Impact of Supercapacitor Ageing Model on Optimal Sizing and Control of a HEV using Combinatorial Optimization

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Impact of Supercapacitor Ageing Model on Impact of Supercapacitor Ageing Model on Impact of Supercapacitor Ageing Model on Impact of Supercapacitor Ageing Model on Optimal Sizing and Control of a HEV using Optimal Sizing and Control of a HEV using Optimal Sizing and Control of a HEV using OptimalCombinatorial Sizing and Control of a HEV using Optimization Combinatorial Optimization Combinatorial Optimization Optimization Combinatorial ∗ ∗ ∗∗ ∗ ∗ , Alaa Hijazi ∗ , Ali Sari ∗∗ , Eric Bideaux ∗ Alan Chauvin Alan Chauvin ∗ , Alaa Hijazi ∗ , Ali Sari ∗∗ , Eric Bideaux ∗ ∗ , Alaa Hijazi ∗ , Ali Sari ∗∗ , Eric Bideaux ∗ Alan Chauvin Alan Chauvin , Alaa Hijazi , Ali Sari , Eric Bideaux ∗ ∗ INSA de Lyon, Universit´ de Lyon, Lyon, Laboratoire Laboratoire Amp` Amp`eere, re, UMR UMR eee de ∗ INSA de Lyon, Universit´ ∗ INSA de Lyon, Lyon, Universit´ de Lyon, Lyon, Cedex, Laboratoire Amp` re, UMR UMR CNRS 5005, 69621 Villeurbanne France (e-mail: INSA de Universit´ e de Laboratoire Amp` eere, CNRS 5005, 69621 Villeurbanne Cedex, France (e-mail: CNRS 69621 Villeurbanne Cedex, France (e-mail: CNRS 5005, 5005, [email protected]). 69621 Villeurbanne Cedex, France (e-mail: [email protected]). ∗∗ [email protected]). ∗∗ Universit´ ee Lyon I, Universit´ eere, [email protected]). Lyon I, Universit´eee de de Lyon, Lyon, Laboratoire Laboratoire Amp` Amp` re, UMR UMR ∗∗ Universit´ ∗∗ Universit´ e Lyon I, Amp` CNRS 5005, 69622 Lyon Lyon Cedex,Laboratoire France (e-mail: (e-mail: Universit´ e Lyon5005, I, Universit´ Universit´ e de de Lyon, Lyon, Laboratoire Amp`eere, re, UMR UMR CNRS 69622 Cedex, France CNRS 5005, 69622 Lyon Cedex, France (e-mail: CNRS 5005, [email protected]) 69622 Lyon Cedex, France (e-mail: [email protected]) [email protected]) [email protected])

