an initial feasible solution s*0 *by assigning nodes to vehicles in* 

 *ascending order of the specified earliest arriving time*; *s* := *s*0; *s*\* := *s*;  *T* := *INITTEMP; Set N as the averaged neighbourhood size*;

Step III, that is the main part of this algorithm, is detailed as follows.

{I. *Preparation*}

*Repeat* 

 *Repeat* 

 *Read input data*;

{IV. *Output*} *Output the best solution s*\**.*

 *trials* := 0*; changes* := 0;

 *trials* := *trials +* 1;

 *Calculate* Δ*E'*

 *If If* (*E'*

 *If* Δ*E'* < *0 Then* 

ACO was introduced by Dorigo et al. in [25]. It was inspired by foraging behaviour of ants. Ants often communicate via chemicals known as pheromones, which are deposited on the ground in the form of trails. With time, paths leading to the more interesting food sources become more frequented and are marked with larger amounts of pheromone. Pheromone trails provide the basis for stochastic trail-following behaviour underlying, for example, the collective ability to find shortest paths between a food source and the nest. ACO is described as follows.

*Initialise pheromone trails*; *While termination criterion is not satisfied Generate population sp of candidate solutions using subsidiary randomised constructive search Perform subsidiary local search on sp*;  *Update pheromone trails based on sp*

In applying ACO to TSP (Travelling Salesman Problem) which is single vehicle version of VRP, details are specified as follows.


$$\frac{[\boldsymbol{\pi}\_{ij}]^{a} \cdot [\boldsymbol{\eta}\_{ij}]^{b}}{\sum\_{l \in \mathcal{N}(i)} [\boldsymbol{\pi}\_{il}]^{a} \cdot [\boldsymbol{\eta}\_{il}]^{b}} \, ^{} \tag{7}$$

(6)

where *N'*(*i*) is the feasible neighbourhood of node *i*, that is, the set of all neighbours of *i* that are not contained in . *a* and *b* are parameters which control the relative impact of the weights *vs.* the heuristic values.

4. After the constructive search, subsidiary local search which is iterative improvement based on standard 2-exchange neighbourhood is operated until local minimum is reached.

5. In the end of loop, pheromone trail is updated according to

$$
\pi\_{i\cdot j}(t+1) \coloneqq (1-\rho)\cdot\pi\_{i\cdot j}(t) + \sum\_{k=1}^{m} \Delta\pi\_{i\cdot j\cdot\prime}^{k} \tag{8}
$$

Improvements in Simulated Quenching Method for Vehicle Routing Problem with Time Windows by Using Search History and Devising Means for Reducing the Number of Vehicles 141

and (*i*, *b*) which are connected with node *i* are frequently contained in the best routes and costs of the edges are small, the value of *ri* becomes small. Because such a situation of node *i* is agreeable to good solutions, it should not be drastically changed in the succeeding search processes. In order to embody the idea stated before, *ri* is used for assigning node *i* the biased small probability with which node *i* is selected for change, instead of obeying uniform distribution as

*i*

*r*

*<sup>r</sup>*

The core part of the basic SQ algorithm (5) is replaced by the revised algorithm which is

*i i*

described in Sec. 4.2. That is to say, node *i* is selected for transformation with probability:

*i*

 *Generate a new state s*´ *from the current state s by applying randomly one of the three transformation rules to the string model of s*, *where posiition i to be changed in the* 

) *and s*´ *is feasible*) *Then s*\* := *s*´

 

, : / *pr r i ii i <sup>i</sup> for all* ;

**5.3. A device for decreasing the number of vehicles in use** 

: (1 ) ; 1.4 : 1.4; 0.2 : 0.2 ( , ); *i j ij ij ij i j i j i j*

*if then elseif then f i j*

When performance of plural solutions for VRPTW is compared, the first measure is the number of vehicles in use, which is denoted by *m*, while the second is total cost *E*. Although *SQph* is expected to utilize characteristics of good solutions already found and to reduce *E*, it could not reduce *m* directly. In order to reduce it, another device should be included in SQ procedures.

 *or all* (13)

*p*

, where *i j d* is traversing cost of edge (*i*, *j*). When edges (*a*, *i*)

(12)

constant. Let / / *i a i ai ib ib rd d*

called *SQph* described as follows.

: 0.2 ( , ) *i j*

 *for all i j* ; : 1/ *p ii <sup>i</sup> <sup>i</sup> for all* ;

 *trials* := 0; *changes* := 0;

 *trials* := *trials +* 1;

 *If* Δ*E* < *0 Then* 

 *If* (*E*(s´) < *E*(*s*\*

 *T* := *T* · *TEMPFACTOR*;

 

:/ / *i a i ai ib ib rd d* 

*string is selected with the probability pi* ;

 *Else s*´ *is accepted with probability* exp(*−*Δ*E*/*T* )  *If s*´ *is accepted Then changes* := *changes +* 1; *s* := *s*´

> 

*Until T* ≤ *INITTEMP / FINDIVISOR*

 *Until trials* ≥ *SIZEFACTOR* · *N or changes* ≥ *CUTOFF* · *N*;

 

*Calculate* Δ*E = E*(*s*´) *− E*(*s*);

 *s*´ *is accepted as a new state*;

*Repeat* 

 *Repeat* 

 where 0 1 is a parameter regarding vaporization of pheromone,

$$
\Delta \pi\_{ij}^k = \begin{cases}
\mathbb{Q} \Big/ \mathbb{E} \{ \mathbf{s}^k \} & \text{if } \{ i, j \} \in \mathbf{s}^k \\
\mathbf{0} & \text{else}
\end{cases} \tag{9}
$$

*E*(*sk* ) is total cost of the *k* th ant's cycle *sk* , *m* is the number of ants (= size of *sp*) and *Q* is a constant. Criterion for weight increase is based on intuition that edges contained in short round trips should be preferably used in subsequent constructions.

