Abstract:
A flow battery system includes a flow cell, a reservoir including an aqueous electrolyte, a memory in which command instructions are stored, a model of the flow cell stored within the memory, and a processor configured to execute the command instructions to obtain a current signal and a voltage signal, estimate a state of charge (SOC) of the flow cell using the obtained current signal, compute a model voltage of the flow cell using the obtained current signal, the obtained voltage signal, and the model, compare the model voltage with the obtained voltage signal, calculate a voltage error based upon the comparison of the model voltage with the obtained voltage signal, and correct the estimated SOC based upon the voltage error.

Description:
[0001]    This application claims the benefit of U.S. Provisional Application No. 61/782,195 filed Mar. 14, 2013, the entirety of which is incorporated herein by reference. 
     
    
     FIELD 
       [0002]    This disclosure relates to batteries, and more particularly, to flow battery system and method thereof. 
       BACKGROUND 
       [0003]    A flow battery is a form of rechargeable battery in which electrolyte containing one or more dissolved electro-active species flows through an electrochemical cell that converts chemical energy directly to electricity. The electrolyte is stored externally, generally in tanks, and is pumped through the cell (or cells) of the reactor. Due to the fact that conversion takes place in the cell of the battery, while the electrolyte with the active species is stored in individual tank or tanks, flow battery systems allow separation between power that can be provided or absorbed by the battery and the amount of energy that can be stored. Power is defined by the properties and the dimensions of the cell, while the amount of energy is defined by capacity of the tanks storing the active ingredients. 
         [0004]    During a charge or discharge process the active species react in the cell and are extracted from or introduced back into the electrolyte solution. Depending on architecture of the battery, the active species can be concentrated on the electrodes of the electrochemical cell during, for example, the battery charge process. In either case, the resulting change in concentration of active species results in a change in the observed impedance of the cell for a given operational condition. 
         [0005]    Flow batteries are a promising technology for storage of electrical energy in stationary applications such as grid-scale renewable bulk energy storage systems, rail regeneration storage systems, and grid-scale frequency regulation systems. These applications require large storage capacities and hence only cost-effective technologies are considered as sustainable long-term solutions. 
         [0006]    Because large scale storage systems are stationary, restrictions on dimensions and weight are less strenuous than for mobile systems. Efficiency requirements are also less strict than those for mobile applications since in most situations the stationary systems provide storage for electrical energy which otherwise would be dissipated (in the case of rail regeneration), not generated due to the lack of load in off-peak hours (in the case with wind and solar) or generated with low-efficiency sources such as oil or gas peaker plants (in the case of frequency and peak regulation). 
         [0007]    Control of flow batteries requires knowledge of the flow rate and State of Charge (SOC). Together, flow rate and SOC determine the concentration and availability of reactants at the electrodes and the current that can be drawn from the cell for the best efficiency and within safe limits. The SOC is thus used to determine how much energy the battery can store or deliver. This can be used to plan the usage of the battery in a device or within a power supply system. It may also determine the power that the battery can produce. 
         [0008]    Estimation of SOC of electro-chemical batteries including flow batteries is considered as one of the most challenging and important technical problems that has to be solved in order to guarantee efficient and reliable operation of an energy storage system. Accurate estimation of SOC of a battery is required for evaluation of the amount of energy that is stored in the battery or can be accumulated by the battery. More importantly SOC is required for correct definition of charge and discharge parameters of the battery such as electric currents and voltages that can be applied to and expected from the battery. These parameters define safe operation margins for the battery and affect its instantaneous and long-term performance and its life span. Accurate estimation of SOC allows optimal operation of a given electrochemical battery and, as a result, provides the most efficient technical and economical utilization of individual battery cells and combined battery systems. 