Abstract: This paper presents an optimization problem for sizing and energy management of aa Abstract: This paper presents an optimization problem for sizing and energy management of Abstract: This paper presents an optimization problem for sizing and energy management of aa hybrid electric vehicle including a fuel cell system and supercapacitors pack. From a predefined Abstract: This paper presents an optimization problem for sizing and energy management of hybrid electric vehicle including a fuel cell system and supercapacitors pack. From aa predefined hybrid electric vehicle including a fuel cell system and supercapacitors pack. From predefined cycle and including ageing and models of supercapacitors, the objective is minimize hybrid electric vehicle including a fuel cell system and supercapacitors pack. From ato cycle and including ageing and thermal thermal models of supercapacitors, the objective is topredefined minimize cycle and including ageing thermal models of supercapacitors, the objective is minimize the of ownership of the hybrid power system. The model is based on cycletotal and cost including ageing and and thermal models ofsupply supercapacitors, theageing objective is to to minimize the total cost of ownership of the hybrid power supply system. The ageing model is based on the total cost of ownership of the hybrid power supply system. The ageing model is based on Eyring law and a thermal model of supercapacitor pack. This problem is solved using a Branch the total cost of ownership of the hybrid power supply system. The ageing model is based on Eyring law and a thermal model of supercapacitor pack. This problem is solved using a Branch Eyring law and a thermal model of supercapacitor pack. This problem is solved using a Branch and Cut algorithm with an iterative strategy. Some results are presented and the impact of Eyring law and a thermal model of supercapacitor pack. This problem is solved using a Branch and Cut algorithm with an iterative strategy. Some results are presented and the impact of and Cut algorithm with an iterative strategy. Some results are presented and the impact of ageing model is discussed. and Cut algorithm with an iterative strategy. Some results are presented and the impact of ageing model is discussed. ageing model model is is discussed. discussed. ageing © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: hybrid Keywords: hybrid electric electric vehicles vehicles (HEV), (HEV), mini-excavator, mini-excavator, Branch Branch and and Cut Cut algorithm, algorithm, optimal optimal Keywords: hybrid electric vehicles (HEV), mini-excavator, Branch and Cut algorithm, sizing, supercapacitor ageing model, optimal energy management, thermal model Keywords: hybrid electric vehicles (HEV), mini-excavator, Branch and Cut algorithm, optimal optimal sizing, supercapacitor ageing model, optimal energy management, thermal model sizing, management, thermal sizing, supercapacitor supercapacitor ageing ageing model, model, optimal optimal energy energy management, thermal model model 1. INTRODUCTION In a classical optimization strategy for solving sizing and 1. In optimization strategy for sizing and 1. INTRODUCTION INTRODUCTION In aaa classical classical optimization strategy for solving solving sizing and optimal control problems, bi-level strategies are often used 1. INTRODUCTION In classical optimization strategy for solving sizing and optimal control problems, bi-level strategies are often used optimal control problems, bi-level strategies are often used as shown in Silvas et al.bi-level (2014) where the is optimal control problems, strategies areproblem often used The development of new powertrain technologies for transas in Silvas et (2014) where the problem is The of technologies for transas shown shown in two Silvas et al. al. (2014) where the problem is The development development of new new powertrain powertrain technologies for ...) transsolved using nested loops: For solving complex nonas shown in Silvas et al. (2014) where the problem is port applications (automotive, delivery, agriculture, alThe development of new powertrain technologies for transsolved using two nested loops: For solving complex nonport applications (automotive, delivery, agriculture, ...) alsolved using two nested loops: For solving complex nonport applications applications (automotive, delivery,inagriculture, agriculture, ...) alal- linear problems, inner loop is used to complex solve optimal solved using two the nested loops: For solving nonlowed manufacturers to offer solutions order to respond port (automotive, delivery, ...) linear the inner loop is lowed manufacturers to offer solutions in order to respond linear problems, problems, the inner loop Programming, is used used to to solve solveLioptimal optimal lowed manufacturers to offer solutions in order to respond control problem (eg Dynamic et al. linear problems, the inner loop is used to solve optimal to more restricting legislative frame (environmental stanlowed manufacturers to offer solutions in order to respond control (eg Programming, Li et to restricting legislative frame (environmental stancontrol problem problem (eg Dynamic Dynamic Programming, Li size et al. al. to more morelimitation restricting legislative framepollution (environmental stan- (2012)) and the outer loop is used to solve the of control problem (eg Dynamic Programming, Li et al. dards, of vehicles during peak). Howto more restricting legislative frame (environmental stan(2012)) and the outer loop is used to solve the size of dards, limitation of vehicles during pollution peak). How(2012)) and and with the outer outer loop is is used used to solve solve inthe theorder size to of dards, limitation of vehicles during pollution peak). Howcomponents metaheuristic algorithms (2012)) the loop to size of ever, although several commercialized products launched dards, limitation of vehicles during pollution peak). Howcomponents with metaheuristic algorithms in order to ever, although several commercialized products launched components with metaheuristic algorithms in order to ever, although several commercialized commercialized products launched limit computation (eg Particle Swarm Optimization components with time metaheuristic algorithms in order to by different manufacturers have a growing interest thanks ever, although several products launched limit computation time (eg Swarm Optimization by manufacturers have interest thanks limit computation time Hegazy (eg Particle Particle Swarm Optimization by adifferent different manufacturers have a a growing growing interest thanks or Genetic Algorithm, and Van Mierlo (2010)), limit computation time (eg Particle Swarm Optimization to substantial fuel consumption economy (hybrid vehiby different manufacturers have a growing interest thanks or Algorithm, Hegazy and (2010)), to fuel consumption economy vehior Genetic Genetic Algorithm, Hegazy and Van Vanis Mierlo Mierlo (2010)), to a a substantial substantial fuel consumption economy (hybrid (hybrid vehialthough the optimality of the solution not insured. In or Genetic Algorithm, Hegazy and Van Mierlo (2010)), cles) and a better respect to the environment (electric veto a substantial fuel consumption economy (hybrid vehialthough the optimality of the solution is not insured. In cles) and a better respect to the environment (electric vealthough the optimality of the solution is not insured. In cles) and a better respect to the environment (electric vea convex problem, an optimal solution exists and some although the optimality of the solution is not insured. In hicle, hydrogen electric vehicles), the initial investment is a cles) and a better respect to the environment (electric vea convex problem, an optimal solution exists and some hicle, hydrogen electric vehicles), the initial investment is a a convex problem, an optimal solution exists and some hicle, hydrogen hydrogen electric vehicles), the the initial investment is aa solvers canproblem, be use to optimal sizing and the optimal asolvers convex anfind optimal solution exists and some major obstacle to the deployment of this new technologies hicle, electric vehicles), initial investment is be find optimal sizing the optimal major to of solvers can can be use use to to find(2011). optimal sizing and andsome the discrete optimal major obstacle obstacle to the the deployment deployment of this this new new technologies technologies control, Murgovski et al. Sometimes, solvers can be use to find optimal sizing and the optimal compared to conventional vehicles. major obstacle to the deployment of this new technologies control, et Sometimes, some discrete compared control, Murgovski