Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 538237 7 pageshttpdxdoiorg1011552013538237

Research ArticleNeural Network Predictive Control forVanadium Redox Flow Battery

Hai-Feng Shen1 Xin-Jian Zhu2 Meng Shao2 and Hong-fei Cao2

1 Automation Department School of Electronic Information and Electrical Engineering Shanghai Jiao Tong UniversityShanghai 200240 China

2 Institute of Fuel Cell School of Mechanical Engineering Shanghai Jiao Tong University Shanghai 200240 China

Correspondence should be addressed to Hai-Feng Shen hfshen999sjtueducn

Received 6 July 2013 Revised 23 September 2013 Accepted 25 September 2013

Academic Editor Baocang Ding

Copyright copy 2013 Hai-Feng Shen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The vanadium redox flow battery (VRB) is a nonlinear system with unknown dynamics and disturbances The flowrate of the elec-trolyte is an important control mechanism in the operation of a VRB system Too low or too high flowrate is unfavorable for thesafety and performance of VRBThis paper presents a neural network predictive control scheme to enhance the overall performanceof the battery A radial basis function (RBF) network is employed to approximate the dynamics of the VRB system The geneticalgorithm (GA) is used to obtain the optimum initial values of the RBF network parametersThe gradient descent algorithm is usedto optimize the objective function of the predictive controller Compared with the constant flowrate the simulation results showthat the flowrate optimized by neural network predictive controller can increase the power delivered by the battery during thedischarge and decrease the power consumed during the charge

1 Introduction

Because of the energy crisis utilization of renewable energysources such as wind and solar energy for electric power sup-ply has received more and more attention in recent yearsHowever the intermittent nature of most renewable energymakes it highly dependent on reliable and economical energystorage systems All-vanadium redox flow battery (VRB) is apromising candidate for the storage of renewable energyCompared with other redox batteries such as zinc brominebattery and lead acid battery VRB has many attractive fea-tures including long cycle life high energy conversion effi-ciency flexible design and low cost [1] Moreover the pro-blem of electrolytes cross-contamination is avoided by usingthe same element in both half cellsThe potential applicationsof VRB include load leveling uninterruptible power supply(UPS) and renewable energy storage [2] Thus it has goodapplication and development prospects

The flowrate of the electrolyte is an important controlmechanism in the operation of a vanadium redox flowbattery

system At low flowrates the electrolyte is provided insuf-ficiently for the chemical reaction and stagnant regions canform in the electrode The higher electrolyte flowrate will in-crease the VRB performance But on the other hand if theflowrate is too high there is a risk of leakage and the pumpconsumption will increase which will reduce the system effi-ciency [3 4] In order to enhance system efficiency the opti-mal electrolyte flow rate should be determined

Until recently most researches are focused on the keymaterials of VRB and there is little information available inthe literature about the optimization of the electrolyte flow-rate An optimal strategy of electrolyte flowrate is proposedin [3] to improve the system efficiency and keep the highcapacity simultaneously At the beginning of the chargedischarge process VRB operates at the lower flowrate andthen increases to higher flowrate when the voltage increasesdecrease to certain value Energy efficiency system efficiencyand capacity at different operating modes are compared andthe optimal electrolyte flowrate is determined Amultiphysicsmodel of the VRB is proposed in [5] The battery power is

2 Journal of Applied Mathematics

represented during the chargedischarge as a function of flowrate states of charge (SOC) and the stack current The opti-mal flow rates are obtained by maximizing the power deliv-ered during the discharge and minimizing the power con-sumed during the charge However these optimal strategiessuffer from a serious drawback in the form of deterioration inthe performance when the system is operated under widerange operating conditions or subjected to disturbance Toovercome these drawbacks controllers based on robust con-trol techniques must have been used

Model predictive control (MPC) is an application of opti-mal control theory In model predictive control processmodel is utilized to predict the future response of a plant Anoptimal control sequence is determined by solving a finitehorizon optimization problem online at each sampling in-stant and the first control in this sequence is applied to theplant [6] Because of its ability to handle the multivariablenonlinear nature of the dynamics constraints and optimalityin an integrated fashion [7] MPC technology can now befound in a wide variety of application areas including chemi-cals food processing automotive and aerospace applications[8] The performance of model predictive controller reliesupon the accuracy of the model on which it is based How-ever the VRB suffers aging reactant crossover and load dis-turbance that cause no well-known effects on the system dy-namics it is difficult to establish accurate mathematicalmodel Moreover the mathematical model is too complex foronline optimization and a simpler model is therefore re-quired An attractive approach to tackle these problems is touse neural networks as nonlinear models of the dynamic be-havior of the process [9] This is because multilayer networkshave a capability to learn and uniformly approximate nonlin-ear functions to a prospected accuracy [10]

In this paper a nonlinear model predictive controlscheme is proposed to maximize the power delivered by thebattery during the discharge and minimize the power con-sumed during the charge

2 VRB System Process Description

TheVRB system consisted of two key elements the cell stackwhere electrochemical reaction occurred and the tanks ofelectrolytes where energy is stored The electrolytes werepumped from the tanks to the stack by a circulation systemA schematic diagram of a vanadium redox flow batter is givenin Figure 1

The main electrode reactions for the VRB are as follows

cathode V3+ + eminus 999448999471 V2+ (1)

anode VO2+ + H2O 999448999471 VO+

2+ eminus + 2H+ (2)

A multiphysics model of a VRB system with 19 cells isintroduced in [11] which is composed of the electrochemicalmodel and the mechanical model

Generator

Load

V5+V4+V5+

V4+

electrolyte

ACDC

tank

V3+

electrolytetank

PumpPump

ee

Stack

V2+V3+

V2+

Figure 1 A schematic diagram of a vanadium redox flow battery

21 Electrochemical Model The equilibrium potential of theindividual cells can be approximated using the Nernst equa-tion (assuming unit activity coefficients) as follows

119864cell = 1198640+

119877119879

119865

ln(

119862V2+119862V5+1198622

H+

119862V3+119862V4+) (3)

where 1198640 is the standard potential 119879 is the cell temperature

119862119894is the molar concentration of species 119894 in the cells For sim-

plicity they assuming that the concentration inside the celland tank is uniform and the time delay of electrolyte flow isnegligible the concentration inside the cell and tank is givenby [12]

119889119862119894

119889119905

=

119887119868 (119905)

119881cell119865+

119876 (119905)

119881cell(119862tank

119894

minus 119862119894)

119889119862tank119894

119889119905

=

1

119881tank(minus119881cell

119889119862119894

119889119905

+

119887119868 (119905)

119865

)

(4)

where 119862tank119894

is the concentration inside the tank 119881cell is thevolume of the cell 119881tank is the volume of the tank 119868(119905) is thecurrent 119876(119905) is the electrolyte flowrate and 119887 is a sign factorthat depends on the considered vanadium species 119894 (minus1 forV2+ and V5+ ions and 1 for V3+ and V4+ ions)

The H+ quantity in the catholyte increases by 1M (afterthe migration) when 1M of vanadium V5+ is produced Sothe H+ concentration in the catholyte at any state of charge is

119862H+ = 119862H+ discharged + 119862VO+2

(5)

where 119862H+discharged is the protons concentration when theelectrolyte is completely discharged

Journal of Applied Mathematics 3

Assuming that each individual cell composing the stackhas the same charging characteristics the equilibrium voltage119880eq of stack can be written as follows

119880eq = 119864cell sdot 119873cell (6)

where 119873cell is the number of cellsThe stack voltage 119880stack is decreased when current flows

through the stack because of several types of internal lossessuch as activation concentration and Ohmic losses Butthese internal losses are difficult to measure here we replacethem with equivalent resistance 119877eq chdisch

119880loss = 120578act + 120578conc + 120578ohm = 119868 sdot 119877eq chdisch (7)

So stack voltage 119880stack is given by

119880stack = 119880eq minus 119880loss (8)

Then the power of stack can be calculated as

119875stake = 119880stack sdot 119868 (9)

22 Mechanical Model The circulation system pumps theelectrolytes from the tanks through the stack and back in thetanksThe power consumed by pumps is expressed as follows

119875mech = 2

(Δ119875pipes + Δ119875stack)119876 (119905)

120578pump (10)

where 120578pumps is the pump efficiency Δ119875pipes is the pressuredrop in the pipes which can be obtained from the extendedBernoulli equation The pressure drop in the stack Δ119875stack isproportional to the flowrate 119876(119905)

Δ119875stack = 119876 (119905)119877 (11)

where 119877 is the hydraulic resistance obtained from FEM sim-

ulations [13]

23 Battery Power In practice 119875mech is provided from theexternal power source during the charge and from the stackduring the discharge [5] By convention the stack current isdefined as positive during the discharge and negative duringthe charge Thus the battery power 119875VRB is given by

119875VRB = 119875stake minus 119875mech (12)

3 Design of Nonlinear ModelPredictive Controllers

The schematic of the neural network predictive control(NNPC) system developed in this research is shown inFigure 2 The main steps of the NNPC algorithm are listed asfollows

(1) Measure the input and output of the VRB system(2) Use the previous calculated control inputs and the

neural network identifier to compute the cost func-tion

yp

yu

ulowast

Optimizer NN model

VRBsystem

Figure 2 Schematics of the NNPC system

(3) Use the optimization algorithm to calculate a newcontrol vector

(4) Repeat steps (2) and (3) till the desired optimal resultis achieved

(5) Apply the first element of the control vector to theVRB system

(6) Update the parameters of the NN with the new train-ing set

(7) Repeat steps (1)ndash(6) for each time step

31 Predictive Model Based on RBF Neural Network Accord-ing to previous section the battery power can be expressed asfollows

119875VRB = 119892 (119876 119868 119879 119905) (13)

Suppose the stack current and temperature keep constant fora certain amount of time So there is only one control vari-able the flowrate119876 The following NARXmodel can be usedto represent the VRB system

119910 (119905) = 119891 (119910 (119905 minus 1) 119910 (119905 minus 2) 119906 (119905 minus 1) 119906 (119905 minus 2)) (14)

where 119910 is the battery power 119906 is the flowrate and 119891(sdot) is anunknown nonlinear function that needs to be identified Ra-dial basis function (RBF) networks having one hidden layerwere proven to be universal approximator [14] Because of theadvantages of easy design and good generalization a RBFnet-work is used to identify the nonlinear function119891(sdot) in this pa-per The structure of the RBF network is shown in Figure 3

A Gaussian function is used as the activation function Soat the hidden layer the output of RBF unit 119894 is

120593119894(119909) = exp(minus

1003817100381710038171003817119909 minus 119888119894

1003817100381710038171003817

2

21205902

119894

) (119894 = 1 2 5) (15)

where 119909(119905) = [119910(119905 minus 1) 119910(119905 minus 2) 119906(119905 minus 1) 119906(119905 minus 2)]119879 is the

input of RBF network 119888119894and 120590119894are the center andwidth of the

119894th unit respectively

4 Journal of Applied Mathematics

x1

x2

x3

x4

w1

w2

w3

w4

w5

120593

120593

120593

120593

120593

Σyp

Input layerwith 4nodes

Hidden layerwith 5nodes

Output layerwith 1node

Figure 3 The structure of the RBF network

The network output is calculated by

119910 =

5

sum

119894=1

119908119894120593119894(119909) (16)

where 119908119894is the weight value on the connection between RBF

unit 119894 and network output The one-step ahead prediction isgiven by

119910 (119905 + 1) = 119891119873119873

(119910 (119905) 119910 (119905 minus 1) 119906 (119905) 119906 (119905 minus 1)) (17)