As mentioned in Sec. 3.2, not so many methods based on ACO for VRPTW are appeared in the literature.

#### **5.2. Application of information obtained in search history to SQ**

One of main drawbacks of SA and SQ which are pointed out by users of other metaheuristics is lack of learning in search history, that is, blind random search often compared to 'stupid fly'. Although the complete ACO belongs population-based SLS methods in which genetic algorithm is also contained, a predecessor of ACO is Ant System which is a single ant version of ACO. It was also proposed by Dorigo et al. in [25] and it is recognized as a member of Adaptive Iterated Construction Search (AICS) methods. In the Ant System single artificial ant works and uses information obtained in its own preceding searches. Utilization of some information about good solutions obtained in the preceding search processes is able to be incorporated into SQ procedures. It would be possible to overcome the blind random searches in SQ. Because traversed arcs in good solutions of VRPTW are recognized as characteristics of them, such arcs are expected to be not drastically changed in the succeeding search processes.

In order to embody the idea described above, artificial pheromone trail *τij* is associated with each edge (*i*, *j*) and *τij* is updated at the end of the loop of temperature *T* in SQ procedures. The 'characteristic of good solution' is embodied in increase of probability of selecting better candidate in random search process in SQ. In end of loop, weight is updated according to

$$
\pi\_{i\bar{j}}(t+1) \coloneqq (1-\rho) \cdot \pi\_{i\bar{j}}(t) + \Delta \pi\_{i\bar{j}} \tag{10}
$$

, where 0 1 is a parameter regarding vaporization of pheromone,

$$
\Delta \pi\_{i\_j} = \begin{cases}
\mathbb{Q} \Big/ \mathbb{E} \{ \overset{\*}{\rm s}^\* \} & \text{if } \{ i, j \} \in \overset{\*}{\rm s}^\* \\
0 & \text{else}
\end{cases} \tag{11}
$$

However, in order to avoid extreme effect of pheromone, lower bound and upper bound of *i j* is set as 0.2 1.4. *i j* \* *E s*( ) is the total cost of the best found solution at the present *s*\* , *Q* is a constant. Let / / *i a i ai ib ib rd d* , where *i j d* is traversing cost of edge (*i*, *j*). When edges (*a*, *i*) and (*i*, *b*) which are connected with node *i* are frequently contained in the best routes and costs of the edges are small, the value of *ri* becomes small. Because such a situation of node *i* is agreeable to good solutions, it should not be drastically changed in the succeeding search processes. In order to embody the idea stated before, *ri* is used for assigning node *i* the biased small probability with which node *i* is selected for change, instead of obeying uniform distribution as described in Sec. 4.2. That is to say, node *i* is selected for transformation with probability:

140 Simulated Annealing – Advances, Applications and Hybridizations

) is total cost of the *k* th ant's cycle *sk*

where 0 1 

*E*(*sk*

the literature.

, where 0 1 

is set as 0.2 1.4. *i j* 

5. In the end of loop, pheromone trail is updated according to

*k i j*

round trips should be preferably used in subsequent constructions.

drastically changed in the succeeding search processes.

**5.2. Application of information obtained in search history to SQ** 

( 1) : (1 ) ( )

0

constant. Criterion for weight increase is based on intuition that edges contained in short

As mentioned in Sec. 3.2, not so many methods based on ACO for VRPTW are appeared in

One of main drawbacks of SA and SQ which are pointed out by users of other metaheuristics is lack of learning in search history, that is, blind random search often compared to 'stupid fly'. Although the complete ACO belongs population-based SLS methods in which genetic algorithm is also contained, a predecessor of ACO is Ant System which is a single ant version of ACO. It was also proposed by Dorigo et al. in [25] and it is recognized as a member of Adaptive Iterated Construction Search (AICS) methods. In the Ant System single artificial ant works and uses information obtained in its own preceding searches. Utilization of some information about good solutions obtained in the preceding search processes is able to be incorporated into SQ procedures. It would be possible to overcome the blind random searches in SQ. Because traversed arcs in good solutions of VRPTW are recognized as characteristics of them, such arcs are expected to be not

In order to embody the idea described above, artificial pheromone trail *τij* is associated with each edge (*i*, *j*) and *τij* is updated at the end of the loop of temperature *T* in SQ procedures. The 'characteristic of good solution' is embodied in increase of probability of selecting better candidate in random search process in SQ. In end of loop, weight is updated according to

( 1) : (1 ) ( ) *i j ij ij*

\* \* ( ) (,)

*else*

*Q E s if i j s*

\* *E s*( ) is the total cost of the best found solution at the present *s*\*

 

 *t t* 

is a parameter regarding vaporization of pheromone,

However, in order to avoid extreme effect of pheromone, lower bound and upper bound of *i j*

0 *i j*

*t t*

*i j i j i j*

is a parameter regarding vaporization of pheromone,

( ) (,)

*Q E s if i j s*

*k k*

1

*k*

*else*

*<sup>m</sup> <sup>k</sup>*

 

, (8)

, *m* is the number of ants (= size of *sp*) and *Q* is a

. (9)

(10)

. (11)

, *Q* is a

$$p\_i = \frac{r\_i}{\sum\_i r\_i} \tag{12}$$

The core part of the basic SQ algorithm (5) is replaced by the revised algorithm which is called *SQph* described as follows.

