Optimization of Train Speed Profile for Minimum Energy Consumption

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1 IEEJ TRANSACTIONS ON ELECTRICAL AND ELECTRONIC ENGINEERING IEEJ Trans 2; : Published online in Wiley InterScience ( DOI:.2/tee.228 Reiew Paper Optimization of Train Speed Profile for Minimum Energy Consumption Masafumi Miyatake a, Member Hideyoshi Ko, Member The optimal operation of railway systems minimizing total energy consumption is discussed in this paper. Firstly, some measures of finding energy-saing train speed profiles are outlined. After the characteristics that should be considered in optimizing train operation are clarified, complete optimization based on optimal control theory is reiewed. Their basic formulations are summarized taking into account most of the difficult characteristics peculiar to railway systems. Three methods of soling the formulation, dynamic programming (DP), gradient method, and sequential quadratic programming (SQP), are introduced. The last two methods can also control the state of charge (SOC) of the energy storage deices. By showing some numerical results of simulations, the significance of soling not only optimal speed profiles but also optimal SOC profiles of energy storage are emphasized, because the numerical results are beyond the conentional qualitatie studies. Future scope for applying the methods to real-time optimal control is also mentioned. 2 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. Keywords: electric railway, rail ehicle, optimization, speed profile, energy-saing operation Receied 2 August 29; Reised 3 Noember 29. Introduction AC motors and inerters hae been used as state-of-the-art dries for energy-saing rail ehicles. They hae enabled strong electrical regeneratie braking that reduces total energy consumption []. Howeer, there are seeral difficulties in efficient utilization of regeneratie energy, especially under DC power feeding systems. One of these difficulties is that the regeneratie ability closely depends on the catenary oltage at the braking train. The other difficulty in utilizing regeneratie energy is the need for another accelerating train or a deice for absorbing the regeneratie energy. If the energy is not absorbed, energy consumption will be larger because of the regeneratie failure. Optimizing the speed profiles of trains is one of the measures to alleiate these difficulties. It is well known that the energy-saing speed profile is composed of maimum acceleration, coasting, and maimum deceleration when regeneratie braking is not used. Howeer, regeneratie braking completely changes the conditions of optimal speed profiles, because regeneratie braking is strongly influenced by the catenary oltage which is determined by the consumed power of the whole train. The use of regeneratie braking makes it difficult to find the optimal solution by qualitatie inestigation. Therefore, numerical and/or analytical approaches to find the optimal speed profile are ineitable for more energy-saing train operation. The optimization can be modeled as an optimal control problem with constraints. These constraints, some of which are nonlinear, make it difficult to sole the problem numerically as well as analytically. Due to recent rapid deelopment of computer ability, seeral studies hae been deoted to sole the problem. The problem of a Correspondence to: Masafumi Miyatake. miyatake@sophia.ac.jp Department of Engineering and Applied Sciences, Sophia Uniersity, 7-, Kioicho, Chiyoda-ku, Tokyo 2-84, Japan Department of Clinical Engineering, Suzuka Uniersity of Medical Science, Kishioka-cho -, Suzuka, Mie -293, Japan soling the optimal speed profile was deeloped in Ref. [2]; howeer, the model did not consider the practical conditions of railway systems sufficiently: for eample, characteristics of the feeding circuits, and so on. The optimization of multiple train scheduling was realized in Ref. [3]; howeer, the conentional speed profile with only maimum acceleration, coasting, and maimum deceleration was used. There are some recent researches on optimal speed profile control with automatic train operation (ATO), traffic management system, etc., such as in Refs [4,]; howeer, only the fundamental shapes of the profiles were gien, i.e. they are not complete optimizations. The complete optimization used in this paper means that the speed profile is deried by using an optimization method without assuming qualitatie knowledge and/or typical patterns of the preferred speed profiles. There hae been some other studies that mention train speed profiles. Energy consumption of Shinkansen trains was reduced by the eperimental studies in Ref. [6]. A simulatie study on energy-saing operation used a precise model of energy loss in Ref. [7]. Howeer, they are not based on optimization. Generally, the preceding researches could not realize both the complete optimization without giing the patterns of the typical speed profiles and consideration of the practical railway conditions. Another measure of the difficulties is the use of energy storage. Regeneratie energy is stored in the energy storage and reused in the net acceleration. Energy-saings as well as preention of regeneratie failure is epected in most cases. Energy storage deices hae achieed sufficient energy and power density to be used in railway systems. There are many types of energy storage deices, such as lead acid, NiMH, and Li-ion batteries; flywheel energy storage; and the electric double layer capacitor (EDLC). In railway applications, a few such attempts can be seen in Refs [8 2]. When energy storage deices are used in railway systems, the control of charging/discharging strongly influences the energy consumption. Howeer, ery few papers deal with the optimal charge/discharge control of the energy storage. The charge/discharge command and speed profile should be optimized together. 2 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