         [0009]    A need exists for a flow battery system which allows for simple and accurate SOC determination. 
       SUMMARY 
       [0010]    In flow batteries, a change in SOC can result a change in the open circuit voltage (OCV) or a change in the observed internal impedance of a flow battery causing the measured output voltage to vary. In this disclosure, a model is used to estimate SOC recursively. The advantage of observing SOC using this technique over direct or indirect SOC measurement techniques or table-based SOC estimation techniques is that the disclosed method is more robust to noise. Furthermore, depending on the design, the observer can reject ‘bad’ data during transients. The method also applies to any generic flow battery. 
         [0011]    In one embodiment, the state of charge (SOC) of a flow battery is estimated using voltage and current measurements, and/or other additional measurements (including but not limited to) flow rate, pressure, and temperature. The disclosed method applies to any generic flow battery which are divided into three classes. The first technique relies on the fact that the internal impedance of a flow battery varies with the SOC of the flow battery. The second technique relies on open circuit voltage (OCV) variation with SOC without significant impedance change. The third class applies to a generic flow battery and carefully estimates impacts of change in OCV due to SOC and change in impedance due to SOC to detect the true SOC. The disclosed method exploits these various facts to design an SOC estimator which uses voltage as a model output, current as a model input, and possibly other measurements such as flow-rate, temperature, and pressure as model inputs. Given the measured inputs, the model output can be computed, i.e. the output voltage of the stack. This computed output voltage is then compared with the measured voltage and a voltage error is calculated. This voltage error is used to correct the current estimate of the SOC of the flow battery that is calculated via coulomb counting using various observer design techniques. 
         [0012]    In one embodiment, a flow battery system includes a flow cell, a reservoir including an aqueous electrolyte, a first sensor configured to generate a current signal indicative of an amplitude of a current passing into or out of the flow cell, a second sensor configured to generate a voltage signal indicative of an output voltage across the flow cell, a memory in which command instructions are stored, a model of the flow cell stored within the memory, and a processor configured to execute the command instructions to obtain the current signal and the voltage signal, estimate a state of charge (SOC) of the flow cell using the obtained current signal, compute a model voltage of the flow cell using the obtained current signal, the obtained voltage signal, and the model, compare the model voltage with the obtained voltage signal, calculate a voltage error based upon the comparison of the model voltage with the obtained voltage signal, and correct the estimated SOC based upon the voltage error. 
         [0013]    In another embodiment, a method of correcting an estimated state of charge (SOC) of a flow battery system includes obtaining a current signal indicative of an amplitude of a current passing into or out of the flow cell, obtaining a voltage signal indicative of an output voltage across the flow cell, estimating with a processor a state of charge (SOC) of the flow cell using the obtained current signal, computing with the processor a model voltage of the flow cell using the obtained current signal and the obtained voltage signal using a model stored in a memory, comparing with the processor the model voltage and the obtained voltage signal, calculating with the processor a voltage error based upon the comparison of the model voltage with the obtained voltage signal, and correcting with the processor the estimated SOC based upon the voltage error. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0014]      FIG. 1  depicts a schematic of a flow battery system in accordance with principles of the present disclosure; 
           [0015]      FIG. 2  depicts a schematic of the battery management system for the battery system of  FIG. 1 ; 
           [0016]      FIG. 3  depicts Gamma functions for a flow-battery; 
           [0017]      FIG. 4  depicts a bulk concentrations as a function of SOC; 
           [0018]      FIG. 5  depicts an equivalent circuit model for flow-battery; and 
           [0019]      FIG. 6  depicts a process controlled by the control system of  FIG. 2  which is various embodiments is used to determine the SOC of the flow battery system of  FIG. 1 . 
       