: 0.2 ( , ) *i j for all i j* ; : 1/ *p ii <sup>i</sup> <sup>i</sup> for all* ; *Repeat trials* := 0; *changes* := 0;  *Repeat trials* := *trials +* 1; *Generate a new state s*´ *from the current state s by applying randomly one of the three transformation rules to the string model of s*, *where posiition i to be changed in the string is selected with the probability pi* ; *Calculate* Δ*E = E*(*s*´) *− E*(*s*);  *If* Δ*E* < *0 Then s*´ *is accepted as a new state*;  *If* (*E*(s´) < *E*(*s*\* ) *and s*´ *is feasible*) *Then s*\* := *s*´  *Else s*´ *is accepted with probability* exp(*−*Δ*E*/*T* )  *If s*´ *is accepted Then changes* := *changes +* 1; *s* := *s*´  *Until trials* ≥ *SIZEFACTOR* · *N or changes* ≥ *CUTOFF* · *N*;  *T* := *T* · *TEMPFACTOR*; (13)

$$\begin{aligned} \tau\_{ij} &:= (1 - \rho) \cdot \tau\_{ij} + \Delta \tau\_{ij}; \text{if } \tau\_{ij} > 1.4 \text{ then } \tau\_{ij} := 1.4; \text{else} \\\sigma\_i &:= d\_{a \, i} \,/\, \tau\_{ai} + d\_{ib \,} \,/\, \tau\_{ib \,} \,. \quad p\_i := r\_i / \,\Sigma\_i r\_i \quad \text{for all } i \; ; \\\text{until } T &\le \text{INITTEMP} / \text{FINDIVISOR} \end{aligned}$$

#### **5.3. A device for decreasing the number of vehicles in use**

When performance of plural solutions for VRPTW is compared, the first measure is the number of vehicles in use, which is denoted by *m*, while the second is total cost *E*. Although *SQph* is expected to utilize characteristics of good solutions already found and to reduce *E*, it could not reduce *m* directly. In order to reduce it, another device should be included in SQ procedures.

In the string model described in Sec. 4.1, successive substring of '0's is interpreted as there is no tours between two '0's, that is to say that there is a vehicle not in use. For example shown in Figure 9, when '0' is replaced by another symbol in the string, one vehicle will become not in use. In order to urge to carry out this kind of transformation, artificial pheromone trail associated with edges (*i*, 0) or (0, *i*) should be decreased.

$$
\tau\_{i0}(t+1) \coloneqq \left\{ (1-\rho)\tau\_{i0}(t) + \Delta\tau\_{i0}(t) \right\} \times \delta\_{\prime\prime} \quad \tau\_{0i}(t+1) \coloneqq \left\{ (1-\rho)\tau\_{0i}(t) + \Delta\tau\_{i0}(t) \right\} \times \delta\_{\prime\prime}
$$

$$
0 \le \delta \le 1 \tag{14}
$$

Improvements in Simulated Quenching Method for Vehicle Routing Problem with Time Windows by Using Search History and Devising Means for Reducing the Number of Vehicles 143

from Solomon's problem sets. In the instances, each position of clients is given as *x*coordinate and *y*-coordinate. Link cost between client *i* and client *j* is calculated with the Euclidian distance. Service time *ci* is also given to each client *i*, in addition to the earliest arriving time *ei* and the latest arriving time *li*. In this benchmark problems, link cost is directly considered as traversing time of (*i*, *j*). Arriving time *ai* of each node *i* is calculated by summing up link cost of traversing edges and service time of traversing nodes. Concerning 100 clients problem sets, the geographical data are clustered in C-series 23 instances, randomly generated in R-series 17 instances, and a mix of random and clustered structures

Gehring & Homberger extended Solomon's benchmark problems to larger scale problem sets including 200, 400, 600, 800 and 1000 clients in [27]. They are provided from their website [28]. Concerning 200 clients problem sets, the geographical data are clustered in Cseries 20instances, randomly generated in R-series 19 instances, and a mix of random and

In this chapter, all of 56 instances with 100 clients from Solomon's problems and all of 59 instances with 200 clients from Gehring & Homberger's problems are chosen for

**6.2. Values of parameters used in the algorithms and specs of the computer in use** 

In the computations, the values of the parameters with respect to SQ that appear in the basic

*N* = 2*L*2 (*L* : length of string, that is *L* = *n* + *vn* - 1, where *n* is the number of clients,

until (*trials* ≥ *SIZEFACTOR* · *N* or *changes* ≥ *CUTOFF* · *N*))

*FINDIVISOR* = 20 (If *T* ≤ *INITTEMP* / *FINDIVISOR*, terminate the whole of the iterations.)

The computational experiments are executed on Windows 7, with Core i7 960, 3.2GHzCPU.

 

and the reference to the recommended values by Johnson et al. in [29-30].

*vn* is the number of vehicles superfluously allocated)

*CUTOFF* = 0.2 (Repeat iterations in the same temperature *T*,

are set commonly as follows according to the preliminary experiments

1 0.95 0.95 *n n TEMPFACTOR T T* (15)

0.5 in 10 and 14 ; 1000 in 11 *Q* (17)

25, 1, 500 ; (16)

in RC-series 16 instances.

computational experiments.

SQ, *SQph* and *SQph\**

*SIZEFACTOR* = 8

*INITTEMP* = 10 (Initial temperature)

Values of parameters appeared in energy function (2)

and values of parameters used in the proposed method

are set according to the preliminary execution.

clustered structures in RC-series 20 instances.

The effect of the device (14) could make the probability *p*0 large according to mechanism described in Sec.5.2. This further revised algorithm in which the device (14) is incorporated with pheromone update (10) is called *SQph\** in this chapter.