2 M. MIYATAKE AND H. KO In this paper, we focus on the complete optimization of multiple trains and energy storage deices based on optimal control theory with constraints. The methods of the optimization of train speed profiles are reiewed on the basis of the studies seen in Refs [3 8]. Future scope for applying the methods to real-time optimal control is also mentioned. 2. Characteristics that Should Be Considered in Optimizing Train Operation Force [N/kg] Tractie force 3[V] [V] 7[V] 2.. Feeder losses To ealuate the feeder losses among substations and trains, it is essential to sole the equations of feeding circuits and ealuate the energy sent from all substations. If the circuit is simple, it can be soled analytically; howeer, soling the circuit equations is irtually impossible. The major reasons for the difficulty are topological changes of the circuit caused by train moements and its compleity. A good way to sole them numerically is by regarding the equations as equality constraints. It should be noted that DC feeding systems only are considered in this paper. Characteristics of substations should be also considered. When a diode rectifier is used in a substation, unidirectional current flow from the grid to the feeder must be modeled. If the modeling is insufficient, the calculated energy loss will be much different from the real one Relation between catenary oltage and tractie effort As an eample shown in Fig., the catenary oltage influences the tractie effort considerably because of the ariation of the maimum applied oltage from the inerters to the motors. It is necessary to consider the characteristics if the train speed is epected to reach to so-called constant power mode Blending electric and kinetic braking systems In the characteristics of Fig. (b), an electro-pneumatic blended braking system with air supplement control is assumed. Only if the regeneratie braking force is not enough for the specific braking force, the air brake works. It must be noted that the control strongly influences the energy consumption Running resistance Mechanical running resistance, such as friction and air drag, is one of the major losses for ealuating energy consumption. Howeer, it is not so difficult to deal with because it only depends on the train speed. 2.. Track profiles Track profiles such as speed limitation mainly caused by cures, switches, and gradient depend on the train position. Such conditions are difficult to analyze because a slight change of the train speed profile greatly affects the train position at each sampling period On-board and/or stationary energy storage When energy storage deices are controlled with an optimal rule, charging/discharging currents should be included into the set of control inputs. It should be noted that the terminal state of charge (SOC) at the final stage as well as the initial one should be specified as illustrated in Fig. 2 in the ealuation of energy consumption. If the constraint is not gien, the SOC of the energy storage will be small in order to reduce the energy from the substation. It is also difficult to compare the solutions under arious conditions with energy consumption Energy efficiency of motors and inerters Motors and inerters hae important roles in electric-kinetic and DC/AC power conersion, respectiely. In order to ealuate the energy loss in a train, it is important to consider their efficiency. The efficiency depends on the speed of the train, the flowing power, the catenary oltage, and so on. An attempt to regard it Force [N/kg].2 Running resistance Velocity [m/s] (a) acceleration characteristic and running resistance Maimum braking force.4 Electrical braking force.2 3[V] [V] 7[V] Velocity [m/s] (b) deceleration characteristic of electric and blended braking (Note: running resistance is the same as (a) in braking and coasting.) Fig.. Acceleration/deceleration characteristics and running resistance Q Initial state of charge Final state of charge With final state constraint Without final state constraint Time Fig. 2. Boundary conditions of the state of charge of energy storage as a ariable was made in Ref. [7], but it is assumed as a constant in this paper for simple formulation Signaling system If many trains run on a track, the interference among trains regulated by the signaling systems must be considered. The apparent behaior of the signaling systems is to impose a ariable speed limitation decided by other train positions. In this paper, signaling systems are not considered for the simple formulation. 264 IEEJ Trans : (2)