    
    
     DESCRIPTION 
       [0020]    For the purposes of promoting an understanding of the principles of the disclosure, reference will now be made to the embodiments illustrated in the drawings and described in the following written specification. It is understood that no limitation to the scope of the disclosure is thereby intended. It is further understood that the present disclosure includes any alterations and modifications to the illustrated embodiments and includes further applications of the principles of the disclosure as would normally occur to one skilled in the art to which this disclosure pertains. 
         [0021]      FIG. 1  depicts a schematic of a battery system  100 . In one embodiment, the battery system  100  is used to power a vehicle motor  102  through a bi-directional inverter  104 . The battery system  100  includes a cell or cell stack  106  which includes a negative electrode  108 , a separator  110 , and a positive electrode  112 . The separator  110  is a dense solid electrolyte that transports metal ions but is a barrier to electrons, liquid electrolyte, and oxygen or other species. 
         [0022]    The positive electrode  112  is a porous matrix of electronically conducting material defining a continuous network of passages. The positive electrode  112  includes carbon or some other electronically conductive material that provides a continuous path for electrons, optionally catalyst materials (especially for oxygen reduction and evolution in aqueous chemistries), and optionally binder material. The electrode is porous, with electrolyte  160  in the pores and, in some embodiments, includes gas channels. The conductive material is contacted to an electronically conductive gas-diffusion layer (e.g., from carbon fiber) at the back side (opposite side from the separator  110 ). The gas-diffusion layer (GDL) is open to oxygen or air (from the environment or from an oxygen tank). 
         [0023]    The positive electrode  112  is in fluid connection with a reservoir  118  through a header  120 . The reservoir  118  includes two outlets  122  and  124 . The outlet  122  is connected to the suction side of a centrifugal pump  126  by a discharge header  128 . A discharge header control valve  130  is located in the discharge header  128 . In one embodiment (not shown), the discharge header control valve  130  is located in an outlet header  132  which connects the centrifugal pump  126  to a supply header  134 . 
         [0024]    The outlet  124  is connected to the suction side of a peristaltic pump  140  by a charging header  142 . A charging header control valve  144  is located in the charging header  142 . An outlet header  146  connects the pump  140  to the supply header  134 . A mixing control valve  148  is located in the outlet header  146 . 
         [0025]    The supply header  134  is in fluid communication with the positive electrode  112 . A mixer  150  and a heat exchanger portion  152  of a temperature control unit  154  are operably positioned within the supply header  134 . Additionally, an exchanger portion  156  of a temperature control unit  158  is operably positioned within the reservoir  118 . 
         [0026]    In the embodiment of  FIG. 6 , the reservoir  118 , positive electrode  112 , and headers  128 ,  132 ,  134 ,  142 , and  146  are substantially filled with an aqueous electrolyte. 
         [0027]    In one embodiment, the battery system  100  is a metal halide flow battery wherein “M” is the active metal and “X” is the halogen species. Operation of the battery system  100 , is controlled by a control system  200  shown in  FIG. 2 . The control system  200  includes a processor  202  and a memory  204  in which command instructions are stored. The processor  202  is operably connected to the various components including the pumps  126  and  140 , discharge header control valve  130 , charging header control valve  144 , mixing control valve  148 , and sensors (not shown) such as coulomb counters, temperature sensors, etc. 
         [0028]    The battery management system  200  further includes a model  206  of the battery system  100  stored in the memory  204 . In different embodiments, the model varies from a very simple equivalent circuit model to very complex electrochemical-based model. The more accurate the model, the better is the accuracy of the disclosed method to compute SOC. 
         [0029]    Denoting ‘u’ as the input current, an embodiment of the electrochemical model  200  for the output voltage for the flow battery of  FIG. 1  is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0000]    where, the first term is a constant open circuit voltage (OCV) measured at a reference concentration c X−   ref , the second term corrects for the OCV due to a changing state-of-charge (SOC), and the remaining terms correspond to various impedances that arise due to an applied current u. The variable ‘T’ is the temperature of the electrolyte. The concentrations at the reaction surface in equation (1) depend on the bulk concentrations in the flow by 
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         [0000]    where the gamma-function is a monotonic function that satisfies 
         [0000]      γ (•) (0)=1  5
 