**Figure 9.** Reduction of *m* as a result of one to one exchange including '0'.

## **6. Computational experiments on the proposed method**

Computational experiments have been attempted for testing the performance of the proposed method compared with basic SQ method. They have been tried on typical instances for VRPTW.

#### **6.1. Solomon's benchmark problems and extended problems for VRPTW**

Solomon's benchmark problem sets are produced by Solomon in [5] and provided from Solomon's own website in [26]. They are extremely popular VRPTW instances, and have been used for testing performance of methods by many researchers. Although in some of instances, optimum solutions have been already found by using exact methods, in others, they have not found yet. In both cases, the best solutions found by heuristics have been presented in the literature. Instances including 25, 50, and 100 clients have been provided from Solomon's problem sets. In the instances, each position of clients is given as *x*coordinate and *y*-coordinate. Link cost between client *i* and client *j* is calculated with the Euclidian distance. Service time *ci* is also given to each client *i*, in addition to the earliest arriving time *ei* and the latest arriving time *li*. In this benchmark problems, link cost is directly considered as traversing time of (*i*, *j*). Arriving time *ai* of each node *i* is calculated by summing up link cost of traversing edges and service time of traversing nodes. Concerning 100 clients problem sets, the geographical data are clustered in C-series 23 instances, randomly generated in R-series 17 instances, and a mix of random and clustered structures in RC-series 16 instances.

Gehring & Homberger extended Solomon's benchmark problems to larger scale problem sets including 200, 400, 600, 800 and 1000 clients in [27]. They are provided from their website [28]. Concerning 200 clients problem sets, the geographical data are clustered in Cseries 20instances, randomly generated in R-series 19 instances, and a mix of random and clustered structures in RC-series 20 instances.

In this chapter, all of 56 instances with 100 clients from Solomon's problems and all of 59 instances with 200 clients from Gehring & Homberger's problems are chosen for computational experiments.

### **6.2. Values of parameters used in the algorithms and specs of the computer in use**

In the computations, the values of the parameters with respect to SQ that appear in the basic SQ, *SQph* and *SQph\** are set commonly as follows according to the preliminary experiments and the reference to the recommended values by Johnson et al. in [29-30].

*N* = 2*L*2 (*L* : length of string, that is *L* = *n* + *vn* - 1, where *n* is the number of clients, *vn* is the number of vehicles superfluously allocated) *SIZEFACTOR* = 8 *CUTOFF* = 0.2 (Repeat iterations in the same temperature *T*, until (*trials* ≥ *SIZEFACTOR* · *N* or *changes* ≥ *CUTOFF* · *N*)) *INITTEMP* = 10 (Initial temperature)

$$T \text{EMPFACTOR} = 0.95 \left( T\_{n+1} = 0.95 T\_n \right) \tag{15}$$

*FINDIVISOR* = 20 (If *T* ≤ *INITTEMP* / *FINDIVISOR*, terminate the whole of the iterations.) Values of parameters appeared in energy function (2)

$$
\alpha = \text{25, } \beta = 1, \gamma = \text{500} \text{ :} \tag{16}
$$

and values of parameters used in the proposed method

$$
\rho = 0.5 \text{ in (10) and (14)}; \text{Q} = 1000 \text{ in (11)} \tag{17}
$$

are set according to the preliminary execution.

142 Simulated Annealing – Advances, Applications and Hybridizations

associated with edges (*i*, 0) or (0, *i*) should be decreased.

 

**Figure 9.** Reduction of *m* as a result of one to one exchange including '0'.

*Vehicle* 3

*E*

*Vehicle* 2

**6. Computational experiments on the proposed method** 

Computational experiments have been attempted for testing the performance of the proposed method compared with basic SQ method. They have been tried on typical

Solomon's benchmark problem sets are produced by Solomon in [5] and provided from Solomon's own website in [26]. They are extremely popular VRPTW instances, and have been used for testing performance of methods by many researchers. Although in some of instances, optimum solutions have been already found by using exact methods, in others, they have not found yet. In both cases, the best solutions found by heuristics have been presented in the literature. Instances including 25, 50, and 100 clients have been provided

**6.1. Solomon's benchmark problems and extended problems for VRPTW** 

01

with pheromone update (10) is called *SQph\**

*<sup>F</sup> <sup>A</sup>*

*C D*

**0**

instances for VRPTW.

*B* 

*Vehicle* 1

In the string model described in Sec. 4.1, successive substring of '0's is interpreted as there is no tours between two '0's, that is to say that there is a vehicle not in use. For example shown in Figure 9, when '0' is replaced by another symbol in the string, one vehicle will become not in use. In order to urge to carry out this kind of transformation, artificial pheromone trail

0 0 00 0 <sup>0</sup> ( 1) : {(1 ) ( ) ( )} , ( 1) : {(1 ) ( ) ( )} , *i i ii i <sup>i</sup>*

The effect of the device (14) could make the probability *p*0 large according to mechanism described in Sec.5.2. This further revised algorithm in which the device (14) is incorporated

in this chapter.

*One to One Exchange*

*B*

*A B C* **0** *D E* **0** *F A B C* **0 0** *E D F* 

 

(14)

*Vehicle* <sup>1</sup> *Vehicle*<sup>3</sup> *<sup>F</sup> <sup>A</sup>*

**0**

*C D*

 

*E* 

*Vehicle* 2 is not in use.

*t tt t tt*

 

The computational experiments are executed on Windows 7, with Core i7 960, 3.2GHzCPU.

## **6.3. Comparison between basic SQ and** *SQph*

Ratio of application of artificial pheromone trail *τij* to all transformations is set to 3 cases regarding 50%, 75% and 100%. In other words, the ratio of random transformation is 50%, 25% and 0% in each case. Computational experiments are performed ten times on all benchmark problems with 100 clients and 200 clients.