The 119895-step ahead prediction of the systemrsquos output is calcu-lated by feeding back themodel outputs (instead of the futuresystemrsquos outputs which do not exist) to the input nodes of thenetwork [15]

Consider the following

119910 (119905 + 119895) = 119891119873119873

(119910 (119905 + 119895 minus 1) 119910 (119905 + 119895 minus 2)

119906 (119905 + 119895 minus 1) 119906 (119905 + 119895 minus 2))

(18)

The computational burden of the optimization problemshowed in next subsection increases with the complexity ofRBF network structure In order to simplify the RBF networkstructure and simultaneously ensure the approximation accu-racy in this study genetic algorithm (GA) is adopted to ob-tain the optimum initial values of the RBF network parame-ters before training the RBF network These parameters in-clude the output weights the centers and widths of the hid-den unit

32 The Objective Function Optimization Algorithm Thereare different forms of the objective function under differentcontrol requirements In this study our purpose is to maxi-mize the power delivered by the battery during the discharge

and minimize the power consumed during the charge whileensuring the control signal is smooth Noticing that the bat-tery power is positive during the discharge and negative dur-ing the charge the objective function is given as follows

min 119869(119905) = minus

119899

sum

119895=1

119910 (119905 + 119895) +

1

2

119898

sum

119894=1

120582Δ1199062(119905 + 119894 minus 1) (19)

subject to constraints

119906min le 119906 (119905 + 119894 minus 1) le 119906max (119894 = 1 2 119898)

119910min le 119910 (119905 + 119895) le 119910max (119895 = 1 2 119899)

(20)

where Δ119906(119905 + 119894 minus 1) = 119906(119905 + 119894 minus 1) minus 119906(119905 + 119894 minus 2) 120582 gt 0 is weightcoefficient and 119899 and 119898 are the predictive horizon and con-trol horizon respectively The vector of the control variablesis obtained from the minimization of the objective functionover the specified horizonThe control vector is available onlywithin the control horizon andmaintains constant afterwardthat is 119906(119905+119894) = 119906(119905+119898minus1) for 119894 = 119898 119899minus1 Only the firstelement of the optimized control sequence is implemented onthe process

Since the function 120593 is nonlinear an analytical solution ofthe objective function is not possible Stochastic optimizationalgorithms such as genetic algorithm and simulated anneal-ing suffer from the drawback of slow convergence whichmake them not suitable for online control Since the objectivefunction surface is simple the gradient based method is anappropriate choice Based on the gradient based method fora given iterative step 119894 the control vector can be calculated asfollows

119906119896(119905 + 119894 minus 1) = 119906

119896minus1(119905 + 119894 minus 1) + Δ119906

119896(119905 + 119894 minus 1)

(119894 = 1 2 119898)

Δ119906119896(119905 + 119894 minus 1) = minus120578

120597119869

120597119906 (119905 + 119894 minus 1)

+ 120572Δ119906119896minus1

(119905 + 119894 minus 1)

(21)

where 120578 is the learning rate and 120572Δ119906119896minus1

(119905+ 119894minus1) is referred toas the additional momentum term The initial value of 119906(119905 +

119894 minus 1) in the iteration at each sampling period is defined as

1199060(119905 + 119894 minus 1) = 119906 (119905 minus 1) (22)

Constraints on control sequence can be handled as followswhen any one of the 119906(119905+119894) reaches its limit this control inputis then set to be equal to its limit [16]

The derivative of the objective function at time 119905 + ℎ minus 1

ℎ = 1 2 119898 can be written as follows

120597119869

120597119906 (119905 + ℎ minus 1)

= minus

119899

sum

119895=1

120597119910 (119905 + 119895)

120597119906 (119905 + ℎ minus 1)

+

119898

sum

119894=0

120582Δ119906 (119905 + 119894)

120597Δ119906 (119905 + 119894)

120597119906 (119905 + ℎ minus 1)

(23)

Journal of Applied Mathematics 5

Table 1 The characteristics of the VRB stack

Name ValueNumber of cells 119873cells 19119877charge 0037Ω

119877discharge 0039Ω

Electrolyte vanadium concentration 2molLInitial 119867+ concentration 5molLTank size 119881tk 83 LFlow resistance

119877 14186843 Pam3

Cell temperature 119879 298KStandard potential 1198640 1255V

The partial derivative can be calculated by the chain rule

120597119910 (119905 + 119895)

120597119906 (119905 + ℎ minus 1)

=

120597119891119873119873

1205971199093

119895 = ℎ

120597119891119873119873

1205971199091

1205971199091

120597119906 (119905 + ℎ minus 1)

+

120597119891119873119873

1205971199094

119895 = ℎ + 1

120597119891119873119873

1205971199091

1205971199091

120597119906 (119905 + ℎ minus 1)

+

120597119891119873119873

1205971199092

1205971199092

120597119906 (119905 + ℎ minus 1)

ℎ + 2 le 119895 le 119899

(24)

where 119909(119905 + 119895) = [119910(119905 + 119895 minus 1) 119910(119905 + 119895 minus 2) 119906(119905 + 119895 minus 1)

119906(119905 + 119895 minus 2)]119879 is the input vector at time 119905 + 119895

120597119891119873119873

120597119909119897

=

5

sum

119894=1

119908119894exp(minus

1003817100381710038171003817119909 minus 119888119894

1003817100381710038171003817

2

21205902

119894

)

119888119894119897minus 119909119897

1205902

119894

119897 = 1 2 3 4

(25)

where 119909119897represents the network input vector of 119910 and 119906

120597Δ119906(119905 + 119894)120597119906(119905 + ℎ minus 1) can be given by

120597Δ119906 (119905 + 119894 minus 1)

120597119906 (119905 + ℎ minus 1)

=

1 119894 = ℎ

minus1 119894 = ℎ + 1

0 else(26)

4 Simulation

To investigate the performance of the proposed controller a19 cells 25 kW 6 kWh VRB is simulated Its main character-istics are listed in Table 1 [5]

41 Identification In order to reduce the online computingtime the RBF network was trained offline before being ap-plied to online control The multiphysics model developed inSection 2 was used for train data generation An input-outputdata set to train the RBF network was obtained by randomlychanging the manipulated variable 119876 within the range of005ndash07 and normalized between minus1 and +1 The sampling

0 1000 2000 3000 4000 5000 6000 7000 80001800

1900

2000

2100

2200

2300

2400

2500

Pow

er (w

)

Time (s)

Physical modelRBF model

Figure 4 Physical model and RBFmodel outputs for battery powerduring the discharge at 100A

time is set as 5 s 1026 samples were used for the trainingwhile 513 samples were used for validation The initial valuesof the RBF network parameters were optimized by GA Afterthe optimum initial values were obtained the Levenberg-Marquardt algorithmwas used as training algorithm to adjustthe network parameters Rootmean square error (RMSE)wasemployed to evaluate the accuracy of RBF network modelThe training was terminated after 500 iterations the obtainedvalue of RMSE is 16591 Figure 4 shows the validation resultsFrom the results of Figure 4 it can be observed that the RBFnetwork can accurately represent the VRB dynamics

The RBF network trained offline works well when thereare no disturbances However it can not accurately representthe VRB dynamics when VRB system is subjected to uncer-tainty So the RBF network requires to train online to adaptwith the change in the process Newest 100 samples were usedfor training

42 Control Results Normally in a charge-discharge cyclethe battery is charged at constant current the battery SoC in-creases from 25 (discharged) to 975 and then it is dis-charged at constant current until it reached its initial SoC [11]The predictive horizon and the control horizons for NMPCare chosen as 4 and 1 respectively The parameter 120582 is set to10000The lower limit and upper limit of flowrate are 005 Lsand 2 Ls respectively In normal working condition the bat-tery is chargeddischarged at constant current Assuming at119905 = 5000 s a disturbance on generator speed causes the chargecurrent to change from 100A to 95A and at 119905 = 12000 sa load disturbance causes the discharge current to changefrom 100A to 110A Figure 5 shows the battery power duringa charge-discharge cycle when influenced by a series of stepchanges in stack currentThe corresponding optimal flowratethat is shown in Figures 6 and 7 shows the comparison of bat-tery power during a charge-discharge cycle at different flow-rate Comparedwith the battery power at119876 = 03 the average

6 Journal of Applied Mathematics

0 5000 10000 150001500

2000

2500

3000

3500

Time (s)

Pow

er (w

)

Figure 5 Battery power during a charge-discharge cycle

0 5000 10000 150000

01

02

03

04

05

06

Time (s)

Flow

rate

(Ls

)

Figure 6 Optimal flowrate during a charge-discharge cycle

power consumed during the charge at optimal flowratedecreased by 1080W and the average power delivered by thebattery during the discharge increased by 1062W

5 Conclusions

The electrolyte flowrate of VRB system was optimized onlineusing model predictive control based on artificial neural net-works An RBF network is built to predict the future batterypower In order to reduce the computational burden of theoptimization problem the hidden layer nodes were chosen as5 The RBF network model was found to be valid for wideflowrate variation with random load disturbances The gra-dient descent algorithm method is used to realize the opti-mization procedure Simulation result at different flowrateindicates that the proposed controller can enhance the outputpower of battery during the discharge and reduce the operat-ing cost during the charge Future works will focus on controlstrategy for VRB and wind farm combined system

References

[1] K-L Huang X-G Li S-Q Liu N Tan and L-Q ChenldquoResearch progress of vanadium redox flow battery for energy

0 5000 10000 150000

1000

2000

3000

4000

5000

Time (s)

Pow

er (w

)

Q = 03

Q = 005

Q = 2

Q = Qopt

Figure 7 Comparison of battery power at different flowrate

storage in Chinardquo Renewable Energy vol 33 no 2 pp 186ndash1922008

[2] M Vynnycky ldquoAnalysis of a model for the operation of a vana-dium redox batteryrdquo Energy vol 36 no 4 pp 2242ndash2256 2011

[3] X Ma H Zhang C Sun Y Zou and T Zhang ldquoAn optimalstrategy of electrolyte flow rate for vanadium redox flow bat-teryrdquo Journal of Power Sources vol 203 pp 153ndash158 2012

[4] H Al-Fetlawi A A Shah and F C Walsh ldquoNon-isothermalmodelling of the all-vanadium redox flow batteryrdquo Electrochim-ica Acta vol 55 no 1 pp 78ndash89 2009

[5] C Blanc and A Rufer ldquoOptimization of the operating point of avanadium redox flow batteryrdquo in Proceedings of the IEEE EnergyConversion Congress and Exposition (ECCE rsquo09) pp 2600ndash2605San Jose Calif USA September 2009

[6] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[7] T Salsbury P Mhaskar and S Joe Qinc ldquoPredictive controlmethods to improve energy efficiency and reduce demand inbuildingsrdquoComputers and Chemical Engineering vol 51 pp 77ndash85 2013

[8] S J Qin and T A Badgwell ldquoA survey of industrial model pre-dictive control technologyrdquoControl Engineering Practice vol 11no 7 pp 733ndash764 2003