         [0030]    The Gamma-functions can vary widely. One embodiment of a flow-battery includes a form that appears qualitatively as in  FIG. 3 . 
         [0031]    Finally, the bulk concentrations in flow given in equations (2)-(4) are described in terms of the bulk concentrations in the reservoir and the flow-rate ‘N’ as 
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         [0000]    where the bulk concentrations are a monotonic function of the SOC ‘x’, ‘C 0 ’ is the nominal capacity, ‘F’ is Faraday&#39;s constant, and η is the coulombic efficiency. Among different examples, one embodiment of the bulk concentration function is as shown in  FIG. 4 . 
         [0032]    The model given by equations (1)-(9) is a complex model that describes the behavior of a generic flow-battery. The model (1)-(9) is too complex to design state-of-charge algorithms. To alleviate this problem, one can for build an equivalent circuit model from (1)-(9) given as 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0033]    where R 0 , R 1 , and C 1  are impedances modeled as shown in  FIG. 5 . These impedances are functions of state-of-charge, current, temperature, and flow-rate, respectively. 
         [0034]    In  FIG. 5 , The capacitor C 1  is the capacitance due to, among other factors, multi-polar electrode structure and the double layer capacitance. Note that for a DC operation, impedance due to C 1  can be assumed infinite, and hence the total impedance is given by R tot =R 0 +R 1 . The impedance R tot  is thus a function of concentration, u, x, N, T. Thus, in the case where the current u does not vary very quickly, one can neglect the dynamics in (11) due to C 1  and approximate (10) further as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       V 
                       cell 
                     
                     = 
                     
                       
                         U 
                         ocv 
                       
                       + 
                       
                         k 
                          
                         
                             
                         
                          
                         
                           ln 
                            
                           
                             ( 
                             
                               x 
                               
                                 x 
                                 ref 
                               
                             
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         [0000]    where x ref  is some reference SOC. Thus, the reduced approximate model is given by equations (9) and (12). The second term in equation (12) indicates the change in OCV for the actual concentration of the active species in the reservoir of the flow battery, and the third term lumps all kinetic, diffusion and ohmic losses into an impedance term. 
         [0035]    Depending on the type of flow-battery, one has to apply different strategies for SOC estimation. We will next classify all flow batteries into three categories and present a strategy to estimate SOC for each type. 
       Class 1: ‘Flat’ Open Circuit Voltage Flow-Battery 
       [0036]    In a “Flat” open circuit voltage flow battery, the open circuit voltage (OCV) does not change significantly with respect to SOC. Thus, one can model the flow-battery with equations (9) and (12), where one identifies the parameter k≈0. Thus, the model is reduced to 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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                 13 
               
             
             
               
                 
                   
                     V 
                     cell 
                   
                   = 
                   
                     
                       U 
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                     + 
                     
                       
                         
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                           tot 
                         
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                 14 
               
             
           
         
       
     
         [0037]    Now since the impedance R tot  is a function of SOC (x), current (u), temperature (T), and flow-rate (N), one can invert this model for given impedance, current, temperature and flow-rate as 
         [0000]        x =ƒ( R   tot   ,u,T,N )  15
 
         [0000]    where ƒ is now a known function from the model and R tot =∥(V cell −U OCV )/u∥. While one can use this directly for SOC estimation, the estimates will be very sensitive to noise and model uncertainties. Also, when current ‘u’ is zero, the method may fail to give a reliable estimate due to division-by-zero. Thus a more robust way of implementing this solution is to estimate R tot  using a least-squares or other recurrence-based method such as gradient estimator. In once such an embodiment, we define {circumflex over (R)} tot  as the estimate of the total resistance. Then, the online estimation of R tot  is defined by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                          
                         
                           
                             R 
                             ^ 
                           
                           tot 
                         
                       
                       
                          
                         t 
                       
                     
                     = 
                     
                       - 
                       
                         pu 
                          
                         
                           ( 
                           
                             
                               