Improvements in Simulated Quenching Method for Vehicle Routing Problem with Time Windows by Using Search History and Devising Means for Reducing the Number of Vehicles 145

*δ* = 0.25 *δ* = 0.5 *δ* = 1

*δ* = 0.5 --- *SQph\**

Computational experiments are performed ten times on all benchmark problems with 100 clients and 200 clients. Regarding the number of vehicles in use *m*, improvement ratio of

In the latter half processes in SQ

3 --- *SQph\**

(*SQph\**

1 *SQph\**

1, *SQph\**

4

2

to that using *SQph*.

.

, comparison of *E* is attempted in

to SQ is illustrated in Figure 13.

4 accompany reduction of pheromone on the edges

4 are better than those by *SQph*100%, *SQph\**

3 and *SQph\**

1 and *SQph\**

1 and *SQph\**

3.

3

4)

2, *SQph\**

3,

clients) is set to 0.25, 0.5 and 1(= not reducing). Four types of *SQph\**

*δ* = 0.25 *SQph\**

*SQph\**

*SQph\**

4) are defined in Table 1.

In the first half processes in SQ

**Table 1.** Four types of *SQph\** for experiments.

*6.4.2. Comparison of traversing cost E* 

Values of *E* by using *SQph\**

As defined in Table 1, *SQph\**

*6.4.1. Comparison of the number of vehicles m* 

to *SQph*100% (also to SQ) is illustrated in Figure 12.

**Figure 12.** Improvement ratio (%) of the number of vehicles in use *m* using *SQph\**

According to this result, severer reducing coefficient *δ* (corresponding to *SQph\**

using *SQph*100% that there is potential to be greatly improved by using *SQph\**

instances further reduction of *m* cannot be brought by *SQph\**

2 and *SQph\**

2 and *SQph\**

connecting depot conducted only in the first processes in *SQph*, while *SQph\**

these instances. Regarding *E*, improvement ratio of *SQph\**

brings smaller number of vehicles in use. Improvement in worst cases is larger than in best cases. This situation might be caused by the fact that the value of *m* is so large in worst case

Comparison of traversing cost *E* is significant only between situations based on same number of vehicles *m*. There are 15 instances with 100 clients and 23 benchmark instances with 200 clients in which optimal *m* is already obtained by basic SQ. Because in these

Concerning the number of vehicles in use *m*, significant difference is not detected. Regarding total traversing cost *E*, improvement ratio of *SQph* to SQ is illustrated in Figure 10. Values corresponding to the best cost solutions and the worst cost solutions of SQ in ten executions are shown by two bars. Computing time consumed by the methods using SQ and *SQph* is illustrated in Figure 11. According to these results, larger ratios of application of artificial pheromone trail bring better total traversing cost and longer computing time.

**Figure 10.** Improvement ratio of total traversing cost *E* using *SQph* to that using SQ.

**Figure 11.** Computing time consumed by the methods using SQ and *SQph*.

## **6.4. Comparison between SQ,** *SQph* **and** *SQph\**

In order to evaluate effect of device for reducing the number of vehicles in use *m*, results of experiments using four types of *SQph\** are compared. Processes in SQ are divided into two parts. The first half processes correspond to higher temperature, the latter processes to lower temperature. *δ (*coefficient of reducing pheromone on the edges connecting depot and other clients) is set to 0.25, 0.5 and 1(= not reducing). Four types of *SQph\** (*SQph\** 1, *SQph\** 2, *SQph\** 3, *SQph\** 4) are defined in Table 1.


**Table 1.** Four types of *SQph\** for experiments.

144 Simulated Annealing – Advances, Applications and Hybridizations

**6.3. Comparison between basic SQ and** *SQph*

benchmark problems with 100 clients and 200 clients.

Ratio of application of artificial pheromone trail *τij* to all transformations is set to 3 cases regarding 50%, 75% and 100%. In other words, the ratio of random transformation is 50%, 25% and 0% in each case. Computational experiments are performed ten times on all

Concerning the number of vehicles in use *m*, significant difference is not detected. Regarding total traversing cost *E*, improvement ratio of *SQph* to SQ is illustrated in Figure 10. Values corresponding to the best cost solutions and the worst cost solutions of SQ in ten executions are shown by two bars. Computing time consumed by the methods using SQ and *SQph* is illustrated in Figure 11. According to these results, larger ratios of application of artificial pheromone trail bring better total traversing cost and longer computing time.

**Figure 10.** Improvement ratio of total traversing cost *E* using *SQph* to that using SQ.

**Figure 11.** Computing time consumed by the methods using SQ and *SQph*.

In order to evaluate effect of device for reducing the number of vehicles in use *m*, results of

parts. The first half processes correspond to higher temperature, the latter processes to lower temperature. *δ (*coefficient of reducing pheromone on the edges connecting depot and other

are compared. Processes in SQ are divided into two

**6.4. Comparison between SQ,** *SQph* **and** *SQph\**

experiments using four types of *SQph\**

#### *6.4.1. Comparison of the number of vehicles m*

Computational experiments are performed ten times on all benchmark problems with 100 clients and 200 clients. Regarding the number of vehicles in use *m*, improvement ratio of *SQph\** to *SQph*100% (also to SQ) is illustrated in Figure 12.

**Figure 12.** Improvement ratio (%) of the number of vehicles in use *m* using *SQph\** to that using *SQph*.

According to this result, severer reducing coefficient *δ* (corresponding to *SQph\** 3 and *SQph\** 4) brings smaller number of vehicles in use. Improvement in worst cases is larger than in best cases. This situation might be caused by the fact that the value of *m* is so large in worst case using *SQph*100% that there is potential to be greatly improved by using *SQph\** .