Murgovski et al. al. (2011). (2011). Sometimes, some discrete compared to to conventional conventional vehicles. vehicles. variables are introduced as gear ratio or cells number for control, Murgovski et al. (2011). Sometimes, some discrete compared to conventional vehicles. variables are introduced as gear ratio or cells number for variables are introduced as gear ratio or cells number for It is necessary to design hybrid powertrain with an approenergy storage system. The combinatorial optimization is variables are introduced as gear ratio or cells number for It is necessary to design hybrid powertrain with an approenergy storage system. The combinatorial optimization is It is is necessary necessary to design design hybrid powertrain with an approapproenergy storage system. The combinatorial optimization is priate components sizing in order to limit the initial cost It to hybrid powertrain with an used to solve problem including mixed-integer variables. energy storage system. The combinatorial optimization is priate components sizing in order to limit the initial cost used to solve problem including mixed-integer variables. priate components sizing in order to limit the initial cost used to solve problem including mixed-integer variables. investment. But, the main issue for manufacturer is to find priate components sizing in order to limit the initial cost In this original non linear problem is variables. split into used topaper solve the problem including mixed-integer investment. the issue is In paper original non linear problem split into investment. But, But, the main main issue for for manufacturer manufacturer is to to find find In this this integer paper the the original non which linear represents problem is is each splitfixed into ainvestment. between aa reasonable initial investment and But, the main issue for manufacturer is to find several linear problems In this paper the original non linear problem is split into aa compromise compromise between reasonable initial investment and several integer linear problems which represents each fixed compromise between reasonable initial investment and several integer linear problems which represents each fixed fuel consumption of vehicles. Indeed, the fuel consumption a compromise between aa reasonable initial investment and sizing combination. Then the method is based on Branch several integer linear problems which represents each fixed fuel consumption of vehicles. Indeed, the fuel consumption sizing combination. Then the is on Branch fuela consumption consumption ofisvehicles. vehicles. Indeed, the fuel fuel cycle consumption sizing combination. Then theetmethod method is based based on all Branch of vehicle linked to the operating and the fuel of Indeed, the consumption and Bound algorithm Gaoua al. (2013) to solve subsizing combination. Then the method is based on Branch of aa hybrid hybrid vehicle is linked to the operating cycle and the and Bound algorithm Gaoua et al. (2013) to solve all subof hybrid vehicle is linked to the operating cycle and the and Bound algorithm Gaoua et al. (2013) to solve all subsizing of powertrain components as presented in Moore of a hybrid vehicle is linked to the operating cycle and the problems by finding the optimal control. and Bound algorithm Gaoua et al. (2013) to solve all subsizing of powertrain components as presented in Moore problems by finding the optimal control. sizing of powertrain components as presented in Moore problems by finding the optimal control. (1997). The fuel consumption is deduced from the energy sizing of powertrain components as presented in Moore problems by finding the optimal control. (1997). consumption is deduced from the energy (1997). The The fuel fuel consumption isthe deduced from theand energy This paper is outlined as follows. The application is management strategy between power source the (1997). The fuel consumption is deduced from the energy This outlined as The is management strategy between the power source and the This paper paperin is issection outlined as follows. follows. The application application is management strategy between the power source and the described 2, then the optimization approach This paper is outlined as follows. The application is energy storage system in a hybrid vehicle. Moreover, while management strategy between the power source and the described in section 2, then the optimization approach energy storage system in a hybrid vehicle. Moreover, while described in section 2, then the optimization approach energy storage system in a hybrid vehicle. Moreover, while is presented and detailed in section 3. The original nondescribed in section 2, then the optimization approach internal combustion engines have got a long experience and energy storage system in a hybrid vehicle. Moreover, while is and detailed in 3. The noninternal combustion engines got experience and is presented presented and detailed in section section 3. The original original noninternal combustion engines have have got a a long long experience and linear problem is reformulated into aa 3. multitude of integer is presented and detailed in section The original nonainternal specialized technological know-how since one hundred combustion engines have got a long experience and linear problem is reformulated into multitude of integer aa specialized technological know-how since one hundred linear problem is reformulated into a multitude of integer specialized technological know-how since systems one hundred hundred linear sub-problems. In sectioninto 4 a adiscrete sub-problem problem is reformulated multitude of integer years, critical components as energy storage have a specialized technological know-how since one linear sub-problems. In sub-problem years, critical components as storage systems have linear sub-problems. In section section 444ofaaa discrete discrete sub-problem years, critical components as energy energy storage systems have is modeled with aa description behaviour model as linear sub-problems. In section discrete sub-problem ayears, limited lifetime which depends on operating conditions critical components as energy storage systems have is modeled with description of behaviour model as aa limited lifetime which depends on operating conditions is modeled modeled with aa models description of behaviour behaviourcells. model as limited lifetime which depends on operating conditions ageing and thermal for supercapacitor Some is with description of model as due to internal cell component degradation. Heat mana limited lifetime which depends on operating conditions ageing and thermal models for supercapacitor cells. Some due to internal cell component degradation. Heat manageing and thermal models for supercapacitor cells. Some due to to internal internal cell component component degradation. Heat man- simulations with different scenarios are presented and ageing and thermal models for supercapacitor cells. Some agement is an other critical problem wich has a major due cell degradation. Heat mansimulations different scenarios are and agement other critical wich has major simulationstowith with different scenarios are presented presented and agement is isin an an other critical problem problem wich has aaasystems major compared study the impact of behaviour models on simulations with different scenarios are presented and influence the degradation of energy storage agement is an other critical problem wich has major compared to study the impact of behaviour models on influence in the degradation of energy storage systems compared to study the impact of behaviour models on influence in the degradation of energy storage systems components sizing. The paper concludes on comparison compared to sizing. study The the impact of behaviour models on performances. This parameter has to be taken into account influence in the degradation of energy storage systems components paper concludes on comparison performances. This parameter has to be taken into account components sizing. The paper concludes on comparison performances. This parameter has to be taken into account results and future outlook. components sizing. The paper concludes on comparison in the optimization problem. has to be taken into account results and future outlook. performances. This parameter in results and and future future outlook. outlook. in the the optimization optimization problem. problem. results in the optimization problem. 2405-8963 © 2015, IFAC IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 403 Copyright © 2015 2015 IFAC 403 Copyright 2015 IFAC 403 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 403Control. 10.1016/j.ifacol.2015.10.058