3 TRAIN SPEED PROFILE FOR MINIMUM ENERGY CONSUMPTION 3. Formulation and Soling the Optimal Speed Profile as an Optimal Control Problem with Constraints It is important to formulate mathematical models for easy handling and accuracy. If the model is too accurate, the number of eplanatory ariables and the degree of the ariables increase and the solution will be numerically unstable and/or not satisfy the optimal condition. The general formulation of optimizing train speed profile is eplained on the basis of Ref. []. In this formulation, most of the important characteristics mentioned in Sections are considered. 3.. Objectie functional and state ariable functions The objectie functional J is gien as the total electrical energy consumption at substations (). J = T E T (t)i (ζ (t), t)dt () Here, the constant T is the total time that defines the range of optimization, and the superscripts T on ectors and matrices indicate their transposes. The ector function ζ (t) represents whole state ariables of the system, e.g. position and speed of trains, node oltages e at the trains pantographs, and control inputs u as in the following equation. ζ (t) = [ T (t) T (t) e T (t) u T (t) ] T (2) The control input ector u consists of acceleration/deceleration commands of all trains and, if any, charging/discharging commands of all energy storage deices. Substation source oltage ector E and substation load current ector I are defined as timedependent functions. I depends on the system ariables ζ, while E is constant or eternally controlled Train characteristics The states and restrictions on train operation are formulated as state equations and inequality constraints, respectiely. The state equations and constraints of the jth train are gien in the following equations using the position j, speed j, catenary oltage at the pantograph e j, departure time t j, arrial time t jf, departure position j, arrial position jf, speed limit j ma, maimum acceleration with electrical torque f j ma, maimum deceleration with electrical and mechanical blending torque f j min, and load current i j of the train. j = { j (t j = j t t jf ) (others) { fj (u j,e j, j ) r j ( j, j ) (t j t t jf ) (others) (t j ) = j, (t j ) = () (t jf ) = jf, (t jf ) = (6) u j (7) { uj f f j (u j,e j, j ) = j ma (e j, j ) (u j ) u j f j min (e j, j ) (u j < ) (8) j j ma ( j ) (9) i j = i j (u j,e j, j ) () Equations (3) and (4) are kinetic equations and () and (6) are boundary conditions on the departure/arrial times defined by the train schedule. Here, the function f j describes acceleration/deceleration and u j is a constrained control input defined as (7). Torque and speed limits of a train are described by (8) and (9). f j ma and f j min ary according to the train speed and catenary oltage. Function r j consists of running and grade resistances. Equation () means the load current of a train depending on the kinetic power and feeding oltage. (3) (4) 3.3. Characteristics of the feeding circuit DC circuit equations of a feeding system are modeled as equality constraints in the optimal control problem shown in () by neglecting fast transient responses of the circuit on the order of milliseconds. C (ζ (t), t) = () As discussed in Section 2., it is to be soled numerically. When a substation with a diode rectifier is assumed, the internal resistance of the substation is not a constant but a function of the substation current. If the current alue is negatie, the resistance is set to a alue large enough to preent current flowing from the feeder to the grid Representation as an optimal control problem Finally, the energy-saing train operating problem is formulated as an optimal control problem (2) with equality and inequality constraints. min J, subj. to (3),...,() (2) {u,e} In order to sole circuit equations numerically, the state ariable e is treated as auiliary state ariables, although it is deried by soling circuit equations in general. 4. Methods of Soling the Optimal Control Problem There are mainly three methods to sole the optimal control problem with constraints. They are compared in Table I. Here, the three methods are eplained briefly. For more details, please refer to the original papers. 4.. Method This method is based on dynamic programming (DP) [3,4,8,9] that has been known as a kind of optimal control theory. It is ery easy to deal with the constraints on the state ariables, such as speed limitations. It also has the adantage that the generated optimal train profile can be reconfigured against disturbances. Howeer, it is difficult to apply this to a system with multiple control inputs, because DP has a serious disadantage that an increase of control input dimension causes an eplosie increase of computation time and use of memory space. In this paper, it is applied only to a single control input optimization. This method discretizes the state space domain [ ]aswellas the time domain and finds the optimal control input at each lattice point as shown in Fig. 3(a). The search process is performed from t = T to t =. In deciding the optimal control input at the stage t = (k ) t, energy consumption from the net stage t = k t to the last stage t = T stored in eery lattice points at the net stage is added to consumed energy between t = (k ) t and t = k t. After searching the initial stage t =, the optimal control inputs on eery lattice points guide the trajectory on the state space to the last stage as shown in Fig. 3(b). It should be noted that the bilinear interpolation technique is essential to do this. Please see Ref. [4] for details. There is another algorithm that can be applied to control two trains [2] with DP. In this method, the dimension of the state space is one for each train, i.e. the train position [] is regarded as the only state ariable. The adantage of this method is that the search domain is drastically reduced to deal with two-train operation. On the other hand, it has the fundamental shortcoming that it is difficult to consider some constraints such as speed limitations, because the train speed should be calculated from the position for each control input at each lattice point. This method is suitable for approimate calculations in timetabling, and so on. Since it is important to handle speed limitations and some other constraints, this paper is focused on the former method based on [3,4,8]. 26 IEEJ Trans : (2)

4 M. MIYATAKE AND H. KO Table I. Comparison of optimization methods for soling the optimal control problem Method Method [3] Method 2 [] Method 3 [7] Based algorithm Dynamic programming Gradient method Sequential quadratic programming Calculation time Slow fast Slow Reconfiguration of profiles against disturbances Very easy Fair Difficult Ease of implementation Good Bad Good Ease of considering loss in feeding circuits Fair Good Good Ease of considering speed limitations Very good Fair Fair Ease of applying to control of energy storage Bad Good Good Ease of epanding to multiple-input optimization Bad Very good Good Ease of applying parallel computing technique Bad Good Bad Possibility of applying online control Fair Good Fair State space at t = k t State space at t = k t u k = u k :optimal u k = Optimal control input at each lattice point t Bilinear interpolation t State space at t = (k ) t (a) in searching backward State space at t = (k ) t (b) in searching forward Fig. 3. Dynamic programming for optimization of speed profile 4.2. Method 2 The method is based on the optimality condition of the optimal control theory soled by the gradient method. It is so fleible that it can be applied to the optimization of not only multiple-train speed profiles but also charging/discharging of energy storage deices and the oltage of substations. Howeer, the shortcoming of this method is the requirement of a deep mathematical background [,6]. The method is deried using some transformation techniques and optimality conditions. The formulated optimal control problem (2) is a kind of two-point boundary-alue problem because the initial and terminal state constraints are gien as shown in () and (6), respectiely. It is generally known to be a difficult problem to sole numerically. A transformation technique is employed to aoid these difficulties. The modified objectie function with penalty terms is deried as (3). J = J + N { pj ((t jf ) jf ) 2 + p j2 (t jf ) 2} (3) j= The parameters p j and p j2 are designed as penalty parameters adjusted by considering the trade-off between energy consumption and position and speed errors at the terminal state. The other transformation from the constrained input (7) to the unconstrained one by employing slack control input u defined as (4) is also used to simplify the formulated optimal control problem (2). u = sin(u ) (4) Finally, the simplified optimal operating problem is deried as () from these transformation techniques. It is also assumed that the solution of the circuit equation () is unique. min J () {u, e} subj. to ξ = g(u, ξ,t) ξ j (t j ) = ξ j (j =, ) C (ζ,t)= Here, the state ector ξ = [ T T ] T and the ector function g are introduced to simplify the epression. The ariable e is introduced as a quasi-control input to sole the circuit equations () numerically. The optimality condition of the problem () is gien as follows: H u = H e = (6) The Hamiltonian H is defined as (7) and (8) by introducing the ariable ectors λ and π. H = E T I + λ T g + π T C (7) λ = H (8) ξ The method is based on the theory of soling Bolza-type optimal control problem. It is finally discretized for numerical solution. The outline of flowchart is shown in Fig IEEJ Trans : (2)