#### *6.4.2. Comparison of traversing cost E*

Comparison of traversing cost *E* is significant only between situations based on same number of vehicles *m*. There are 15 instances with 100 clients and 23 benchmark instances with 200 clients in which optimal *m* is already obtained by basic SQ. Because in these instances further reduction of *m* cannot be brought by *SQph\** , comparison of *E* is attempted in these instances. Regarding *E*, improvement ratio of *SQph\** to SQ is illustrated in Figure 13.

Values of *E* by using *SQph\** 2 and *SQph\** 4 are better than those by *SQph*100%, *SQph\** 1 and *SQph\** 3. As defined in Table 1, *SQph\** 2 and *SQph\** 4 accompany reduction of pheromone on the edges connecting depot conducted only in the first processes in *SQph*, while *SQph\** 1 and *SQph\** 3

accompany reduction of the pheromone in all processes in *SQph*. These results are interpreted to mean that most of reduction of the number of vehicles in use is likely attained in the first processes in *SQph\** , while convergence of *E* mainly depends on the latter processes in *SQph*. Computing time consumed by the methods using SQ, *SQph* and *SQph\** is compared in Figure 14.

Improvements in Simulated Quenching Method for Vehicle Routing Problem with Time Windows by Using Search History and Devising Means for Reducing the Number of Vehicles 147

discussed in this chapter by taking account of both of reduction of the number of vehicles in use *m* and reduction of total traversing cost *E*. Moreover, regarding computing time, *SQph\**

that by basic SQ, it takes about only 1.6 min for solving 100 clients problems, and it takes about 6.5 min even in 200 clients problems. It is applicable to make actual vehicle routing

In this chapter, in order to relieve blind searches in Simulated Quenching (SQ), that is are a practical variant of Simulated Annealing (SA), utilization good characteristics of history records during SQ search processes is attempted. Two new devices which are inspired by the effect of pheromone in ant colony optimization (ACO) are adjusted and incorporated into SQ procedures to solve VRPTW. The one is a device to reduce total traversing cost *E* and the other is a device to reduce the number of vehicles in use *m*. By computational experiments on all of 56 benchmark instances with 100 clients and all of 59 benchmark instances with 200 clients, it is shown that both of two devices are effective. However, there is a trade-off between effects for reducing *E* and for reducing *m*. Taking account of putting

half processes and the device for reducing *E* is set in all processes in SQ seems to be the best method. Moreover, it is moderate in computing time consumed. Reducing *m* in the first half processes is corresponding to diversification, while reducing *E* in all processes is corresponding to intensification of search. Hence, it is considered that this method improves

As mentioned before, ease of implementation and flexibility are very important as well as quality of solution, running time and robustness in practical application of VRPTW algorithms to real-world problems. The proposed method is composed by a simple data model and straightforward one-phase algorithm to solve VRPTW. Therefore, the proposed

Two devices incorporated in SQ procedures in this chapter are able to be incorporated into SQ procedures in other routing problems which are embodied in the string model. As introduced in Sec.4.4, VRP with backhaouls (VRPB), Pickup and Delivery Problem (PDP), Capacitated Arc Routing Problem (CARP) and a general routing problem with nodes, edges, and arcs (NEARP) have aleady been embodied in string model and solved by SQ method.

4 is the most well-balanced method among methods

4 in which the device for reducing *m* is set in the first

4 is about two times longer than

4

To summarize these experiments, *SQph\**

the right device in the right place, *SQph\**

both diversification and intensification in SQ procedures.

method has comparative ease of implementation and much flexibility.

Application of two devices to these problems are left for future study.

*Department of Administration Engineering, Keio University, Japan* 

, Yuji Shimazaki, Shuichi Matsumoto,

plans in freight carriers.

**7. Conclusion** 

**Author details** 

Corresponding Author

 \*

Hisafumi Kokubugata\*

Hironao Kawashima and Tatsuru Daimon

is moderate. Although computing time consumed by *SQph\**

**Figure 13.** Improvement ratio of total traversing cost *E* using *SQph* and *SQph\** to that using SQ.

**Figure 14.** Computing time consumed by the methods using SQ, *SQph* and *SQph\** .

To summarize these experiments, *SQph\** 4 is the most well-balanced method among methods discussed in this chapter by taking account of both of reduction of the number of vehicles in use *m* and reduction of total traversing cost *E*. Moreover, regarding computing time, *SQph\** 4 is moderate. Although computing time consumed by *SQph\** 4 is about two times longer than that by basic SQ, it takes about only 1.6 min for solving 100 clients problems, and it takes about 6.5 min even in 200 clients problems. It is applicable to make actual vehicle routing plans in freight carriers.

## **7. Conclusion**

146 Simulated Annealing – Advances, Applications and Hybridizations

the first processes in *SQph\**

Figure 14.

accompany reduction of the pheromone in all processes in *SQph*. These results are interpreted to mean that most of reduction of the number of vehicles in use is likely attained in

*SQph*. Computing time consumed by the methods using SQ, *SQph* and *SQph\**

**Figure 13.** Improvement ratio of total traversing cost *E* using *SQph* and *SQph\**

**Figure 14.** Computing time consumed by the methods using SQ, *SQph* and *SQph\**

, while convergence of *E* mainly depends on the latter processes in

is compared in

to that using SQ.

.