                                 V 
                                 ^ 
                               
                               cell 
                             
                             - 
                             
                               U 
                               OCV 
                             
                             + 
                             
                               u 
                                
                               
                                 
                                   R 
                                   ^ 
                                 
                                 tot 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 16 
               
             
             
               
                 
                   
                     
                        
                       p 
                     
                     
                        
                       t 
                     
                   
                   = 
                   
                     
                       β 
                        
                       
                           
                       
                        
                       p 
                     
                     - 
                     
                       p 
                       2 
                     
                     - 
                     
                       
                         p 
                         2 
                       
                        
                       
                         
                           u 
                           2 
                         
                         . 
                       
                     
                   
                 
               
               
                 17 
               
             
           
         
       
     
         [0000]    where {circumflex over (V)} cell  is the measured voltage, U OCV  is the known open-circuit voltage and p is a positive number. The SOC estimate is then defined as 
         [0000]        {circumflex over (x)} =ƒ( {circumflex over (R)}   tot   ,u,T,N )  18
 
         [0038]    Thus, for “Flat” open circuit voltage flow-batteries, the strategy to estimate SOC is to use equations (16)-(18). 
       Class 2: ‘Flat’ Impedance Characteristic Flow-Battery 
       [0039]    In a “Flat” impedance characteristic flow-battery, the impedance does not vary significantly with respect to SOC. For such a case, the model is given by equations (9) and (12), where R tot  is not a function of x anymore. Then one can estimate SOC indirectly by observing the error between measured voltage and model voltage, and designing a standard observer (eg. Kalman Filter, Luenberger observer, etc.). Thus, the estimator is directly defined for the SOC ‘x’ in contrast to class 1, where the estimator was defined for the impedance. An embodiment of such an observer is given as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       
                         x 
                         ^ 
                       
                     
                     
                        
                       t 
                     
                   
                   = 
                   
                     
                       
                         η 
                          
                         
                           ( 
                           
                             u 
                             , 
                             T 
                           
                           ) 
                         
                       
                        
                       
                         u 
                         
                           C 
                           0 
                         
                       
                     
                     + 
                     
                       
                         L 
                          
                         
                           ( 
                           
                             
                               V 
                               cell 
                             
                             - 
                             
                               
                                 V 
                                 ^ 
                               
                               cell 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 19 
               
             
           
         
       
     
         [0000]    where L&gt;0 is a constant gain, {circumflex over (x)} represents the estimated SOC and {circumflex over (V)} cell  is the model output. 
       Class 3: Generic Flow-Battery Model 
       [0040]    In the most general class of flow-battery models, all parameters of the battery model can vary with SOC. Thus, both OCV and the impedance are functions of SOC, in addition to other variables such as current u, temperature T, flow-rate N, etc. Now for a flow-battery, with everything else constant, if SOC increases then R tot  monotonically decreases. This can be mathematically expressed as for every x 1 ≠x 2 , there exists M 1 , M 2 &gt;0 such that 
         [0000]      − M   1 ( x   1   −x   2 ) 2   &gt;[R   tot ( x   1   ,u,T,N )− R   tot ( x   2   ,u,T,N )]( x   1   −x   2 )&gt;− M   2 ( x   1   −x   2 ) 2   20
 
         [0041]    Additionally note that since the OCV is not flat, k&gt;0. Then, an embodiment of the estimator to estimate SOC for a generic model of flow-battery is as follows. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       
                         x 
                         ^ 
                       
                     
                     
                        
                       t 
                     
                   
                   = 
                   
                     
                       
                         η 
                          
                         
                           ( 
                           
                             u 
                             , 
                             T 
                           
                           ) 
                         
                       
                        
                       
                         u 
                         
                           C 
                           0 
                         
                       
                     
                     + 
                     
                       
                         L 
                          
                         
                           ( 
                           
                             
                               V 
                               cell 
                             
                             - 
                             
                               
                                 V 
                                 ^ 
                               
                               cell 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 21 
               
             
             