5 TRAIN SPEED PROFILE FOR MINIMUM ENERGY CONSUMPTION uk ek u e Initial estimates Searching better control Inputs Operation(i): The operation is a conentional one, without optimization. It just consists of maimum acceleration, coasting, and maimum deceleration. Operation(ii): The operation is optimized without considering interference between trains. k++ No uk+ s Soling circuit equations ek+ ek+ ek < ε Yes Node oltages Optimal solution Fig. 4. Flowchart of the numerical algorithm The formulated optimal train operating problem often becomes a large-scale problem. Parallel computing algorithms are aluable because they are faster to perform large computing tasks. A parallel computing algorithm for the formulated problem has also been proposed. Please see Ref. [6] for details Method 3 This method is based on the sequential quadratic problem (SQP). It is easier to understand because it uses one of the commonly used nonlinear optimization methods. Calculation time is longer than for Method 2, but it is applicable to optimizations of a few control inputs [7] The method is based on a commonly used optimization technique for easy software deelopment. The continuous time formulation must be transformed to a discrete one in order to apply mathematical programming to the optimal control problem. Therefore, t is defined as the constant sampling interal. State ariables are discretized as (9). ζ D = [ζ () ζ ( t) ζ (T )] (9) A general optimal control problem with equality and inequality constraints can be deried from the discretized formulation as min J D (ζ D ) (2) subj. to d (ζ D ) =, h(ζ D ) where J is the objectie function, and d and h are equality and inequality constraints. It is a typical nonlinear optimization problem that can be soled with a few methods. The optimal control problem is soled by the SQP method. SQP is an optimization method to sole general nonlinear programming problems. Details of soling the problem with SQP are described in Ref. [7].. Eamples of Numerical Analysis In this section, some eamples of numerical analyses are introduced to erify the significance of the numerical study of optimal train speed profiles... Results of analyses with running two trains In this simulation, two trains run with a time interal t s under the gien conditions [4]on a straight line without speed limitations and gradients. Three cases of t s were tested, because t s greatly influences the speed profiles. Three operation types (i) (iii) were assumed as follows. Operation(iii): The operation is globally optimized by considering all trains behaior. The program was deeloped with C/C++, and each optimizing procedure conerged in less than min with Method 2. The optimization results are shown in Fig.. It is obious from Fig. that the train speed profiles and control inputs change dynamically. This fact indicates that the numerical analyses are important to realize energy-saing operation. The operation (iii) can reduce energy consumption by % compared to operation (i). From the results, it is confirmed that the echange of regeneratie energy plays an important role in saing energy consumption. Detailed discussions of these results can be seen in Ref. [4]..2. Results of analyses of a train with complicated speed limitations and gradients The methods were also applied to the optimization on a track with complicated speed limitations and gradients. It can be soled with Methods, 2, and 3. The precise conditions can be seen in Refs [4 and 6]. The result obtained by Method is depicted in Fig. 6 as case D. From the result, it is obious that the constant speed mode is used in sections of seere speed limitations. Constant speed mode consumes energy but it can reduce time to go through the speed limitations. The saed time in the speed limitations can decrease the maimum speed and energy loss by air drag. The proposed methods can cope with speed limitations and proide remarkable results, which cannot be deried from qualitatie studies..3. Results of analyses of a train with energy storage on board In these simulations, the electric double layer capacitor (EDLC) was assumed in the modeling of the energy storage as in Ref. [7] because of its high power density. A train shown in Fig.7 ran on a straight line without speed limitations and gradients. The final EDLC oltage V T was taken to be equal to the initial one. Two cases with and without EDLC were simulated. The optimization program was deeloped with optimization toolbo on MATLAB software. By using such a conenient toolbo, the program could be implemented with a shorter deelopment period. The optimization results are shown in Fig. 8. The trend of speed profiles without and with EDLC, cases E and F, respectiely, are almost identical because the assumed energy capacity of the installed EDLC was small. The profile of V T that represents SOC shows that the discharge at acceleration is not steep in order to preent energy loss caused by the internal resistance of the EDLC. In train coasting, the train collects a small current for charging the EDLC. Regarding energy consumption, the energysaing effect by introducing EDLC is.67%. The EDLC enables obious energy-saing operation in this case. 6. Future Scope 6.. Consideration of more realistic conditions The introduced methods ignore some important conditions, especially those mentioned in Sections 2.7 and 2.8. The optimal departure and arrial times that are specified by the train schedule are also not considered, because the methods define them as constant alues. There is still much room for improing the methods from this iewpoint. 267 IEEJ Trans : (2)