[9] A Draeger S Engell and H Ranke ldquoModel predictive controlusing neural networksrdquo IEEE Control SystemsMagazine vol 15no 5 pp 61ndash66 1995

[10] M T Hagan and H B Demuth ldquoNeural networks for controlrdquoin Proceedings of the American Control Conference (ACC rsquo99)pp 1642ndash1656 June 1999

[11] C Blanc and A Rufer ldquoMultiphysics and energetic modelingof a vanadium redox flow batteryrdquo in Proceedings of the IEEEInternational Conference on Sustainable Energy Technologies(ICSET rsquo08) pp 696ndash701 Singapore November 2008

[12] M Li and T Hikihara ldquoA coupled dynamical model of redoxflow battery based on chemical reaction fluid flow and electri-cal circuitrdquo IEICE Transactions on Fundamentals of ElectronicsCommunications and Computer Sciences A vol 91 no 7 pp1741ndash1747 2008

Journal of Applied Mathematics 7

[13] C BlancModeling of a VanadiumRedox Flow Battery ElectricityStorage System Ecole Polytechnique Federale de Lausanne2009

[14] J Park and I W Sandberg ldquoUniversal approximation usingradial-basis-function networksrdquoNeural Computation vol 3 no2 pp 246ndash257 1991

[15] P H Soslashrensen M Noslashrgaard O Ravn and N K Poulsen ldquoIm-plementation of neural network based non-linear predictivecontrolrdquo Neurocomputing vol 28 no 1ndash3 pp 37ndash51 1999

[16] C Venkateswarlu and K Venkat Rao ldquoDynamic recurrent ra-dial basis function networkmodel predictive control of unstablenonlinear processesrdquo Chemical Engineering Science vol 60 no23 pp 6718ndash6732 2005

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2 Journal of Applied Mathematics

represented during the chargedischarge as a function of flowrate states of charge (SOC) and the stack current The opti-mal flow rates are obtained by maximizing the power deliv-ered during the discharge and minimizing the power con-sumed during the charge However these optimal strategiessuffer from a serious drawback in the form of deterioration inthe performance when the system is operated under widerange operating conditions or subjected to disturbance Toovercome these drawbacks controllers based on robust con-trol techniques must have been used

Model predictive control (MPC) is an application of opti-mal control theory In model predictive control processmodel is utilized to predict the future response of a plant Anoptimal control sequence is determined by solving a finitehorizon optimization problem online at each sampling in-stant and the first control in this sequence is applied to theplant [6] Because of its ability to handle the multivariablenonlinear nature of the dynamics constraints and optimalityin an integrated fashion [7] MPC technology can now befound in a wide variety of application areas including chemi-cals food processing automotive and aerospace applications[8] The performance of model predictive controller reliesupon the accuracy of the model on which it is based How-ever the VRB suffers aging reactant crossover and load dis-turbance that cause no well-known effects on the system dy-namics it is difficult to establish accurate mathematicalmodel Moreover the mathematical model is too complex foronline optimization and a simpler model is therefore re-quired An attractive approach to tackle these problems is touse neural networks as nonlinear models of the dynamic be-havior of the process [9] This is because multilayer networkshave a capability to learn and uniformly approximate nonlin-ear functions to a prospected accuracy [10]

In this paper a nonlinear model predictive controlscheme is proposed to maximize the power delivered by thebattery during the discharge and minimize the power con-sumed during the charge

2 VRB System Process Description

TheVRB system consisted of two key elements the cell stackwhere electrochemical reaction occurred and the tanks ofelectrolytes where energy is stored The electrolytes werepumped from the tanks to the stack by a circulation systemA schematic diagram of a vanadium redox flow batter is givenin Figure 1

The main electrode reactions for the VRB are as follows

cathode V3+ + eminus 999448999471 V2+ (1)

anode VO2+ + H2O 999448999471 VO+

2+ eminus + 2H+ (2)

A multiphysics model of a VRB system with 19 cells isintroduced in [11] which is composed of the electrochemicalmodel and the mechanical model

Generator

Load

V5+V4+V5+

V4+

electrolyte

ACDC

tank

V3+

electrolytetank

PumpPump

ee

Stack

V2+V3+

V2+

Figure 1 A schematic diagram of a vanadium redox flow battery

21 Electrochemical Model The equilibrium potential of theindividual cells can be approximated using the Nernst equa-tion (assuming unit activity coefficients) as follows

119864cell = 1198640+

119877119879

119865

ln(

119862V2+119862V5+1198622

H+

119862V3+119862V4+) (3)

where 1198640 is the standard potential 119879 is the cell temperature

119862119894is the molar concentration of species 119894 in the cells For sim-

plicity they assuming that the concentration inside the celland tank is uniform and the time delay of electrolyte flow isnegligible the concentration inside the cell and tank is givenby [12]

119889119862119894

119889119905

=

119887119868 (119905)

119881cell119865+

119876 (119905)

119881cell(119862tank

119894

minus 119862119894)

119889119862tank119894

119889119905

=

1

119881tank(minus119881cell

119889119862119894

119889119905

+

119887119868 (119905)

119865

)

(4)

where 119862tank119894

is the concentration inside the tank 119881cell is thevolume of the cell 119881tank is the volume of the tank 119868(119905) is thecurrent 119876(119905) is the electrolyte flowrate and 119887 is a sign factorthat depends on the considered vanadium species 119894 (minus1 forV2+ and V5+ ions and 1 for V3+ and V4+ ions)

The H+ quantity in the catholyte increases by 1M (afterthe migration) when 1M of vanadium V5+ is produced Sothe H+ concentration in the catholyte at any state of charge is

119862H+ = 119862H+ discharged + 119862VO+2

(5)

where 119862H+discharged is the protons concentration when theelectrolyte is completely discharged

Journal of Applied Mathematics 3

Assuming that each individual cell composing the stackhas the same charging characteristics the equilibrium voltage119880eq of stack can be written as follows

119880eq = 119864cell sdot 119873cell (6)

where 119873cell is the number of cellsThe stack voltage 119880stack is decreased when current flows

through the stack because of several types of internal lossessuch as activation concentration and Ohmic losses Butthese internal losses are difficult to measure here we replacethem with equivalent resistance 119877eq chdisch

119880loss = 120578act + 120578conc + 120578ohm = 119868 sdot 119877eq chdisch (7)

So stack voltage 119880stack is given by

119880stack = 119880eq minus 119880loss (8)

Then the power of stack can be calculated as

119875stake = 119880stack sdot 119868 (9)

22 Mechanical Model The circulation system pumps theelectrolytes from the tanks through the stack and back in thetanksThe power consumed by pumps is expressed as follows

119875mech = 2

(Δ119875pipes + Δ119875stack)119876 (119905)

120578pump (10)

where 120578pumps is the pump efficiency Δ119875pipes is the pressuredrop in the pipes which can be obtained from the extendedBernoulli equation The pressure drop in the stack Δ119875stack isproportional to the flowrate 119876(119905)

Δ119875stack = 119876 (119905)119877 (11)

where 119877 is the hydraulic resistance obtained from FEM sim-

ulations [13]

23 Battery Power In practice 119875mech is provided from theexternal power source during the charge and from the stackduring the discharge [5] By convention the stack current isdefined as positive during the discharge and negative duringthe charge Thus the battery power 119875VRB is given by

119875VRB = 119875stake minus 119875mech (12)

3 Design of Nonlinear ModelPredictive Controllers

The schematic of the neural network predictive control(NNPC) system developed in this research is shown inFigure 2 The main steps of the NNPC algorithm are listed asfollows

(1) Measure the input and output of the VRB system(2) Use the previous calculated control inputs and the

neural network identifier to compute the cost func-tion

yp

yu

ulowast

Optimizer NN model

VRBsystem

Figure 2 Schematics of the NNPC system

(3) Use the optimization algorithm to calculate a newcontrol vector

(4) Repeat steps (2) and (3) till the desired optimal resultis achieved

(5) Apply the first element of the control vector to theVRB system

(6) Update the parameters of the NN with the new train-ing set

(7) Repeat steps (1)ndash(6) for each time step

31 Predictive Model Based on RBF Neural Network Accord-ing to previous section the battery power can be expressed asfollows

119875VRB = 119892 (119876 119868 119879 119905) (13)

Suppose the stack current and temperature keep constant fora certain amount of time So there is only one control vari-able the flowrate119876 The following NARXmodel can be usedto represent the VRB system

119910 (119905) = 119891 (119910 (119905 minus 1) 119910 (119905 minus 2) 119906 (119905 minus 1) 119906 (119905 minus 2)) (14)

where 119910 is the battery power 119906 is the flowrate and 119891(sdot) is anunknown nonlinear function that needs to be identified Ra-dial basis function (RBF) networks having one hidden layerwere proven to be universal approximator [14] Because of theadvantages of easy design and good generalization a RBFnet-work is used to identify the nonlinear function119891(sdot) in this pa-per The structure of the RBF network is shown in Figure 3

A Gaussian function is used as the activation function Soat the hidden layer the output of RBF unit 119894 is

120593119894(119909) = exp(minus

1003817100381710038171003817119909 minus 119888119894

1003817100381710038171003817

2

21205902

119894

) (119894 = 1 2 5) (15)

where 119909(119905) = [119910(119905 minus 1) 119910(119905 minus 2) 119906(119905 minus 1) 119906(119905 minus 2)]119879 is the

input of RBF network 119888119894and 120590119894are the center andwidth of the

119894th unit respectively

4 Journal of Applied Mathematics

x1

x2

x3

x4

w1

w2

w3

w4

w5

120593

120593

120593

120593

120593

Σyp

Input layerwith 4nodes

Hidden layerwith 5nodes

Output layerwith 1node

Figure 3 The structure of the RBF network

The network output is calculated by

119910 =

5

sum

119894=1

119908119894120593119894(119909) (16)

where 119908119894is the weight value on the connection between RBF

unit 119894 and network output The one-step ahead prediction isgiven by

119910 (119905 + 1) = 119891119873119873

(119910 (119905) 119910 (119905 minus 1) 119906 (119905) 119906 (119905 minus 1)) (17)

The 119895-step ahead prediction of the systemrsquos output is calcu-lated by feeding back themodel outputs (instead of the futuresystemrsquos outputs which do not exist) to the input nodes of thenetwork [15]

Consider the following

119910 (119905 + 119895) = 119891119873119873

(119910 (119905 + 119895 minus 1) 119910 (119905 + 119895 minus 2)

119906 (119905 + 119895 minus 1) 119906 (119905 + 119895 minus 2))

(18)

The computational burden of the optimization problemshowed in next subsection increases with the complexity ofRBF network structure In order to simplify the RBF networkstructure and simultaneously ensure the approximation accu-racy in this study genetic algorithm (GA) is adopted to ob-tain the optimum initial values of the RBF network parame-ters before training the RBF network These parameters in-clude the output weights the centers and widths of the hid-den unit