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2. EXCAVATOR OVERVIEW Excavators are off-road machines used for earthmoving tasks as digging a trench, grading soil or loading material in a truck. A wide variety of machines are commercialized by manufacturers with an embedded power from some kW to several thousand kW.

Fig. 1. Hybrid Power Supply System overview For this reason, excavators are classified into classes following their weight. In this paper, a machine from the compact-excavator class (machines from 800kg to 8.5 tonnes) is studied as shown on the figure 1 (at left). In a conventional excavator, a diesel engine with 40kW net power size leads a hydraulic pump in order to supply energy to several hydraulic actuators and hydraulic motors. The study is focused on the hybridization and the electrification of a conventional machine. Hydraulic architecture is replaced by electromechanical actuators and electrical motors (not represented on figure 1). The diesel engine is replaced by a Hybrid Power Supply System (HPSS) including a fuel cell system (FCS) and supercapacitor bank (SC) as energy storage system. The scheme of the hybrid architecture is shown on figure 1 (at right). The aim of the study is to optimize HPSS components sizing and energy management strategy in order to minimize the Total Cost of Ownership (TCO) of HPSS.

Fig. 2. Optimization approach method

•

•

3. OPTIMIZATION APPROACH The approach developed in this paper is based on an optimization method from operational research, combinatorial optimization and Branch and Bound algorithm, Winston and Goldberg (1994). The original non-linear problem (N LP ) is split into a multitude of integer linear sub-problems (ILPk ) where sizing variables are fixed for each sub-problem. The optimization process is shown in figure 2. This approach uses a discretization of component sizing range and power control range. An integer-linear subproblem is modeled for each sizing combination. Then each sub-problem is solved using a modified Branch and Bound algorithm with a method to reduce the space domain, Gomory (1958). The algorithm is run on Matlab 2012a and with the solver Cplex 12.4, Linderoth and Ralphs (2005). The optimization approach is described as follows: • System modeling and mathematical formulation into a (N LP ): The problem is described as 404

•

•

a mathematical optimization problem by modeling components. This problem includes sizing variables and control variables. This is a non linear problem. Discretization process and combinatorial subproblems (ILPk ) modeling: By discretizing the sizing range of each component and the FCS control, an integer linear problem (ILPk ) is modeled for each sizing combination k. ndim sub-problems are considered. Initialization: An upper bound ub is seek. It represents an integer solution of the problem, not necessary optimal. The algorithm starts with the first sub-problem (ILP1 ). Loop: For each sub-problem (ILPk ), integer constraints on variables are removed in order to obtain a linear problem (LPk ) where a lower bound is seek. If the solution x∗k is lower than L, a better solution could exists. A Branch and Cut algorithm is executed using Cplex solver. The solver is stopped if all explored nodes lead to a solution worst than ub or if there is no nodes to explore. If an integer solution xk is found and is better than L, so this is the new upper bound stored in L. Solution: The algorithm is stopped when all subproblems have been explored. The solution contained in L is optimal

4. COMBINATORIAL SUB-PROBLEM MODELING From the components sizing discretization, the maximum power of the FCS, Pfmax cs , and the number of cells in the supercapacitor bank, Nsc , are choosen. The power profile

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Preq which represents the total electrical power consumed by all actuators and electrical auxiliairies on the cycle is discretized from t0 to tf into T points with a constant interval time ∆t. The power balance equation for the HPSS is described by (1). Preq (t) = Pf cs (t) + Psc (t)

(1)

4.1 Fuel Cell System (FCS) The Fuel Cell System is a power source system which supplies electrical energy from hydrogen with a chemical reaction with air-oxygen through a membrane Chan (2007). The output voltage from the stack is adapted to the high voltage network with a power converter. Some auxiliaries supply air and hydrogen and exhaust the water from the stack. A cooling system evacuates the heat. 60

η

fcs

[%]

50 40 30 20 10 0 0

10

20

30

40 50 60 Percentage of Pmax [%]

70

80

90

100

fcs

Fig. 3. FCS global efficiency with Pfmax cs The FCS is characterized by a global efficiency ηf cs between the equivalent input power PH2 and the electrical output power Pf cs . This relation is modeled on the figure 3 (blue line) and represented by (2). PH2 (t) =

Pf cs (t) ηf cs (Pf cs (t))

(2)

The FCS power control is limited by the FCS maximum power Pfmax cs according to the constraint (3). 0 ≤ Pf cs (t) ≤ Pfmax cs

Supercapacitors are components used to store energy in electric charges. The technology studied is called EDLC (Electrochemical Double Layer Capacitor), Khaligh and Li (2010). This components have a low output voltage and are assemblied into series and parallel configuration and a power converter is added to adapt the output cell voltage to the high voltage network. All cells are assumed to be similar. A cell is modeled as a constant high value capacitor coupled to an ohmic resistance in serie. In an energetic model, transient dynamics are neglected. The equivalent serie resistance (ESR), noted Resr [Ω], and the supercapacitor cell capacitance Ccell [F] are assumed to be independent of environmental and operating conditions. The state of charge is proportional to the square of the nominal voltage of the cell. The output voltage range is limited by boundaries Umin and Umax in order to improve the power converter efficiency. The internal power supplied by the cell Pcell [W] depends on the converter efficiency ηconv [%], the number of supercapacitor cells Nsc in the bank, and the power loss function Ploss [W]. This relation is described by (9). Pcell (t) =

N �

x(u, t) = 1 and

x(u, t) = {0, 1}

(6)

(7)

u=1

405

1 · [Psc (t) + Ploss (Psc (t))] ηconv (Psc ) · Nsc

(9)

min Pcell is limited by thresholds respectively noted Pcell and max Pcell , and defined according to manufacturer data for technical reasons (heat management). This constrained is described by (10) min max ≤ Pcell (t) ≤ Pcell Pcell

(10)

The energy evolution in a cell Ecell [J] for each time t from t0 to tf is described by (11) Ecell (t) = Ecell (t0 ) −