In this chapter, in order to relieve blind searches in Simulated Quenching (SQ), that is are a practical variant of Simulated Annealing (SA), utilization good characteristics of history records during SQ search processes is attempted. Two new devices which are inspired by the effect of pheromone in ant colony optimization (ACO) are adjusted and incorporated into SQ procedures to solve VRPTW. The one is a device to reduce total traversing cost *E* and the other is a device to reduce the number of vehicles in use *m*. By computational experiments on all of 56 benchmark instances with 100 clients and all of 59 benchmark instances with 200 clients, it is shown that both of two devices are effective. However, there is a trade-off between effects for reducing *E* and for reducing *m*. Taking account of putting the right device in the right place, *SQph\** 4 in which the device for reducing *m* is set in the first half processes and the device for reducing *E* is set in all processes in SQ seems to be the best method. Moreover, it is moderate in computing time consumed. Reducing *m* in the first half processes is corresponding to diversification, while reducing *E* in all processes is corresponding to intensification of search. Hence, it is considered that this method improves both diversification and intensification in SQ procedures.

As mentioned before, ease of implementation and flexibility are very important as well as quality of solution, running time and robustness in practical application of VRPTW algorithms to real-world problems. The proposed method is composed by a simple data model and straightforward one-phase algorithm to solve VRPTW. Therefore, the proposed method has comparative ease of implementation and much flexibility.

Two devices incorporated in SQ procedures in this chapter are able to be incorporated into SQ procedures in other routing problems which are embodied in the string model. As introduced in Sec.4.4, VRP with backhaouls (VRPB), Pickup and Delivery Problem (PDP), Capacitated Arc Routing Problem (CARP) and a general routing problem with nodes, edges, and arcs (NEARP) have aleady been embodied in string model and solved by SQ method. Application of two devices to these problems are left for future study.

## **Author details**

Hisafumi Kokubugata\* , Yuji Shimazaki, Shuichi Matsumoto, Hironao Kawashima and Tatsuru Daimon *Department of Administration Engineering, Keio University, Japan* 

<sup>\*</sup> Corresponding Author

#### **8. References**

[1] Crescenzi, P. & Kann, V. (2000). A Compendium of NP Optimization Problem, Web site: http://www.nada.kth.se/~viggo/wwwcompendium/node103.html

Improvements in Simulated Quenching Method for Vehicle Routing Problem with Time Windows by Using Search History and Devising Means for Reducing the Number of Vehicles 149

[16] Hasama, T.; Kokubugata, H. & Kawashima H. (1999). A Heuristic Approach Based on the String Model to Solve Vehicle Routing Problem with Various Conditions, *Preprint for World Congress on Intelligent Transport Systems*, No.3027, Toronto, Canada, Nov. 1999. [17] Gendreau, M.; Laporte, G. & Potvin, J.-Y. (2002). Metaheuristics for the Capacitated VRP, In: *The Vehicle Routing Problem*, Toth P. & Vigo, D. (Ed), pp. 129–154, SIAM,

[18] Van Breedam, A. (1996). An Analysis of the Effect of Local Improvement Operators in GA and SA for the Vehicle Routing Problem, *RUCA working paper 96/14*, University of

[19] Kokubugata, H.; Itoyama, H. & Kawashima, H. (1997). Vehicle Routing Methods for City Logistics Operations, *Preprint for 8th IFAC Symposium on Transportation Systems*,

[20] Ingber, A. L. (1993). Simulated annealing: Practice versus theory, *J Mathl. Comput.* 

[21] Hasama, T.; Kokubugata H. & Kawashima H. (1998). A Heuristic Approach Based on the String Model to Solve Vehicle Routing Problem with Backhauls, *Preprint for 5th Annual World Congress on Intelligent Transport Systems*, No. 3025, Seoul, Korea, Oct.

[22] Kokubugata, H.; Hirashima, K. & Kawashima, H. (2006). A Practical Solution of Capacitated Arc Routing for City Logistics, *Proceeding of 11th IFAC Symposium on* 

[23] Kokubugata, H.; Moriyama, A. & Kawashima, H. (2007). A Practical Solution Using Simulated Annealing for General Routing Problems with Nodes, Edges, and Arcs, *Lecture Notes in Computer Science*, Vol. 4638 (SLS2007), pp. 136-149, Springer,

[24] Kokubugata, H. & Kawashima, H. (2008). Application of Simulated Annealing to Routing Problems in City Logistics, in *Simulated Annealing*, Cher Ming Tan (Ed.),

[25] Dorigo, M.; Maniezzo, V. & Colorni, A. (1996).Ant System: Optimization by a Colony of Cooperating Agents. *IEEE Transactions on Systems, Man, and Cybernetics - Part B*, Vol. 26

[27] Gehring, H. & Homberger J. (2001). A Parallel Two-phase Metaheuristic for Routing Problems with Time Windows, *Asia-Pacific Journal of Operational Research*, 18, 35-47.

[29] Johnson, D.S.; Aragon, C.R.; MacGeoch, L.A. & Schevon, C. (1989). Optimization by Simulated Annealing : An Experimental Evaluation, Part I, Graph Partitioning,

[26] Solomon, M., (2005). Web site: http://w.cba.neu.edu/~msolomon/home.htm

http://www.fernuni-hagen.de/WINF/touren/inhalte/probinst.htm

Philadelphia, USA.

Antwerp, Belgium.

1998.

Berlin.

No.1, pp.29-41.

pp.727-732, Hania, Greece, June 1997.

*Control in Transportation Systems,* No.222.

[28] Gehring, H. & Homberger J. (2001). Web site:

*Operations research*, Vol. 37, pp. 865-892.

pp.131-154, I-Tech Education and Publishing, Vienna.

*Modelling*, Vol.18, No.11, pp.29-57.


[16] Hasama, T.; Kokubugata, H. & Kawashima H. (1999). A Heuristic Approach Based on the String Model to Solve Vehicle Routing Problem with Various Conditions, *Preprint for World Congress on Intelligent Transport Systems*, No.3027, Toronto, Canada, Nov. 1999.