6 M. MIYATAKE AND H. KO Speed [m/sec] 2 Speed [m/sec] 2 Speed [m/sec] Operation (i) Operation (ii) Operation (iii) 2 2 Case A ts = 9 [sec] Case B ts = 6 [sec] Case C ts = 4 [sec] st train (traditional operating patterns of a train) st train (optimal operating patterns of a train) st train (optimal operating patterns) Fig.. Optimization results with different departing phases 2nd train (traditional operating patterns of a train) 2nd train (optimal operating patterns of a train) 2nd train (optimal operating patterns) Speed[m/sec] e 2 2 Optimal speed profile (case D) 9 [km/h] Steed decending slope Speed limitations 7 [km/h] 4 [km/h] 2 [km/h] 2 Position[m] Fig. 6. Graph of optimal train speed profile INV. IM DC link Chopper V c I c EDLC Fig. 7. Circuit model of a train with an on-board electric double layer capacitor Howeer, the larger the number of control inputs and ariable conditions is, the longer will be the computation time for the optimization. If it is difficult with a practical computation time, meta-heuristic approaches may hae to be introduced. Although such approaches do not guarantee finding the global optimum point, it is useful to know the quasi-global optimum in most cases Real-time Control The introduced methods cannot be applied to real-time control of trains. It should be etended for the application in order to realize the control cycle of s or less, which was used in the methods already shown. There are some potential etensions to the real-time control. Receding horizon control (RHC) [2] is one of the methods that can be applied. It is in close agreement with Method 2 because both of them are based on optimal control theory. There has already been an attempt to apply RHC to minimum energy operation of biped robots [22] as well as simpler robots. Preious papers on robotics realized the control cycle of about ms or less, which is sufficient for railway applications. Howeer, realizing RHC on train control depends on how to cope with some difficult nonlinear characteristics and inequality constraints peculiar to railway systems. We suppose that some inequality constraints may increase slack ariables and lead to a far longer control cycle. Another method for real-time control is model predictie control (MPC) [23]. Like Method 3, it discretizes the model and soles a nonlinear optimization problem or a linearized optimization problem. It will be easier to formulate the nonlinear characteristics; howeer, there may be some problems in calculation time and conergence of controller outputs. MPC was already applied to a current controller of permanent magnet synchronous motors (PMSMs) and realized the control cycle of 2 µs [24]; howeer, it used a kind of lookup table where offline calculation data were stored. We think that combining offline calculation and online control is the only way to realize the control cycle suitable for railway applications of MPC, although the lookup table must be ery large because of arious running conditions. There is another problem from the iewpoint of controller implementation. It is to be carefully considered as to how many state ariables are obserable and which configuration of concentrated or distributed control system is better. 7. Conclusion In this paper, some measures of finding energy-saing train speed profiles were outlined. In the measures, some proposals based on complete optimization were introduced, and their basic formulations and soling methods are summarized. By showing 268 IEEJ Trans : (2)

7 TRAIN SPEED PROFILE FOR MINIMUM ENERGY CONSUMPTION Speed [m/s] 2 2 Case E (acceleration). Case E EDLC oltage V C Fig. 8. Graphs of optimal control inputs and state ariables (charging) some numerical results of simulations, the significance of soling not only optimal speed profiles but also optimal SOC profiles of energy storage were emphasized, because the numerical results are beyond the conentional qualitatie studies. References () Sone S. Optimisation of regeneratie train operation Pt. contents of optima. Proceedings of IEEJ JIASC2 (in Japanese). ol. 3, 2; (2) Khmelnitsky E. On an optimal control problem of train operation. IEEE Transactions on Automatic Control 2; 4(7): (3) Albrecht T. Reducing power peaks and energy consumption in rail transit systems by simultaneous train running time control. Computers in Railways IX. WIT Press: Southampton, UK, 24; (4) Dominguez M, Fernandez A, Cucala AP, Cayuela LP. 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Proceedings of IEEJ Industry Applications Society Conference (in Japanese), ol. I, No. 22, 28; Masafumi Miyatake (Member) receied the B.S. and M.S. degrees in electrical engineering in 994 and 996, respectiely, and the Ph.D. degree in information and communication engineering in 999, all from the Uniersity of Tokyo. In 999, he joined the Tokyo Uniersity of Science. In 2, he joined Sophia Uniersity. From 24, he has been an Associate Professor at Sophia Uniersity. His research interests include energy management control as well as renewable energy generation and their applications to transportation systems. Hideyoshi Ko (Member) receied the B.E., M.E., and Dr Eng. degrees from Sophia Uniersity, Japan, in 2, 24, and 27, respectiely. He joined the Suzuka Uniersity of Medical Science, Japan, in 27. He is currently an Assistant Professor of the Department of Clinical Engineering at the same uniersity. His research interests include mathematical optimization as well as metaheuristics and their applications. 269 IEEJ Trans : (2)