�t

Pcell (τ )dτ

(11)

t0

where Ecell (t0 ) is the energy in one cell at the initial time. The relation between the energy stored in a cell and the terminal voltage Ucell [V] is described by the equation (12). Ecell (t) =

1 2 · Ccell · Ucell (t) 2

(12)

The energy stored is limited by voltage boundaries as shown in the equation (13). min max ≤ Ecell (t) ≤ Ecell Ecell

u=1

N � Pf cs (u) PH2 (t) = η (u) u=1 f cs

(8)

4.2 Supercapacitor bank (SC)

(4)

In order to obtain a linear model of the FCS, the power control range Pf cs is split into N equidistant points from 0 to Pfmax cs . For each discretization point u and each time t, a binary activation variable x(u, t) is assigned. ηf cs is discretized according to Pf cs (u), as shown on figure 2 (red points). Relations (5) to (8) are the new equations of FCS in a discrete format. N � Pf cs (t) = x(u, t) · Pf cs (u) (5)

(x(u, t + 1) − x(u, t)) · Pf cs (u) ≤ Amax · ∆t

u=1

(3)

FCS has slow dynamics due to the inertia of auxiliaries. For this reason, a threshold on increase power is introduced by the constraint (4) where Amax depends on the FCS size. dPf cs (t) ≤ Amax (Pfmax cs ) dt

N �

405

(13)

where  1  2 min  Ecell = · Ccell · Umin 2 1  max 2  Ecell = · Ccell · Umax 2

(14)

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To insure the continuity of the cycle, the energy balance is assumed to be null on the complete cycle from t0 to tf and described by (15). Ecell (t0 ) = Ecell (tf )

(15)

The equation (16) describes the electrical power supplied by the supercapacitor bank Psc [W] which depends on the FCS control Pf cs (u) and the required power Preq (t) from the power balance equation (1). (16) Psc (u, t) = Preq (t) − Pf cs (u) The internal cell power Pcell for each time t and control u is determined from the equation (17). The power losses are estimated from Psc (u, t). 1 · [Psc (u, t) + Ploss (u, t)] (17) Pcell (u, t) = ηconv (u, t) · Nsc The internal cell power for each time t is obtained by the sum function presented in (18). Pcell (t) =

N 

x(u, t) · Pcell (u, t)

(18)

u=1

Power cell boundaries constraints in (10) are replaced by the new discretization in (19). min ≤ Pcell

N 

max x(u, t) · Pcell (u, t) ≤ Pcell

(19)

u=1

φ = Pth

1 = t f − t0

N τ  

x(u, t) · Pcell (u, t) · ∆t (20)

t=1 u=1

For a complete cycle, the energy variation is null, so this condition is obtained by (21).

tf

2 Pth (t)dt = Resr · Irms

(22)

t0

where Irms [A] is the root mean square (RMS) current through the cell. This value is estimated from the cell current Icell [A] with the equation (23).   tf  1  2 (τ )dτ Icell (23) Irms =  (tf − t0 ) t0

The thermal conduction resistance value is linked to the cell material, whereas thermal convection resistance value is linked to the cooling flow and the geometry of the cell. The heat equation (24) describes the evolution of the temperature in the cell, Incropera (2011). Cth · Rth

The energy evolution is represented as a sum of accumulated energy evaluated from the internal cell power Pcell . Ecell (τ ) = Ecell (t0 ) −

where φ [W] is the total amount of heat energy per time unit dissipated by the cell, Cth [J/kg] is the thermal capacity of the cell, Rcond and Rconv [K/W] are respectively the thermal resistance by conduction and convection phenomenon. θc , θh and θa [◦ C] are respectively the core temperature, the temperature of the case and the ambient temperature. For the complete cycle defined between t0 to tf , φ is deduced from the average heat power dissipated by the cell.

dθc (t) = Rth · Pth (t) + θa − θc (t) dt

(24)

where Rth [K/W] is the sum of thermal resistances Rcond and Rconv . Pth (t) represents the average heat power from initial time t0 to t. At steady-state, the temperature in the cell θc [◦ C] reaches its maximal temperature θcmax [◦ C]. φ=

θcmax − θa Rth

N T  

(25)

The supercapacitor bank contains a set of similar superx(u, t) · Pcell (u, t) · ∆t = 0(21) capacitor cells. Cells are placed into an enclosed box as t=1 u=1 shown on figure 5. A column contains 6 supercapacitor cells. Then columns are disposed into staggered rows. A 4.3 Supercapacitor thermal model fan supplies an air flow at constant speed Vair [m/s] and constant ambient temperature θa . Supercapacitors are thermal sensitive components due to the high capabilities of current through the cell. Thermal models have been proposed by Al Sakka et al. (2009) in order to evaluate the impact of temperature on cell degradation. Most of studies are focused on a single supercapacitor cell as Gualous et al. (2011). On figure 4, the thermal model is represented as an equivalent electrical circuit. Ecell (tf ) = Ecell (t0 ) ⇒

Fig. 4. Equivalent circuit for supercapacitor thermal model 406

Fig. 5. Supercapacitor bank with forced cooling

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At the first column, the ambient temperature is θa . The airflow transfers a part of dissipated heat energy by convection phenomenon. The ambient air temperature after the first column increases from θa to θa + ǫ due to the heat mass transfer. For each column, the air temperature increases and the temperature difference is similar (Rth and φ are constant), so the core temperature increases. A thermal network model validated by experiments has been developed by Hijazi et al. (2012) to take into account this phenomenon. Hot spots are detected inside the box. Each supercapacitor cell is assumed to have a similar control on energy management strategy. A numerical model can estimate the maximal core temperature θcmax reached in the bank according to several parameters as cell configuration, initial airflow temperature, airflow velocity and RMS current. The figure 6 shows the maximal cell temperature with the RMS current and for different air ambient temperature in a 6 × 10 supercapacitors module. 200 θ =10°C a