148 Simulated Annealing – Advances, Applications and Hybridizations

& Smith J. (Eds.), pp. 65-85, Springer, Berlin.

[1] Crescenzi, P. & Kann, V. (2000). A Compendium of NP Optimization Problem, Web

[2] Prins, C. & Bouchenoua, S. (2004). A Memetic Algorithm Solving the VRP, the CARP and more General Routing Problems with Nodes, Edges and Arcs, In: *Recent Advances in Memetic Algorithms, Studies in Fuzziness and Soft Computing 166*, Hart W.; Kranogor N.

[3] Bräysy, O. & Gendreau, M. (2005). Vehicle Routing Problem with Time Windows Part I: Route construction and local search algorithms, *Trans. Sci.*, Vol.39, No.1, pp.104–

[4] Ioannou, G.; Kritikos M. & Prastacos G. (2001). A Greedy Look-ahead Heuristic for the Vehicle Routing Problem with Time Windows, *J. Oper. Res. Soc.* Vol.52, pp.523–537. [5] Solomon, M. (1987). Algorithms for the Vehicle Routing and Scheduling Problems with

[6] Bräysy, O. (2003). Fast Local Searches for the Vehicle Routing Problem with Time

[7] Cordeau, J.-F.; Laporte G., Mercier. A. (2001). A unified tabu search heuristic for vehicle

[8] Homberger, J. & Gehring, H. (2005). A two-phase hybrid metaheuristic for the vehicle

[9] Berger, J.; Barkaoui, M. & Bräysy, O. (2003). A Route-Directed Hybrid Genetic Approach for the Vehicle Routing Problem with Time Windows, *Inform. Systems Oper.* 

[10] Gambardella, L. M.; Taillard, E. & Agazzi, G. (1999). MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows. In: New Ideas in Optimization, Corne, D., Dorigo, M. & Glover, F. (Ed). pp. 63–76, McGraw-Hill,

[11] Chiang, W. C. & Russell, R. A. (1996). Simulated Annealing Metaheuristics for the Vehicle Routing Problem with Time Windows. *Ann. Oper. Res*. Vol. 63 pp.3–27. [12] Bräysy, O. & Gendreau, M. (2005). Vehicle Routing Problem with Time Windows Part

[13] Bent, R. & P. Van Hentenryck. (2004). A two-stage hybrid local search for the vehicle routing problem with time windows, *Transportation Sci.*, Vol.38, No.4, pp.515–530. [14] Bräysy, O. (2003b). A Reactive Variable Neighborhood Search for the Vehicle Routing

[15] Ibaraki, T.; Imahori, S., Kubo, M., Masuda, T., Uno, T. & Yagiura, M. (2005). Effective Local Search Algorithms for Routing and Scheduling Problems with General Time

Problem with Time Windows, *INFORMS J. Comput.*, Vol.15, pp.347–368.

Window Constraints, Transportation Science, Vol. 39, No. 2, pp.206-232.

Time Window Constraints, *Operations Research*, Vol. 35, No. 2, pp. 254-265.

routing problems with time windows, *J. Oper. Res. Soc.* Vol.52, pp.928–936.

routing problem with time windows. *Eur. J. Oper. Res.* Vol.162, pp.220–238.

Windows. *Inform. Systems Oper. Res.* Vol.41, pp.179–194.

II: Metaheuristics, *Trans. Sci.*, No.39, No.1, pp.119–139.

site: http://www.nada.kth.se/~viggo/wwwcompendium/node103.html

**8. References** 

118.

*Res*., Vol.41, pp.179–194.

London, UK.

	- [30] Johnson, D.S.; Aragon, C.R.; MacGeoch, L.A. & Schevon, C. (1991). Optimization by Simulated Annealing : An Experimental Evaluation, Part II, Graph Colouring and Number Partitioning, *Operations research*, Vol. 39, pp. 378-406.

**Chapter 8** 

© 2012 Charrua Santos et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Lot Sizing and Scheduling in** 

Additional information is available at the end of the chapter

obtain "nearly optimal" solutions for the problem.

http://dx.doi.org/10.5772/50975

**1. Introduction** 

setup and tardiness time.

(Mokotoff, 2001).

*T*ppk, can be considered.

**Parallel Uniform Machines – A Case Study** 

This chapter presents the problem of scheduling tasks in uniform parallel machines with sequence-dependent setup times. The problem under analysis has a particular set of constraints, including equipment capacity, precedence tasks, lot sizing and task delivery plan. This kind of problem should be considered an operational planning problem and is

The main goal of this work is to minimise the total production time, including processing,

The complexity of the studied model does not allow to locate the optimal solution. To solve this, the authors used a meta-heuristic known as the simulated annealing algorithm to

The results obtained through the application of this particular heuristic show the importance of using a structured approach in production scheduling when compared with the results

The classical parallel machine scheduling problem considers *p* tasks with a fixed processing time that must be processed in *k* machines. This type of problem involves two kinds of decisions: i) which tasks are assigned to a specific machine and ii) tasks processing order

This problem becomes more complex as the number and characteristics of the available machines increases. The following levels of complexity can be considered: i) identical

In the case of uniform parallel machines, each machine or group of machines has a different processing velocity. A processing time *T*p for each task *p* in each machine *k*, described by

and reproduction in any medium, provided the original work is properly cited.

parallel machines; ii) uniform parallel machines and iii) unrelated parallel machines.

obtained from a reference plant where mainly "ad hoc" actions were normally taken.

F. Charrua Santos, Francisco Brojo and Pedro M. Vilarinho

found in different types of industries, namely, in the textile industry.

**Chapter 8** 