32 The Objective Function Optimization Algorithm Thereare different forms of the objective function under differentcontrol requirements In this study our purpose is to maxi-mize the power delivered by the battery during the discharge

and minimize the power consumed during the charge whileensuring the control signal is smooth Noticing that the bat-tery power is positive during the discharge and negative dur-ing the charge the objective function is given as follows

min 119869(119905) = minus

119899

sum

119895=1

119910 (119905 + 119895) +

1

2

119898

sum

119894=1

120582Δ1199062(119905 + 119894 minus 1) (19)

subject to constraints

119906min le 119906 (119905 + 119894 minus 1) le 119906max (119894 = 1 2 119898)

119910min le 119910 (119905 + 119895) le 119910max (119895 = 1 2 119899)

(20)

where Δ119906(119905 + 119894 minus 1) = 119906(119905 + 119894 minus 1) minus 119906(119905 + 119894 minus 2) 120582 gt 0 is weightcoefficient and 119899 and 119898 are the predictive horizon and con-trol horizon respectively The vector of the control variablesis obtained from the minimization of the objective functionover the specified horizonThe control vector is available onlywithin the control horizon andmaintains constant afterwardthat is 119906(119905+119894) = 119906(119905+119898minus1) for 119894 = 119898 119899minus1 Only the firstelement of the optimized control sequence is implemented onthe process

Since the function 120593 is nonlinear an analytical solution ofthe objective function is not possible Stochastic optimizationalgorithms such as genetic algorithm and simulated anneal-ing suffer from the drawback of slow convergence whichmake them not suitable for online control Since the objectivefunction surface is simple the gradient based method is anappropriate choice Based on the gradient based method fora given iterative step 119894 the control vector can be calculated asfollows

119906119896(119905 + 119894 minus 1) = 119906

119896minus1(119905 + 119894 minus 1) + Δ119906

119896(119905 + 119894 minus 1)

(119894 = 1 2 119898)

Δ119906119896(119905 + 119894 minus 1) = minus120578

120597119869

120597119906 (119905 + 119894 minus 1)

+ 120572Δ119906119896minus1

(119905 + 119894 minus 1)

(21)

where 120578 is the learning rate and 120572Δ119906119896minus1

(119905+ 119894minus1) is referred toas the additional momentum term The initial value of 119906(119905 +

119894 minus 1) in the iteration at each sampling period is defined as

1199060(119905 + 119894 minus 1) = 119906 (119905 minus 1) (22)

Constraints on control sequence can be handled as followswhen any one of the 119906(119905+119894) reaches its limit this control inputis then set to be equal to its limit [16]

The derivative of the objective function at time 119905 + ℎ minus 1

ℎ = 1 2 119898 can be written as follows

120597119869

120597119906 (119905 + ℎ minus 1)

= minus

119899

sum

119895=1

120597119910 (119905 + 119895)

120597119906 (119905 + ℎ minus 1)

+

119898

sum

119894=0

120582Δ119906 (119905 + 119894)

120597Δ119906 (119905 + 119894)

120597119906 (119905 + ℎ minus 1)

(23)

Journal of Applied Mathematics 5

Table 1 The characteristics of the VRB stack

Name ValueNumber of cells 119873cells 19119877charge 0037Ω

119877discharge 0039Ω

Electrolyte vanadium concentration 2molLInitial 119867+ concentration 5molLTank size 119881tk 83 LFlow resistance

119877 14186843 Pam3

Cell temperature 119879 298KStandard potential 1198640 1255V

The partial derivative can be calculated by the chain rule

120597119910 (119905 + 119895)

120597119906 (119905 + ℎ minus 1)

=

120597119891119873119873

1205971199093

119895 = ℎ

120597119891119873119873

1205971199091

1205971199091

120597119906 (119905 + ℎ minus 1)

+

120597119891119873119873

1205971199094

119895 = ℎ + 1

120597119891119873119873

1205971199091

1205971199091

120597119906 (119905 + ℎ minus 1)

+

120597119891119873119873

1205971199092

1205971199092

120597119906 (119905 + ℎ minus 1)

ℎ + 2 le 119895 le 119899

(24)

where 119909(119905 + 119895) = [119910(119905 + 119895 minus 1) 119910(119905 + 119895 minus 2) 119906(119905 + 119895 minus 1)

119906(119905 + 119895 minus 2)]119879 is the input vector at time 119905 + 119895

120597119891119873119873

120597119909119897

=

5

sum

119894=1

119908119894exp(minus

1003817100381710038171003817119909 minus 119888119894

1003817100381710038171003817

2

21205902

119894

)

119888119894119897minus 119909119897

1205902

119894

119897 = 1 2 3 4

(25)

where 119909119897represents the network input vector of 119910 and 119906

120597Δ119906(119905 + 119894)120597119906(119905 + ℎ minus 1) can be given by

120597Δ119906 (119905 + 119894 minus 1)

120597119906 (119905 + ℎ minus 1)

=

1 119894 = ℎ

minus1 119894 = ℎ + 1

0 else(26)

4 Simulation

To investigate the performance of the proposed controller a19 cells 25 kW 6 kWh VRB is simulated Its main character-istics are listed in Table 1 [5]

41 Identification In order to reduce the online computingtime the RBF network was trained offline before being ap-plied to online control The multiphysics model developed inSection 2 was used for train data generation An input-outputdata set to train the RBF network was obtained by randomlychanging the manipulated variable 119876 within the range of005ndash07 and normalized between minus1 and +1 The sampling

0 1000 2000 3000 4000 5000 6000 7000 80001800

1900

2000

2100

2200

2300

2400

2500

Pow

er (w

)

Time (s)

Physical modelRBF model

Figure 4 Physical model and RBFmodel outputs for battery powerduring the discharge at 100A

time is set as 5 s 1026 samples were used for the trainingwhile 513 samples were used for validation The initial valuesof the RBF network parameters were optimized by GA Afterthe optimum initial values were obtained the Levenberg-Marquardt algorithmwas used as training algorithm to adjustthe network parameters Rootmean square error (RMSE)wasemployed to evaluate the accuracy of RBF network modelThe training was terminated after 500 iterations the obtainedvalue of RMSE is 16591 Figure 4 shows the validation resultsFrom the results of Figure 4 it can be observed that the RBFnetwork can accurately represent the VRB dynamics

The RBF network trained offline works well when thereare no disturbances However it can not accurately representthe VRB dynamics when VRB system is subjected to uncer-tainty So the RBF network requires to train online to adaptwith the change in the process Newest 100 samples were usedfor training

42 Control Results Normally in a charge-discharge cyclethe battery is charged at constant current the battery SoC in-creases from 25 (discharged) to 975 and then it is dis-charged at constant current until it reached its initial SoC [11]The predictive horizon and the control horizons for NMPCare chosen as 4 and 1 respectively The parameter 120582 is set to10000The lower limit and upper limit of flowrate are 005 Lsand 2 Ls respectively In normal working condition the bat-tery is chargeddischarged at constant current Assuming at119905 = 5000 s a disturbance on generator speed causes the chargecurrent to change from 100A to 95A and at 119905 = 12000 sa load disturbance causes the discharge current to changefrom 100A to 110A Figure 5 shows the battery power duringa charge-discharge cycle when influenced by a series of stepchanges in stack currentThe corresponding optimal flowratethat is shown in Figures 6 and 7 shows the comparison of bat-tery power during a charge-discharge cycle at different flow-rate Comparedwith the battery power at119876 = 03 the average

6 Journal of Applied Mathematics

0 5000 10000 150001500

2000

2500

3000

3500

Time (s)

Pow

er (w

)

Figure 5 Battery power during a charge-discharge cycle

0 5000 10000 150000

01

02

03

04

05

06

Time (s)

Flow

rate

(Ls

)

Figure 6 Optimal flowrate during a charge-discharge cycle

power consumed during the charge at optimal flowratedecreased by 1080W and the average power delivered by thebattery during the discharge increased by 1062W

5 Conclusions

The electrolyte flowrate of VRB system was optimized onlineusing model predictive control based on artificial neural net-works An RBF network is built to predict the future batterypower In order to reduce the computational burden of theoptimization problem the hidden layer nodes were chosen as5 The RBF network model was found to be valid for wideflowrate variation with random load disturbances The gra-dient descent algorithm method is used to realize the opti-mization procedure Simulation result at different flowrateindicates that the proposed controller can enhance the outputpower of battery during the discharge and reduce the operat-ing cost during the charge Future works will focus on controlstrategy for VRB and wind farm combined system

References

[1] K-L Huang X-G Li S-Q Liu N Tan and L-Q ChenldquoResearch progress of vanadium redox flow battery for energy

0 5000 10000 150000

1000

2000

3000

4000

5000

Time (s)

Pow

er (w

)

Q = 03

Q = 005

Q = 2

Q = Qopt

Figure 7 Comparison of battery power at different flowrate

storage in Chinardquo Renewable Energy vol 33 no 2 pp 186ndash1922008

[2] M Vynnycky ldquoAnalysis of a model for the operation of a vana-dium redox batteryrdquo Energy vol 36 no 4 pp 2242ndash2256 2011

[3] X Ma H Zhang C Sun Y Zou and T Zhang ldquoAn optimalstrategy of electrolyte flow rate for vanadium redox flow bat-teryrdquo Journal of Power Sources vol 203 pp 153ndash158 2012

[4] H Al-Fetlawi A A Shah and F C Walsh ldquoNon-isothermalmodelling of the all-vanadium redox flow batteryrdquo Electrochim-ica Acta vol 55 no 1 pp 78ndash89 2009

[5] C Blanc and A Rufer ldquoOptimization of the operating point of avanadium redox flow batteryrdquo in Proceedings of the IEEE EnergyConversion Congress and Exposition (ECCE rsquo09) pp 2600ndash2605San Jose Calif USA September 2009

[6] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[7] T Salsbury P Mhaskar and S Joe Qinc ldquoPredictive controlmethods to improve energy efficiency and reduce demand inbuildingsrdquoComputers and Chemical Engineering vol 51 pp 77ndash85 2013

[8] S J Qin and T A Badgwell ldquoA survey of industrial model pre-dictive control technologyrdquoControl Engineering Practice vol 11no 7 pp 733ndash764 2003

[9] A Draeger S Engell and H Ranke ldquoModel predictive controlusing neural networksrdquo IEEE Control SystemsMagazine vol 15no 5 pp 61ndash66 1995

[10] M T Hagan and H B Demuth ldquoNeural networks for controlrdquoin Proceedings of the American Control Conference (ACC rsquo99)pp 1642ndash1656 June 1999

[11] C Blanc and A Rufer ldquoMultiphysics and energetic modelingof a vanadium redox flow batteryrdquo in Proceedings of the IEEEInternational Conference on Sustainable Energy Technologies(ICSET rsquo08) pp 696ndash701 Singapore November 2008

[12] M Li and T Hikihara ldquoA coupled dynamical model of redoxflow battery based on chemical reaction fluid flow and electri-cal circuitrdquo IEICE Transactions on Fundamentals of ElectronicsCommunications and Computer Sciences A vol 91 no 7 pp1741ndash1747 2008

Journal of Applied Mathematics 7

[13] C BlancModeling of a VanadiumRedox Flow Battery ElectricityStorage System Ecole Polytechnique Federale de Lausanne2009

[14] J Park and I W Sandberg ldquoUniversal approximation usingradial-basis-function networksrdquoNeural Computation vol 3 no2 pp 246ndash257 1991