θa=20°C

θmax c

150

θa=30°C

100

θa=40°C

50 0

20

40

60

80

I

100 [A]

120

140

160

180

200

rms

Fig. 6. Maximal core temperature with Irms for different air ambient temperatures and Vair =0.23m/s, Nsc = 6 × 10 cells

main physical factors in the degradation effect of the cell. But the current is an other important parameter which accelerates the effect. In this paper, the ageing model is based on Eyring law taking into account the voltage, the core temperature and the RMS current Kreczanik et al. (2014). Then a combinatorial model is developed for our optimization problem. For a supercapacitor cell, the End-of-Life (EoL) is considered as reached when Resr increases two times compared 0 to its initial value Resr , or for a decrease of 20% for initial 0 capacitance Ccell , MaxwellTechnologies (2015). Kreczanik et al. (2014) have realized ageing experiments on similar supercapacitor cells, BCAP3000 model from Maxwell Technologies. This component has an initial ca0 pacitance Ccell of 3000F, a rated voltage of 2.7V and an initial ESR of 0.27mΩ. The degradation behaviour is linked to two main phenomenon: the calendar effect and the power cycling effect. ¯cell and The calendar effect is due to the average voltage U the core temperature θc . The Eyring law describing the velocity of a chemical reaction is shown in (27).   ¯ ¯cell , θc , Irms ) = τ0 · exp − Ucell − θc − Irms (27) τd (U U0 θ0 I0 ¯cell [V] is where τd [day] is the supercapacitor lifetime, U the average voltage across the terminal cell, θc [◦ C] is the core temperature in the cell, and Irms is the RMS current through the cell for the studied cycle. The effect of the voltage and the core temperature on the lifetime τd is represented on the figure 7. The cycling effect is represented on the figure 8.

This thermal phenomenon is modeled with a polynomial function as described by the following equation = θa + a 1 ·

2 Irms

5

θ =30°C

10

c

θ =40°C

(26)

where a1 is a parameter which depends on the stack configuration (Nsc ) and the airflow inlet (θa , Vair ).

c

θ =50°C c

4

θc=60°C

10 τd [day]

θcmax

407

3

10

4.4 Supercapacitor ageing law

407

1.8

2

2.2 ¯cell [V] U

2.4

2.6

Fig. 7. Voltage and core temperature effects on supercapacitor lifetime 9

10

8

10

d

Arrhenius law describes the velocity of a chemical reaction according to the temperature of the system. This law is used to estimate the reliability and the lifetime of devices, Kreczanik et al. (2009); German et al. (2014). In oxydoreduction reactions, the voltage across the terminal of the electrode is a decisive factor in the velocity of the reaction. Eyring law is an extension of Arrhenius law for several physical independant parameters, German et al. (2014). Most of studies are focused on a calendar ageing effect where the core temperature and the rated voltage are

2

10 1.6

τ [day]

Supercapacitors cells are devices which accumulate charges between an electrode and an electrolyte. The electrostatic storage induces chemical reaction (oxydo-reduction) which affects internal components and changes technical characteristics of electrolyte and electrodes during the power cycling. Two phenomenon are observed: the capacitance decreases and the ESR increases. The consequences are a decrease of the total energy which could be stored in the cell and an increase of ESR value which have a negative effect on heat losses and efficiency, Gualous et al. (2010).

7

10

6

10

0

20

40

60

80

100 Irms [A]

120

140

160

180

Fig. 8. Cycling effect on the supercapacitor lifetime

200

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For the studied supercapacitor (BCAP 3000F Maxwell Technologies), following data are used 30 I0 = ln(2)

3

In the optimization problem, the supercapacitor lifetime τd [day] of the more constrained cell needs to be superior to the working machine lifetime L [h] in order to respect the reliability condition. � � ¯ L θcmax Irms Ucell (30) ≥ τ0 · exp − − − U0 θ0 I0 24 The relation (26) about the maximal core temperature θcmax is included in (30) and with the discrete expression of Irms . The constraints leads to solve the inequation (31)

  α=      β=      γ =

a1 θ0 1 I0 � � ¯cell θa U L + + ln θ0 24 · τ0 U0

(31)

req

[W]

It means that Irms is a sum of square current values Icell (u, t) as shown on the equation (23). � � N T � �1 � � 2 (u, t) x(u, t) · Icell (29) Irms = T t=1 u=1

with

2

P

In this problem, it is assumed that the core temperature θc is constant in a steady-state. The study is focused on the more constrained cell, where the maximal temperature θcmax is reached in the stack. In the combinatorial model, the RMS current is deduced from the internal cell power ¯cell if voltage fluctuations Pcell and the average voltage U are neglected. ¯cell Icell (u, t) = Pcell (u, t)/U (28)

2 + β · Irms + γ ≤ 0 α · Irms

4

1 0

−1 −2 10

20

30

40

50

60 70 Time [s]

80

90

100

110

120

Fig. 9. Required power cycle supplied by the HPSS 5.2 Cost function The objective function to minimize represents the Total Cost of Ownership (TCO) of the HPSS. This cost includes the purchase component cost J1 , the fuel cost J2 and the maintenance cost J3 . This cost is calculated for the total lifetime L where the machine is run. TCO depends on the maximum FCS power Pfmax cs , the number of SC cells Nsc , and the energy management strategy applied on the HPSS Pf cs (t). JT CO = J1 + J2 + J3

(35)

The purchase cost J1 depends on components sizing. max J1 = f1 (Pfmax cs ) + Nsc · E0 · f2 (Nsc ) + ·f3 (Nsc · Psc0 ) (36) max where E0 is the energy stored in one cell, Psc0 is the maximum power delivered by a cell, f1 , f2 and f3 represent respectively the FCS cost function, the supercapacitor cost function and the power converter cost function. This nonlinear functions are represented on figures 10, 11 and 12. Cost functions are estimated from different manufacturers data. 4

(32)

RMS current is positive (Irms ≥ 0) and the function from (31) is monotonically increasing. The analytical solution of this inequation gives the following constraint (33). � −β + β 2 − 4 · α · γ (33) Irms ≤ 2·α The combinatorial constraint is represented on the equation (34). �2 � � N T 2−4·α·γ 1 �� β −β + 2 (34) x(u, t) · Icell (u, t) ≤ T t=1 u=1 2·α 5. COMBINATORIAL FORMULATION

6 FCS cost [euro]

10 θ0 = ln(2)

x 10

Piece−wise linear function Approximated polynomial function

4

2

0

1

3

P

4 [W]

5

6

7 4

x 10

Fig. 10. FCS cost function 11000 10000 9000 8000 7000 6000

5.1 Operating cycle

2

fcs

Cost [euro/kWh]

τ0 = 1.6 × 108

0.2 U0 = ln(2)

4

x 10

50

100

150

200 N

250

300

350

400

sc

The power cycle Preq represents the total electrical power used by actuators and auxiliaries to run a representative power cycle. This cycle is represented on figure 9. This cycle is assumed to be used and repeated during the whole working machine lifetime L. 408

Fig. 11. Supercapacitor cost per energy unit with the number of cells in the SC bank and including BMS J2 =

T � N � Pf cs (u) cH2 · L · · ∆t (37) x(u, t) · QH2 · (tf − t0 ) t=1 u=1 ηf cs (u)

Cost [euro]

IFAC E-COSM 2015 August 23-26, 2015. Columbus, OH, USA Alan Chauvin et al. / IFAC-PapersOnLine 48-15 (2015) 403–410

409

8000

6.2 Impact of ESR degradation

6000

Several scenarios are presented for several supercapacitor state-of-health and air ambient temperature. At first, the optimization problem is solved taking into account different supercapacitor state-of-health where the Resr increases. At initial time, SoHsc = 100% and there is no degradation of ESR. At the end-of-life, SoHsc = 0% and ESR value reaches two times the initial ESR value. Wear scenarios are presented on table 4.

4000 2000 0

0.5

1

1.5

2

2.5 3 Power [W]

3.5

4

4.5

5 4

x 10

Fig. 12. Cost function for the DC/DC converter

Table 4. Supercapacitor wear scenarios

where cH2 [euro/kg] represents the hydrogen cost and QH2 [J/kg] is the energy fuel content of the hydrogen. J3 = h1 (L, Pfmax cs )

SoHsc Resr

100% R0esr

50% 1.5 · R0esr

0% 2 · R0esr

(38)

where h1 is a non linear-function which represents FCS maintenance cost. 6. SIMULATION AND RESULTS 6.1 Technical data The problem is solved following some predefined data and discretization parameters. Main data are presented in table 1 and 2. Table 1. Discretization parameters FCS sizing FCS range FCS power step

Compared to the solution without ageing constraints, there is a gap of 2% at initial state (SoHsc = 100%). This is due to the ageing constraint. The degradation of ESR induces an increase of supercapacitor cells in order to limit the maximal core temperature in the stack. θcmax is around 50◦ C for all SoH conditions.

10-20kW 0.5kW

SC sizing SC range 30-150 cells SC cells step 6 cells

The difference on the TCO between the initial stateof-health and the end-of-life is 2.5%. The purchase cost increase is counterbalanced by hydrogen fuel economies due to a more precise energy management strategy.

Other parameters FCS control N 30 pts Time discretization T 120 (∆t = 1s)

Table 2. Supercapacitor data SC model 0 Initial capacitance Ccell Initial ESR R0esr Ucell (t0 ) max Ucell min Ucell ηconv L

6.3 Impact of ambient temperature θa

BCAP3000 3000F 0.27mΩ 2.5V 2.7V 1.25V 0.95 if Psc ≥ 0, else 1/0.95 6000h

At first, the initial configuration is presented in table 3. This solution represents the optimal configuration of the HPSS without ageing model. It is assumed that there is no influence from the ambient temperature or the RMS current. The capacity loss Ccell is not considered in this study. Table 3. Initial optimal solution Pfmax cs Nsc EH2 TCO max θC

Fig. 13. Impact of state-of-health on optimal solution

11.5kW 54 cells 2.706MJ 54 680 euro 61◦ C

Some parameters as the supercapacitor cells disposal configuration, the airflow volume or the ambient temperature have an influence on the evolution of the maximal temperature in a stack. A second scenario is studied for different air ambient temperature θa at steady-state. The model of supercapacitor bank is based on the configuration and results from Hijazi et al. (2012). The temperature gap ∆θ is defined as the difference between the hot point θcmax and the ambient temperature θa . Table 5. Ambient air temperature scenarios Climate conditions Cold Normal Hot Extremely hot

θa 10◦ C 20◦ C 30◦ C 40◦ C

The air ambient temperature has a major impact for the sizing and the control of the hybrid system. For cold climate conditions (θa = 10◦ C), the optimal solution is similar to the no ageing model with high temperature gap (∆θ ≈ 34◦ C) and high RMS current (Irms =81A). For 409

IFAC E-COSM 2015 410 August 23-26, 2015. Columbus, OH, USA Alan Chauvin et al. / IFAC-PapersOnLine 48-15 (2015) 403–410

hot climate conditions, the capacitor stack is oversized in order to limit the temperature gap (∆θ ≤13◦ C) and RMS current (Irms ≤56A). The TCO deviation reaches 10% compared to extremum climate conditions.