[15] P H Soslashrensen M Noslashrgaard O Ravn and N K Poulsen ldquoIm-plementation of neural network based non-linear predictivecontrolrdquo Neurocomputing vol 28 no 1ndash3 pp 37ndash51 1999

[16] C Venkateswarlu and K Venkat Rao ldquoDynamic recurrent ra-dial basis function networkmodel predictive control of unstablenonlinear processesrdquo Chemical Engineering Science vol 60 no23 pp 6718ndash6732 2005

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

Assuming that each individual cell composing the stackhas the same charging characteristics the equilibrium voltage119880eq of stack can be written as follows

119880eq = 119864cell sdot 119873cell (6)

where 119873cell is the number of cellsThe stack voltage 119880stack is decreased when current flows

through the stack because of several types of internal lossessuch as activation concentration and Ohmic losses Butthese internal losses are difficult to measure here we replacethem with equivalent resistance 119877eq chdisch

119880loss = 120578act + 120578conc + 120578ohm = 119868 sdot 119877eq chdisch (7)

So stack voltage 119880stack is given by

119880stack = 119880eq minus 119880loss (8)

Then the power of stack can be calculated as

119875stake = 119880stack sdot 119868 (9)

22 Mechanical Model The circulation system pumps theelectrolytes from the tanks through the stack and back in thetanksThe power consumed by pumps is expressed as follows

119875mech = 2

(Δ119875pipes + Δ119875stack)119876 (119905)

120578pump (10)

where 120578pumps is the pump efficiency Δ119875pipes is the pressuredrop in the pipes which can be obtained from the extendedBernoulli equation The pressure drop in the stack Δ119875stack isproportional to the flowrate 119876(119905)

Δ119875stack = 119876 (119905)119877 (11)

where 119877 is the hydraulic resistance obtained from FEM sim-

ulations [13]

23 Battery Power In practice 119875mech is provided from theexternal power source during the charge and from the stackduring the discharge [5] By convention the stack current isdefined as positive during the discharge and negative duringthe charge Thus the battery power 119875VRB is given by

119875VRB = 119875stake minus 119875mech (12)

3 Design of Nonlinear ModelPredictive Controllers

The schematic of the neural network predictive control(NNPC) system developed in this research is shown inFigure 2 The main steps of the NNPC algorithm are listed asfollows

(1) Measure the input and output of the VRB system(2) Use the previous calculated control inputs and the

neural network identifier to compute the cost func-tion

yp

yu

ulowast

Optimizer NN model

VRBsystem

Figure 2 Schematics of the NNPC system

(3) Use the optimization algorithm to calculate a newcontrol vector

(4) Repeat steps (2) and (3) till the desired optimal resultis achieved

(5) Apply the first element of the control vector to theVRB system

(6) Update the parameters of the NN with the new train-ing set

(7) Repeat steps (1)ndash(6) for each time step

31 Predictive Model Based on RBF Neural Network Accord-ing to previous section the battery power can be expressed asfollows

119875VRB = 119892 (119876 119868 119879 119905) (13)

Suppose the stack current and temperature keep constant fora certain amount of time So there is only one control vari-able the flowrate119876 The following NARXmodel can be usedto represent the VRB system

119910 (119905) = 119891 (119910 (119905 minus 1) 119910 (119905 minus 2) 119906 (119905 minus 1) 119906 (119905 minus 2)) (14)

where 119910 is the battery power 119906 is the flowrate and 119891(sdot) is anunknown nonlinear function that needs to be identified Ra-dial basis function (RBF) networks having one hidden layerwere proven to be universal approximator [14] Because of theadvantages of easy design and good generalization a RBFnet-work is used to identify the nonlinear function119891(sdot) in this pa-per The structure of the RBF network is shown in Figure 3

A Gaussian function is used as the activation function Soat the hidden layer the output of RBF unit 119894 is

120593119894(119909) = exp(minus

1003817100381710038171003817119909 minus 119888119894

1003817100381710038171003817

2

21205902

119894

) (119894 = 1 2 5) (15)

where 119909(119905) = [119910(119905 minus 1) 119910(119905 minus 2) 119906(119905 minus 1) 119906(119905 minus 2)]119879 is the

input of RBF network 119888119894and 120590119894are the center andwidth of the

119894th unit respectively

4 Journal of Applied Mathematics

x1

x2

x3

x4

w1

w2

w3

w4

w5

120593

120593

120593

120593

120593

Σyp

Input layerwith 4nodes

Hidden layerwith 5nodes

Output layerwith 1node

Figure 3 The structure of the RBF network

The network output is calculated by

119910 =

5

sum

119894=1

119908119894120593119894(119909) (16)

where 119908119894is the weight value on the connection between RBF

unit 119894 and network output The one-step ahead prediction isgiven by

119910 (119905 + 1) = 119891119873119873

(119910 (119905) 119910 (119905 minus 1) 119906 (119905) 119906 (119905 minus 1)) (17)

The 119895-step ahead prediction of the systemrsquos output is calcu-lated by feeding back themodel outputs (instead of the futuresystemrsquos outputs which do not exist) to the input nodes of thenetwork [15]

Consider the following

119910 (119905 + 119895) = 119891119873119873

(119910 (119905 + 119895 minus 1) 119910 (119905 + 119895 minus 2)

119906 (119905 + 119895 minus 1) 119906 (119905 + 119895 minus 2))

(18)

The computational burden of the optimization problemshowed in next subsection increases with the complexity ofRBF network structure In order to simplify the RBF networkstructure and simultaneously ensure the approximation accu-racy in this study genetic algorithm (GA) is adopted to ob-tain the optimum initial values of the RBF network parame-ters before training the RBF network These parameters in-clude the output weights the centers and widths of the hid-den unit

32 The Objective Function Optimization Algorithm Thereare different forms of the objective function under differentcontrol requirements In this study our purpose is to maxi-mize the power delivered by the battery during the discharge

and minimize the power consumed during the charge whileensuring the control signal is smooth Noticing that the bat-tery power is positive during the discharge and negative dur-ing the charge the objective function is given as follows

min 119869(119905) = minus

119899

sum

119895=1

119910 (119905 + 119895) +

1

2

119898

sum

119894=1

120582Δ1199062(119905 + 119894 minus 1) (19)

subject to constraints

119906min le 119906 (119905 + 119894 minus 1) le 119906max (119894 = 1 2 119898)

119910min le 119910 (119905 + 119895) le 119910max (119895 = 1 2 119899)

(20)

where Δ119906(119905 + 119894 minus 1) = 119906(119905 + 119894 minus 1) minus 119906(119905 + 119894 minus 2) 120582 gt 0 is weightcoefficient and 119899 and 119898 are the predictive horizon and con-trol horizon respectively The vector of the control variablesis obtained from the minimization of the objective functionover the specified horizonThe control vector is available onlywithin the control horizon andmaintains constant afterwardthat is 119906(119905+119894) = 119906(119905+119898minus1) for 119894 = 119898 119899minus1 Only the firstelement of the optimized control sequence is implemented onthe process

Since the function 120593 is nonlinear an analytical solution ofthe objective function is not possible Stochastic optimizationalgorithms such as genetic algorithm and simulated anneal-ing suffer from the drawback of slow convergence whichmake them not suitable for online control Since the objectivefunction surface is simple the gradient based method is anappropriate choice Based on the gradient based method fora given iterative step 119894 the control vector can be calculated asfollows

119906119896(119905 + 119894 minus 1) = 119906

119896minus1(119905 + 119894 minus 1) + Δ119906

119896(119905 + 119894 minus 1)

(119894 = 1 2 119898)

Δ119906119896(119905 + 119894 minus 1) = minus120578

120597119869

120597119906 (119905 + 119894 minus 1)

+ 120572Δ119906119896minus1

(119905 + 119894 minus 1)

(21)

where 120578 is the learning rate and 120572Δ119906119896minus1

(119905+ 119894minus1) is referred toas the additional momentum term The initial value of 119906(119905 +

119894 minus 1) in the iteration at each sampling period is defined as

1199060(119905 + 119894 minus 1) = 119906 (119905 minus 1) (22)

Constraints on control sequence can be handled as followswhen any one of the 119906(119905+119894) reaches its limit this control inputis then set to be equal to its limit [16]

The derivative of the objective function at time 119905 + ℎ minus 1

ℎ = 1 2 119898 can be written as follows

120597119869

120597119906 (119905 + ℎ minus 1)

= minus

119899

sum

119895=1

120597119910 (119905 + 119895)

120597119906 (119905 + ℎ minus 1)

+

119898

sum

119894=0

120582Δ119906 (119905 + 119894)

120597Δ119906 (119905 + 119894)

120597119906 (119905 + ℎ minus 1)

(23)

Journal of Applied Mathematics 5

Table 1 The characteristics of the VRB stack

Name ValueNumber of cells 119873cells 19119877charge 0037Ω

119877discharge 0039Ω

Electrolyte vanadium concentration 2molLInitial 119867+ concentration 5molLTank size 119881tk 83 LFlow resistance

119877 14186843 Pam3

Cell temperature 119879 298KStandard potential 1198640 1255V

The partial derivative can be calculated by the chain rule

120597119910 (119905 + 119895)

120597119906 (119905 + ℎ minus 1)

=

120597119891119873119873

1205971199093

119895 = ℎ

120597119891119873119873

1205971199091

1205971199091

120597119906 (119905 + ℎ minus 1)

+

120597119891119873119873

1205971199094

119895 = ℎ + 1

120597119891119873119873

1205971199091

1205971199091

120597119906 (119905 + ℎ minus 1)

+

120597119891119873119873

1205971199092

1205971199092

120597119906 (119905 + ℎ minus 1)

ℎ + 2 le 119895 le 119899

(24)

where 119909(119905 + 119895) = [119910(119905 + 119895 minus 1) 119910(119905 + 119895 minus 2) 119906(119905 + 119895 minus 1)

119906(119905 + 119895 minus 2)]119879 is the input vector at time 119905 + 119895

120597119891119873119873

120597119909119897

=

5

sum

119894=1

119908119894exp(minus

1003817100381710038171003817119909 minus 119888119894

1003817100381710038171003817

2

21205902

119894

)

119888119894119897minus 119909119897

1205902

119894

119897 = 1 2 3 4

(25)

where 119909119897represents the network input vector of 119910 and 119906

120597Δ119906(119905 + 119894)120597119906(119905 + ℎ minus 1) can be given by

120597Δ119906 (119905 + 119894 minus 1)

120597119906 (119905 + ℎ minus 1)

=

1 119894 = ℎ

minus1 119894 = ℎ + 1

0 else(26)

4 Simulation

To investigate the performance of the proposed controller a19 cells 25 kW 6 kWh VRB is simulated Its main character-istics are listed in Table 1 [5]

41 Identification In order to reduce the online computingtime the RBF network was trained offline before being ap-plied to online control The multiphysics model developed inSection 2 was used for train data generation An input-outputdata set to train the RBF network was obtained by randomlychanging the manipulated variable 119876 within the range of005ndash07 and normalized between minus1 and +1 The sampling

0 1000 2000 3000 4000 5000 6000 7000 80001800

1900

2000

2100

2200

2300

2400

2500

Pow

er (w

)

Time (s)