Fig. 14. Impact of air ambient temperature on optimal solution 7. CONCLUSION AND FUTURE OUTLOOKS In this paper, an optimization problem of a hybrid electric power supply system for a mini-excavator has been studied. The aim of this problem is to minimize the global cost (TCO) by optimizing fuel cell and supercapacitor bank sizing and optimal energy management of the fuel cell following a predefined cycle. The problem includes ageing model for supercapacitors bank. The optimization approach is based on the reformulation of initial non linear problem into a multitude of integer linear sub-problems. The optimal solution is solved with a Branch & Cut algorithm and a smart iterative strategy. Results show that ageing effect has an effect on components sizing for critical scenario as ESR degradation and climate conditions. In the future, a model including a double buffer as a battery and a supercapacitor pack with their own ageing model will be developed and studied for a similar optimization problem. REFERENCES Al Sakka, M., Gualous, H., Van Mierlo, J., and Culcu, H. (2009). Thermal modeling and heat management of supercapacitor modules for vehicle applications. Journal of Power Sources, 194(2), 581–587. Chan, C. (2007). The state of the art of electric, hybrid, and fuel cell vehicles. Proceedings of the IEEE, 95(4), 704–718. Gaoua, Y., Caux, S., and Lopez, P. (2013). A combinatorial optimization approach for the electrical energy management in a multi-source system. In SciTePress (ed.), 2nd International Conference on Operations Research and Enterprise Systems (ICORES 2013), 55–59. German, R., Sari, A., Venet, P., Zitouni, Y., Briat, O., and Vinassa, J.M. (2014). Ageing law for supercapacitors floating ageing. In Industrial Electronics (ISIE), 2014 IEEE 23rd International Symposium on, 1773–1777. IEEE. Gomory, R.E. (1958). Outline of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Society, 64(5), 275–278. 410

Gualous, H., Gallay, R., Alcicek, G., Tala-Ighil, B., Oukaour, A., Boudart, B., and Makany, P. (2010). Supercapacitor ageing at constant temperature and constant voltage and thermal shock. Microelectronics Reliability, 50(9), 1783–1788. Gualous, H., Louahlia, H., and Gallay, R. (2011). Supercapacitor characterization and thermal modelling with reversible and irreversible heat effect. Power Electronics, IEEE Transactions on, 26(11), 3402–3409. Hegazy, O. and Van Mierlo, J. (2010). Particle swarm optimization for optimal powertrain component sizing and design of fuel cell hybrid electric vehicle. In Optimization of Electrical and Electronic Equipment (OPTIM), 2010 12th International Conference on, 601– 609. IEEE. Hijazi, A., Kreczanik, P., Bideaux, E., Venet, P., Clerc, G., and Di Loreto, M. (2012). Thermal network model of supercapacitors stack. Industrial Electronics, IEEE Transactions on, 59(2), 979–987. Incropera, F.P. (2011). Fundamentals of heat and mass transfer. John Wiley & Sons. Khaligh, A. and Li, Z. (2010). Battery, ultracapacitor, fuel cell, and hybrid energy storage systems for electric, hybrid electric, fuel cell, and plug-in hybrid electric vehicles: State of the art. Vehicular Technology, IEEE Transactions on, 59(6), 2806–2814. Kreczanik, P., Martin, C., Venet, P., Clerc, G., Rojat, G., and Zitouni, Y. (2009). Constant power cycling for accelerated ageing of supercapacitors. In Power Electronics and Applications, 2009. EPE’09. 13th European Conference on, 1–10. IEEE. Kreczanik, P., Venet, P., Hijazi, A., and Clerc, G. (2014). Study of supercapacitor aging and lifetime estimation according to voltage, temperature, and rms current. Industrial Electronics, IEEE Transactions on, 61(9), 4895–4902. Li, D.G., Zou, Y., Hu, X.S., and Sun, F.C. (2012). Optimal sizing and control strategy design for heavy hybrid electric truck. In Vehicle Power and Propulsion Conference, 1100–1106. IEEE. Linderoth, J.T. and Ralphs, T.K. (2005). Noncommercial software for mixed-integer linear programming. Integer programming: theory and practice, 3, 253–303. MaxwellTechnologies (2015). Application note, life duration estimation. URL http://www.maxwell.com. Moore, T.C. (1997). Hev control strategy: Implications of performance criteria, system configuration and design, and component selection. In American Control Conference, 1997. Proceedings of the 1997, volume 1, 679–683. IEEE. Murgovski, N., Johannesson, L., Hellgren, J., Egardt, B., and Sj¨oberg, J. (2011). Convex optimization of charging infrastructure design and component sizing of a plug-in series hev powertrain. In World Congress, volume 18, 13052–13057. Silvas, E., Bergshoeff, E., Hofman, T., and Steinbuch, M. (2014). Comparison of bi-level optimization frameworks for sizing and control of a hybrid electric vehicle. In Vehicle Power Propulsion Conference (VPPC), 2014 IEEE. Winston, W.L. and Goldberg, J.B. (1994). Operations research: applications and algorithms. Duxbury press Belmont, CA.