Physical modelRBF model

Figure 4 Physical model and RBFmodel outputs for battery powerduring the discharge at 100A

time is set as 5 s 1026 samples were used for the trainingwhile 513 samples were used for validation The initial valuesof the RBF network parameters were optimized by GA Afterthe optimum initial values were obtained the Levenberg-Marquardt algorithmwas used as training algorithm to adjustthe network parameters Rootmean square error (RMSE)wasemployed to evaluate the accuracy of RBF network modelThe training was terminated after 500 iterations the obtainedvalue of RMSE is 16591 Figure 4 shows the validation resultsFrom the results of Figure 4 it can be observed that the RBFnetwork can accurately represent the VRB dynamics

The RBF network trained offline works well when thereare no disturbances However it can not accurately representthe VRB dynamics when VRB system is subjected to uncer-tainty So the RBF network requires to train online to adaptwith the change in the process Newest 100 samples were usedfor training

42 Control Results Normally in a charge-discharge cyclethe battery is charged at constant current the battery SoC in-creases from 25 (discharged) to 975 and then it is dis-charged at constant current until it reached its initial SoC [11]The predictive horizon and the control horizons for NMPCare chosen as 4 and 1 respectively The parameter 120582 is set to10000The lower limit and upper limit of flowrate are 005 Lsand 2 Ls respectively In normal working condition the bat-tery is chargeddischarged at constant current Assuming at119905 = 5000 s a disturbance on generator speed causes the chargecurrent to change from 100A to 95A and at 119905 = 12000 sa load disturbance causes the discharge current to changefrom 100A to 110A Figure 5 shows the battery power duringa charge-discharge cycle when influenced by a series of stepchanges in stack currentThe corresponding optimal flowratethat is shown in Figures 6 and 7 shows the comparison of bat-tery power during a charge-discharge cycle at different flow-rate Comparedwith the battery power at119876 = 03 the average

6 Journal of Applied Mathematics

0 5000 10000 150001500

2000

2500

3000

3500

Time (s)

Pow

er (w

)

Figure 5 Battery power during a charge-discharge cycle

0 5000 10000 150000

01

02

03

04

05

06

Time (s)

Flow

rate

(Ls

)

Figure 6 Optimal flowrate during a charge-discharge cycle

power consumed during the charge at optimal flowratedecreased by 1080W and the average power delivered by thebattery during the discharge increased by 1062W

5 Conclusions

The electrolyte flowrate of VRB system was optimized onlineusing model predictive control based on artificial neural net-works An RBF network is built to predict the future batterypower In order to reduce the computational burden of theoptimization problem the hidden layer nodes were chosen as5 The RBF network model was found to be valid for wideflowrate variation with random load disturbances The gra-dient descent algorithm method is used to realize the opti-mization procedure Simulation result at different flowrateindicates that the proposed controller can enhance the outputpower of battery during the discharge and reduce the operat-ing cost during the charge Future works will focus on controlstrategy for VRB and wind farm combined system

References

[1] K-L Huang X-G Li S-Q Liu N Tan and L-Q ChenldquoResearch progress of vanadium redox flow battery for energy

0 5000 10000 150000

1000

2000

3000

4000

5000

Time (s)

Pow

er (w

)

Q = 03

Q = 005

Q = 2

Q = Qopt

Figure 7 Comparison of battery power at different flowrate

storage in Chinardquo Renewable Energy vol 33 no 2 pp 186ndash1922008

[2] M Vynnycky ldquoAnalysis of a model for the operation of a vana-dium redox batteryrdquo Energy vol 36 no 4 pp 2242ndash2256 2011

[3] X Ma H Zhang C Sun Y Zou and T Zhang ldquoAn optimalstrategy of electrolyte flow rate for vanadium redox flow bat-teryrdquo Journal of Power Sources vol 203 pp 153ndash158 2012

[4] H Al-Fetlawi A A Shah and F C Walsh ldquoNon-isothermalmodelling of the all-vanadium redox flow batteryrdquo Electrochim-ica Acta vol 55 no 1 pp 78ndash89 2009

[5] C Blanc and A Rufer ldquoOptimization of the operating point of avanadium redox flow batteryrdquo in Proceedings of the IEEE EnergyConversion Congress and Exposition (ECCE rsquo09) pp 2600ndash2605San Jose Calif USA September 2009

[6] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[7] T Salsbury P Mhaskar and S Joe Qinc ldquoPredictive controlmethods to improve energy efficiency and reduce demand inbuildingsrdquoComputers and Chemical Engineering vol 51 pp 77ndash85 2013

[8] S J Qin and T A Badgwell ldquoA survey of industrial model pre-dictive control technologyrdquoControl Engineering Practice vol 11no 7 pp 733ndash764 2003

[9] A Draeger S Engell and H Ranke ldquoModel predictive controlusing neural networksrdquo IEEE Control SystemsMagazine vol 15no 5 pp 61ndash66 1995

[10] M T Hagan and H B Demuth ldquoNeural networks for controlrdquoin Proceedings of the American Control Conference (ACC rsquo99)pp 1642ndash1656 June 1999

[11] C Blanc and A Rufer ldquoMultiphysics and energetic modelingof a vanadium redox flow batteryrdquo in Proceedings of the IEEEInternational Conference on Sustainable Energy Technologies(ICSET rsquo08) pp 696ndash701 Singapore November 2008

[12] M Li and T Hikihara ldquoA coupled dynamical model of redoxflow battery based on chemical reaction fluid flow and electri-cal circuitrdquo IEICE Transactions on Fundamentals of ElectronicsCommunications and Computer Sciences A vol 91 no 7 pp1741ndash1747 2008

Journal of Applied Mathematics 7

[13] C BlancModeling of a VanadiumRedox Flow Battery ElectricityStorage System Ecole Polytechnique Federale de Lausanne2009

[14] J Park and I W Sandberg ldquoUniversal approximation usingradial-basis-function networksrdquoNeural Computation vol 3 no2 pp 246ndash257 1991

[15] P H Soslashrensen M Noslashrgaard O Ravn and N K Poulsen ldquoIm-plementation of neural network based non-linear predictivecontrolrdquo Neurocomputing vol 28 no 1ndash3 pp 37ndash51 1999

[16] C Venkateswarlu and K Venkat Rao ldquoDynamic recurrent ra-dial basis function networkmodel predictive control of unstablenonlinear processesrdquo Chemical Engineering Science vol 60 no23 pp 6718ndash6732 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

x1

x2

x3

x4

w1

w2

w3

w4

w5

120593

120593

120593

120593

120593

Σyp

Input layerwith 4nodes

Hidden layerwith 5nodes

Output layerwith 1node

Figure 3 The structure of the RBF network

The network output is calculated by

119910 =

5

sum

119894=1

119908119894120593119894(119909) (16)

where 119908119894is the weight value on the connection between RBF

unit 119894 and network output The one-step ahead prediction isgiven by

119910 (119905 + 1) = 119891119873119873

(119910 (119905) 119910 (119905 minus 1) 119906 (119905) 119906 (119905 minus 1)) (17)

The 119895-step ahead prediction of the systemrsquos output is calcu-lated by feeding back themodel outputs (instead of the futuresystemrsquos outputs which do not exist) to the input nodes of thenetwork [15]

Consider the following

119910 (119905 + 119895) = 119891119873119873

(119910 (119905 + 119895 minus 1) 119910 (119905 + 119895 minus 2)

119906 (119905 + 119895 minus 1) 119906 (119905 + 119895 minus 2))

(18)

The computational burden of the optimization problemshowed in next subsection increases with the complexity ofRBF network structure In order to simplify the RBF networkstructure and simultaneously ensure the approximation accu-racy in this study genetic algorithm (GA) is adopted to ob-tain the optimum initial values of the RBF network parame-ters before training the RBF network These parameters in-clude the output weights the centers and widths of the hid-den unit

32 The Objective Function Optimization Algorithm Thereare different forms of the objective function under differentcontrol requirements In this study our purpose is to maxi-mize the power delivered by the battery during the discharge

and minimize the power consumed during the charge whileensuring the control signal is smooth Noticing that the bat-tery power is positive during the discharge and negative dur-ing the charge the objective function is given as follows

min 119869(119905) = minus

119899

sum

119895=1

119910 (119905 + 119895) +

1

2

119898

sum

119894=1

120582Δ1199062(119905 + 119894 minus 1) (19)

subject to constraints

119906min le 119906 (119905 + 119894 minus 1) le 119906max (119894 = 1 2 119898)

119910min le 119910 (119905 + 119895) le 119910max (119895 = 1 2 119899)

(20)

where Δ119906(119905 + 119894 minus 1) = 119906(119905 + 119894 minus 1) minus 119906(119905 + 119894 minus 2) 120582 gt 0 is weightcoefficient and 119899 and 119898 are the predictive horizon and con-trol horizon respectively The vector of the control variablesis obtained from the minimization of the objective functionover the specified horizonThe control vector is available onlywithin the control horizon andmaintains constant afterwardthat is 119906(119905+119894) = 119906(119905+119898minus1) for 119894 = 119898 119899minus1 Only the firstelement of the optimized control sequence is implemented onthe process

Since the function 120593 is nonlinear an analytical solution ofthe objective function is not possible Stochastic optimizationalgorithms such as genetic algorithm and simulated anneal-ing suffer from the drawback of slow convergence whichmake them not suitable for online control Since the objectivefunction surface is simple the gradient based method is anappropriate choice Based on the gradient based method fora given iterative step 119894 the control vector can be calculated asfollows

119906119896(119905 + 119894 minus 1) = 119906

119896minus1(119905 + 119894 minus 1) + Δ119906

119896(119905 + 119894 minus 1)

(119894 = 1 2 119898)

Δ119906119896(119905 + 119894 minus 1) = minus120578

120597119869

120597119906 (119905 + 119894 minus 1)

+ 120572Δ119906119896minus1

(119905 + 119894 minus 1)

(21)

where 120578 is the learning rate and 120572Δ119906119896minus1

(119905+ 119894minus1) is referred toas the additional momentum term The initial value of 119906(119905 +

119894 minus 1) in the iteration at each sampling period is defined as

1199060(119905 + 119894 minus 1) = 119906 (119905 minus 1) (22)

Constraints on control sequence can be handled as followswhen any one of the 119906(119905+119894) reaches its limit this control inputis then set to be equal to its limit [16]

The derivative of the objective function at time 119905 + ℎ minus 1

ℎ = 1 2 119898 can be written as follows

120597119869

120597119906 (119905 + ℎ minus 1)

= minus

119899

sum

119895=1

120597119910 (119905 + 119895)

120597119906 (119905 + ℎ minus 1)

+

119898

sum

119894=0

120582Δ119906 (119905 + 119894)

120597Δ119906 (119905 + 119894)

120597119906 (119905 + ℎ minus 1)

(23)

Journal of Applied Mathematics 5

Table 1 The characteristics of the VRB stack

Name ValueNumber of cells 119873cells 19119877charge 0037Ω

119877discharge 0039Ω

Electrolyte vanadium concentration 2molLInitial 119867+ concentration 5molLTank size 119881tk 83 LFlow resistance

119877 14186843 Pam3

Cell temperature 119879 298KStandard potential 1198640 1255V

The partial derivative can be calculated by the chain rule

120597119910 (119905 + 119895)

120597119906 (119905 + ℎ minus 1)

=

120597119891119873119873

1205971199093

119895 = ℎ

120597119891119873119873

1205971199091

1205971199091

120597119906 (119905 + ℎ minus 1)

+

120597119891119873119873

1205971199094

119895 = ℎ + 1

120597119891119873119873

1205971199091

1205971199091

120597119906 (119905 + ℎ minus 1)

+

120597119891119873119873

1205971199092

1205971199092

120597119906 (119905 + ℎ minus 1)

ℎ + 2 le 119895 le 119899

(24)

where 119909(119905 + 119895) = [119910(119905 + 119895 minus 1) 119910(119905 + 119895 minus 2) 119906(119905 + 119895 minus 1)

119906(119905 + 119895 minus 2)]119879 is the input vector at time 119905 + 119895

120597119891119873119873

120597119909119897

=

5

sum

119894=1

119908119894exp(minus

1003817100381710038171003817119909 minus 119888119894

1003817100381710038171003817

2

21205902

119894

)

119888119894119897minus 119909119897

1205902

119894

119897 = 1 2 3 4

(25)

where 119909119897represents the network input vector of 119910 and 119906

120597Δ119906(119905 + 119894)120597119906(119905 + ℎ minus 1) can be given by

120597Δ119906 (119905 + 119894 minus 1)

120597119906 (119905 + ℎ minus 1)

=

1 119894 = ℎ

minus1 119894 = ℎ + 1

0 else(26)

4 Simulation

To investigate the performance of the proposed controller a19 cells 25 kW 6 kWh VRB is simulated Its main character-istics are listed in Table 1 [5]

41 Identification In order to reduce the online computingtime the RBF network was trained offline before being ap-plied to online control The multiphysics model developed inSection 2 was used for train data generation An input-outputdata set to train the RBF network was obtained by randomlychanging the manipulated variable 119876 within the range of005ndash07 and normalized between minus1 and +1 The sampling

0 1000 2000 3000 4000 5000 6000 7000 80001800

1900

2000

2100

2200

2300

2400

2500

Pow

er (w

)

Time (s)

Physical modelRBF model

Figure 4 Physical model and RBFmodel outputs for battery powerduring the discharge at 100A

time is set as 5 s 1026 samples were used for the trainingwhile 513 samples were used for validation The initial valuesof the RBF network parameters were optimized by GA Afterthe optimum initial values were obtained the Levenberg-Marquardt algorithmwas used as training algorithm to adjustthe network parameters Rootmean square error (RMSE)wasemployed to evaluate the accuracy of RBF network modelThe training was terminated after 500 iterations the obtainedvalue of RMSE is 16591 Figure 4 shows the validation resultsFrom the results of Figure 4 it can be observed that the RBFnetwork can accurately represent the VRB dynamics

The RBF network trained offline works well when thereare no disturbances However it can not accurately representthe VRB dynamics when VRB system is subjected to uncer-tainty So the RBF network requires to train online to adaptwith the change in the process Newest 100 samples were usedfor training

42 Control Results Normally in a charge-discharge cyclethe battery is charged at constant current the battery SoC in-creases from 25 (discharged) to 975 and then it is dis-charged at constant current until it reached its initial SoC [11]The predictive horizon and the control horizons for NMPCare chosen as 4 and 1 respectively The parameter 120582 is set to10000The lower limit and upper limit of flowrate are 005 Lsand 2 Ls respectively In normal working condition the bat-tery is chargeddischarged at constant current Assuming at119905 = 5000 s a disturbance on generator speed causes the chargecurrent to change from 100A to 95A and at 119905 = 12000 sa load disturbance causes the discharge current to changefrom 100A to 110A Figure 5 shows the battery power duringa charge-discharge cycle when influenced by a series of stepchanges in stack currentThe corresponding optimal flowratethat is shown in Figures 6 and 7 shows the comparison of bat-tery power during a charge-discharge cycle at different flow-rate Comparedwith the battery power at119876 = 03 the average

6 Journal of Applied Mathematics

0 5000 10000 150001500

2000

2500

3000

3500

Time (s)

Pow

er (w

)

Figure 5 Battery power during a charge-discharge cycle

0 5000 10000 150000

01

02

03

04

05

06

Time (s)

Flow

rate

(Ls

)

Figure 6 Optimal flowrate during a charge-discharge cycle

power consumed during the charge at optimal flowratedecreased by 1080W and the average power delivered by thebattery during the discharge increased by 1062W

5 Conclusions

The electrolyte flowrate of VRB system was optimized onlineusing model predictive control based on artificial neural net-works An RBF network is built to predict the future batterypower In order to reduce the computational burden of theoptimization problem the hidden layer nodes were chosen as5 The RBF network model was found to be valid for wideflowrate variation with random load disturbances The gra-dient descent algorithm method is used to realize the opti-mization procedure Simulation result at different flowrateindicates that the proposed controller can enhance the outputpower of battery during the discharge and reduce the operat-ing cost during the charge Future works will focus on controlstrategy for VRB and wind farm combined system

References

[1] K-L Huang X-G Li S-Q Liu N Tan and L-Q ChenldquoResearch progress of vanadium redox flow battery for energy

0 5000 10000 150000

1000

2000

3000

4000

5000

Time (s)

Pow

er (w

)

Q = 03

Q = 005

Q = 2

Q = Qopt

Figure 7 Comparison of battery power at different flowrate

storage in Chinardquo Renewable Energy vol 33 no 2 pp 186ndash1922008

[2] M Vynnycky ldquoAnalysis of a model for the operation of a vana-dium redox batteryrdquo Energy vol 36 no 4 pp 2242ndash2256 2011

[3] X Ma H Zhang C Sun Y Zou and T Zhang ldquoAn optimalstrategy of electrolyte flow rate for vanadium redox flow bat-teryrdquo Journal of Power Sources vol 203 pp 153ndash158 2012

[4] H Al-Fetlawi A A Shah and F C Walsh ldquoNon-isothermalmodelling of the all-vanadium redox flow batteryrdquo Electrochim-ica Acta vol 55 no 1 pp 78ndash89 2009

[5] C Blanc and A Rufer ldquoOptimization of the operating point of avanadium redox flow batteryrdquo in Proceedings of the IEEE EnergyConversion Congress and Exposition (ECCE rsquo09) pp 2600ndash2605San Jose Calif USA September 2009

[6] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[7] T Salsbury P Mhaskar and S Joe Qinc ldquoPredictive controlmethods to improve energy efficiency and reduce demand inbuildingsrdquoComputers and Chemical Engineering vol 51 pp 77ndash85 2013

[8] S J Qin and T A Badgwell ldquoA survey of industrial model pre-dictive control technologyrdquoControl Engineering Practice vol 11no 7 pp 733ndash764 2003

[9] A Draeger S Engell and H Ranke ldquoModel predictive controlusing neural networksrdquo IEEE Control SystemsMagazine vol 15no 5 pp 61ndash66 1995

[10] M T Hagan and H B Demuth ldquoNeural networks for controlrdquoin Proceedings of the American Control Conference (ACC rsquo99)pp 1642ndash1656 June 1999

[11] C Blanc and A Rufer ldquoMultiphysics and energetic modelingof a vanadium redox flow batteryrdquo in Proceedings of the IEEEInternational Conference on Sustainable Energy Technologies(ICSET rsquo08) pp 696ndash701 Singapore November 2008

[12] M Li and T Hikihara ldquoA coupled dynamical model of redoxflow battery based on chemical reaction fluid flow and electri-cal circuitrdquo IEICE Transactions on Fundamentals of ElectronicsCommunications and Computer Sciences A vol 91 no 7 pp1741ndash1747 2008

Journal of Applied Mathematics 7

[13] C BlancModeling of a VanadiumRedox Flow Battery ElectricityStorage System Ecole Polytechnique Federale de Lausanne2009

[14] J Park and I W Sandberg ldquoUniversal approximation usingradial-basis-function networksrdquoNeural Computation vol 3 no2 pp 246ndash257 1991

[15] P H Soslashrensen M Noslashrgaard O Ravn and N K Poulsen ldquoIm-plementation of neural network based non-linear predictivecontrolrdquo Neurocomputing vol 28 no 1ndash3 pp 37ndash51 1999

[16] C Venkateswarlu and K Venkat Rao ldquoDynamic recurrent ra-dial basis function networkmodel predictive control of unstablenonlinear processesrdquo Chemical Engineering Science vol 60 no23 pp 6718ndash6732 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Table 1 The characteristics of the VRB stack

Name ValueNumber of cells 119873cells 19119877charge 0037Ω

119877discharge 0039Ω

Electrolyte vanadium concentration 2molLInitial 119867+ concentration 5molLTank size 119881tk 83 LFlow resistance

119877 14186843 Pam3

Cell temperature 119879 298KStandard potential 1198640 1255V

The partial derivative can be calculated by the chain rule

120597119910 (119905 + 119895)

120597119906 (119905 + ℎ minus 1)

=

120597119891119873119873

1205971199093

119895 = ℎ

120597119891119873119873

1205971199091

1205971199091

120597119906 (119905 + ℎ minus 1)

+

120597119891119873119873

1205971199094

119895 = ℎ + 1

120597119891119873119873

1205971199091

1205971199091

120597119906 (119905 + ℎ minus 1)

+

120597119891119873119873

1205971199092

1205971199092

120597119906 (119905 + ℎ minus 1)

ℎ + 2 le 119895 le 119899

(24)

where 119909(119905 + 119895) = [119910(119905 + 119895 minus 1) 119910(119905 + 119895 minus 2) 119906(119905 + 119895 minus 1)

119906(119905 + 119895 minus 2)]119879 is the input vector at time 119905 + 119895

120597119891119873119873

120597119909119897

=

5

sum

119894=1

119908119894exp(minus

1003817100381710038171003817119909 minus 119888119894

1003817100381710038171003817

2

21205902

119894

)

119888119894119897minus 119909119897

1205902

119894

119897 = 1 2 3 4

(25)

where 119909119897represents the network input vector of 119910 and 119906

120597Δ119906(119905 + 119894)120597119906(119905 + ℎ minus 1) can be given by

120597Δ119906 (119905 + 119894 minus 1)

120597119906 (119905 + ℎ minus 1)

=

1 119894 = ℎ

minus1 119894 = ℎ + 1

0 else(26)

4 Simulation

To investigate the performance of the proposed controller a19 cells 25 kW 6 kWh VRB is simulated Its main character-istics are listed in Table 1 [5]

41 Identification In order to reduce the online computingtime the RBF network was trained offline before being ap-plied to online control The multiphysics model developed inSection 2 was used for train data generation An input-outputdata set to train the RBF network was obtained by randomlychanging the manipulated variable 119876 within the range of005ndash07 and normalized between minus1 and +1 The sampling

0 1000 2000 3000 4000 5000 6000 7000 80001800

1900

2000

2100

2200

2300

2400

2500

Pow

er (w

)

Time (s)

Physical modelRBF model

Figure 4 Physical model and RBFmodel outputs for battery powerduring the discharge at 100A

time is set as 5 s 1026 samples were used for the trainingwhile 513 samples were used for validation The initial valuesof the RBF network parameters were optimized by GA Afterthe optimum initial values were obtained the Levenberg-Marquardt algorithmwas used as training algorithm to adjustthe network parameters Rootmean square error (RMSE)wasemployed to evaluate the accuracy of RBF network modelThe training was terminated after 500 iterations the obtainedvalue of RMSE is 16591 Figure 4 shows the validation resultsFrom the results of Figure 4 it can be observed that the RBFnetwork can accurately represent the VRB dynamics

The RBF network trained offline works well when thereare no disturbances However it can not accurately representthe VRB dynamics when VRB system is subjected to uncer-tainty So the RBF network requires to train online to adaptwith the change in the process Newest 100 samples were usedfor training

42 Control Results Normally in a charge-discharge cyclethe battery is charged at constant current the battery SoC in-creases from 25 (discharged) to 975 and then it is dis-charged at constant current until it reached its initial SoC [11]The predictive horizon and the control horizons for NMPCare chosen as 4 and 1 respectively The parameter 120582 is set to10000The lower limit and upper limit of flowrate are 005 Lsand 2 Ls respectively In normal working condition the bat-tery is chargeddischarged at constant current Assuming at119905 = 5000 s a disturbance on generator speed causes the chargecurrent to change from 100A to 95A and at 119905 = 12000 sa load disturbance causes the discharge current to changefrom 100A to 110A Figure 5 shows the battery power duringa charge-discharge cycle when influenced by a series of stepchanges in stack currentThe corresponding optimal flowratethat is shown in Figures 6 and 7 shows the comparison of bat-tery power during a charge-discharge cycle at different flow-rate Comparedwith the battery power at119876 = 03 the average

6 Journal of Applied Mathematics

0 5000 10000 150001500

2000

2500

3000

3500

Time (s)

Pow

er (w

)

Figure 5 Battery power during a charge-discharge cycle

0 5000 10000 150000

01

02

03

04

05

06

Time (s)

Flow

rate

(Ls

)

Figure 6 Optimal flowrate during a charge-discharge cycle

power consumed during the charge at optimal flowratedecreased by 1080W and the average power delivered by thebattery during the discharge increased by 1062W

5 Conclusions

The electrolyte flowrate of VRB system was optimized onlineusing model predictive control based on artificial neural net-works An RBF network is built to predict the future batterypower In order to reduce the computational burden of theoptimization problem the hidden layer nodes were chosen as5 The RBF network model was found to be valid for wideflowrate variation with random load disturbances The gra-dient descent algorithm method is used to realize the opti-mization procedure Simulation result at different flowrateindicates that the proposed controller can enhance the outputpower of battery during the discharge and reduce the operat-ing cost during the charge Future works will focus on controlstrategy for VRB and wind farm combined system

References

[1] K-L Huang X-G Li S-Q Liu N Tan and L-Q ChenldquoResearch progress of vanadium redox flow battery for energy

0 5000 10000 150000

1000

2000

3000

4000

5000

Time (s)

Pow

er (w

)

Q = 03

Q = 005

Q = 2

Q = Qopt

Figure 7 Comparison of battery power at different flowrate

storage in Chinardquo Renewable Energy vol 33 no 2 pp 186ndash1922008

[2] M Vynnycky ldquoAnalysis of a model for the operation of a vana-dium redox batteryrdquo Energy vol 36 no 4 pp 2242ndash2256 2011

[3] X Ma H Zhang C Sun Y Zou and T Zhang ldquoAn optimalstrategy of electrolyte flow rate for vanadium redox flow bat-teryrdquo Journal of Power Sources vol 203 pp 153ndash158 2012

[4] H Al-Fetlawi A A Shah and F C Walsh ldquoNon-isothermalmodelling of the all-vanadium redox flow batteryrdquo Electrochim-ica Acta vol 55 no 1 pp 78ndash89 2009

[5] C Blanc and A Rufer ldquoOptimization of the operating point of avanadium redox flow batteryrdquo in Proceedings of the IEEE EnergyConversion Congress and Exposition (ECCE rsquo09) pp 2600ndash2605San Jose Calif USA September 2009

[6] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[7] T Salsbury P Mhaskar and S Joe Qinc ldquoPredictive controlmethods to improve energy efficiency and reduce demand inbuildingsrdquoComputers and Chemical Engineering vol 51 pp 77ndash85 2013

[8] S J Qin and T A Badgwell ldquoA survey of industrial model pre-dictive control technologyrdquoControl Engineering Practice vol 11no 7 pp 733ndash764 2003

[9] A Draeger S Engell and H Ranke ldquoModel predictive controlusing neural networksrdquo IEEE Control SystemsMagazine vol 15no 5 pp 61ndash66 1995

[10] M T Hagan and H B Demuth ldquoNeural networks for controlrdquoin Proceedings of the American Control Conference (ACC rsquo99)pp 1642ndash1656 June 1999

[11] C Blanc and A Rufer ldquoMultiphysics and energetic modelingof a vanadium redox flow batteryrdquo in Proceedings of the IEEEInternational Conference on Sustainable Energy Technologies(ICSET rsquo08) pp 696ndash701 Singapore November 2008

[12] M Li and T Hikihara ldquoA coupled dynamical model of redoxflow battery based on chemical reaction fluid flow and electri-cal circuitrdquo IEICE Transactions on Fundamentals of ElectronicsCommunications and Computer Sciences A vol 91 no 7 pp1741ndash1747 2008

Journal of Applied Mathematics 7

[13] C BlancModeling of a VanadiumRedox Flow Battery ElectricityStorage System Ecole Polytechnique Federale de Lausanne2009

[14] J Park and I W Sandberg ldquoUniversal approximation usingradial-basis-function networksrdquoNeural Computation vol 3 no2 pp 246ndash257 1991

[15] P H Soslashrensen M Noslashrgaard O Ravn and N K Poulsen ldquoIm-plementation of neural network based non-linear predictivecontrolrdquo Neurocomputing vol 28 no 1ndash3 pp 37ndash51 1999

[16] C Venkateswarlu and K Venkat Rao ldquoDynamic recurrent ra-dial basis function networkmodel predictive control of unstablenonlinear processesrdquo Chemical Engineering Science vol 60 no23 pp 6718ndash6732 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

0 5000 10000 150001500

2000

2500

3000

3500

Time (s)

Pow

er (w

)

Figure 5 Battery power during a charge-discharge cycle

0 5000 10000 150000

01

02

03

04

05

06

Time (s)

Flow

rate

(Ls

)

Figure 6 Optimal flowrate during a charge-discharge cycle

power consumed during the charge at optimal flowratedecreased by 1080W and the average power delivered by thebattery during the discharge increased by 1062W

5 Conclusions

The electrolyte flowrate of VRB system was optimized onlineusing model predictive control based on artificial neural net-works An RBF network is built to predict the future batterypower In order to reduce the computational burden of theoptimization problem the hidden layer nodes were chosen as5 The RBF network model was found to be valid for wideflowrate variation with random load disturbances The gra-dient descent algorithm method is used to realize the opti-mization procedure Simulation result at different flowrateindicates that the proposed controller can enhance the outputpower of battery during the discharge and reduce the operat-ing cost during the charge Future works will focus on controlstrategy for VRB and wind farm combined system

References

[1] K-L Huang X-G Li S-Q Liu N Tan and L-Q ChenldquoResearch progress of vanadium redox flow battery for energy

0 5000 10000 150000

1000

2000

3000

4000

5000

Time (s)

Pow

er (w

)

Q = 03

Q = 005

Q = 2

Q = Qopt

Figure 7 Comparison of battery power at different flowrate

storage in Chinardquo Renewable Energy vol 33 no 2 pp 186ndash1922008

[2] M Vynnycky ldquoAnalysis of a model for the operation of a vana-dium redox batteryrdquo Energy vol 36 no 4 pp 2242ndash2256 2011

[3] X Ma H Zhang C Sun Y Zou and T Zhang ldquoAn optimalstrategy of electrolyte flow rate for vanadium redox flow bat-teryrdquo Journal of Power Sources vol 203 pp 153ndash158 2012

[4] H Al-Fetlawi A A Shah and F C Walsh ldquoNon-isothermalmodelling of the all-vanadium redox flow batteryrdquo Electrochim-ica Acta vol 55 no 1 pp 78ndash89 2009

[5] C Blanc and A Rufer ldquoOptimization of the operating point of avanadium redox flow batteryrdquo in Proceedings of the IEEE EnergyConversion Congress and Exposition (ECCE rsquo09) pp 2600ndash2605San Jose Calif USA September 2009

[6] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[7] T Salsbury P Mhaskar and S Joe Qinc ldquoPredictive controlmethods to improve energy efficiency and reduce demand inbuildingsrdquoComputers and Chemical Engineering vol 51 pp 77ndash85 2013

[8] S J Qin and T A Badgwell ldquoA survey of industrial model pre-dictive control technologyrdquoControl Engineering Practice vol 11no 7 pp 733ndash764 2003

[9] A Draeger S Engell and H Ranke ldquoModel predictive controlusing neural networksrdquo IEEE Control SystemsMagazine vol 15no 5 pp 61ndash66 1995

[10] M T Hagan and H B Demuth ldquoNeural networks for controlrdquoin Proceedings of the American Control Conference (ACC rsquo99)pp 1642ndash1656 June 1999

[11] C Blanc and A Rufer ldquoMultiphysics and energetic modelingof a vanadium redox flow batteryrdquo in Proceedings of the IEEEInternational Conference on Sustainable Energy Technologies(ICSET rsquo08) pp 696ndash701 Singapore November 2008

[12] M Li and T Hikihara ldquoA coupled dynamical model of redoxflow battery based on chemical reaction fluid flow and electri-cal circuitrdquo IEICE Transactions on Fundamentals of ElectronicsCommunications and Computer Sciences A vol 91 no 7 pp1741ndash1747 2008

Journal of Applied Mathematics 7

[13] C BlancModeling of a VanadiumRedox Flow Battery ElectricityStorage System Ecole Polytechnique Federale de Lausanne2009

[14] J Park and I W Sandberg ldquoUniversal approximation usingradial-basis-function networksrdquoNeural Computation vol 3 no2 pp 246ndash257 1991

[15] P H Soslashrensen M Noslashrgaard O Ravn and N K Poulsen ldquoIm-plementation of neural network based non-linear predictivecontrolrdquo Neurocomputing vol 28 no 1ndash3 pp 37ndash51 1999

[16] C Venkateswarlu and K Venkat Rao ldquoDynamic recurrent ra-dial basis function networkmodel predictive control of unstablenonlinear processesrdquo Chemical Engineering Science vol 60 no23 pp 6718ndash6732 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

[13] C BlancModeling of a VanadiumRedox Flow Battery ElectricityStorage System Ecole Polytechnique Federale de Lausanne2009

[14] J Park and I W Sandberg ldquoUniversal approximation usingradial-basis-function networksrdquoNeural Computation vol 3 no2 pp 246ndash257 1991

[15] P H Soslashrensen M Noslashrgaard O Ravn and N K Poulsen ldquoIm-plementation of neural network based non-linear predictivecontrolrdquo Neurocomputing vol 28 no 1ndash3 pp 37ndash51 1999

[16] C Venkateswarlu and K Venkat Rao ldquoDynamic recurrent ra-dial basis function networkmodel predictive control of unstablenonlinear processesrdquo Chemical Engineering Science vol 60 no23 pp 6718ndash6732 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of