               
                 
                   L 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 0 
                                  
                                 
                                     
                                 
                                  
                                 if 
                                  
                                 
                                     
                                 
                                  
                                 u 
                               
                               &gt; 
                               0 
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   L 
                                   0 
                                 
                                  
                                 
                                     
                                 
                                  
                                 if 
                                  
                                 
                                     
                                 
                                  
                                 u 
                               
                               ≤ 
                               0 
                             
                             , 
                           
                         
                       
                     
                   
                 
               
               
                 22 
               
             
           
         
       
     
         [0000]    and L 0 &gt;0. To show that this estimator indeed estimates SOC correctly, we investigate the error dynamics given by the nonlinear system 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       
                         x 
                         ^ 
                       
                     
                     
                        
                       t 
                     
                   
                   = 
                   
                     
                       
                         - 
                         Lk 
                       
                        
                       
                           
                       
                        
                       
                         ln 
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 x 
                                 ~ 
                               
                               
                                 x 
                                 ^ 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         Lu 
                          
                         
                           [ 
                           
                             
                               
                                 R 
                                 tot 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       x 
                                       ^ 
                                     
                                     + 
                                     
                                       x 
                                       ~ 
                                     
                                   
                                   , 
                                   u 
                                   , 
                                   T 
                                   , 
                                   N 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 R 
                                 tot 
                               
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   , 
                                   u 
                                   , 
                                   T 
                                   , 
                                   N 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       . 
                     
                   
                 
               
               
                 23 
               
             
           
         
       
     
         [0042]    It is easy to show that the error dynamics identified in equation (23) is stable. To see this, we calculate the derivative of the Lyapunov function V(x)=x 2 /2 given as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       V 
                     
                     
                        
                       t 
                     
                   
                   = 
                   
                     
                       
                         - 
                         Lk 
                       
                        
                       
                           
                       
                        
                       
                         ln 
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 x 
                                 ~ 
                               
                               
                                 x 
                                 ^ 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         x 
                         ~ 
                       
                     
                     - 
                     
                       Lu 
                        
                       
                           
                       
                        
                       
                         
                           
                             x 
                             ~ 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   R 
                                   tot 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         x 
                                         ^ 
                                       
                                       + 
                                       
                                         x 
                                         ~ 
                                       
                                     
                                     , 
                                     u 
                                     , 
                                     T 
                                     , 
                                     N 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   R 
                                   tot 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       ^ 
                                     
                                     , 
                                     u 
                                     , 
                                     T 
                                     , 
                                     N 
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 24 
               
             
           
         
       
     
         [0043]    Clearly the derivative is zero for u&gt;0. For u≦0, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                              
                             V 
                           
                           
                              
                             t 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 L 
                                 0 
                               
                             
                              
                             k 
                              
                             
                                 
                             
                              
                             
                               ln 
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       x 
                                       ~ 
                                     
                                     
                                       x 
                                       ^ 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               x 
                               ~ 
                             
                           
                           + 
                           
                             
                               L 
                               0 
                             
                              
                             
                                
                               u 
                                
                             
                              
                             
                               
                                 x 
                                 ~ 
                               
                               [ 
                               
                                 
                                   
                                     R 
                                     tot 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           x 
                                           ^ 
                                         
                                         + 
                                         
                                           x 
                                           ~ 
                                         
                                       
                                       , 
                                       u 
                                       , 
                                       T 
                                       , 
                                       N 
                                     
                                     ) 
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                             
                            
                           
                             
                               R 
                               tot 
                             
                              
                             
                               ( 
                               
                                 
                                   x 
                                   ^ 
                                 
                                 , 
                                 u 
                                 , 
                                 T 
                                 , 
                                 N 
                               
                               ) 
                             
                           
                           ] 
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           ≤ 
                             
                            
                           
                             
                               
                                 - 
                                 
                                   L 
                                   0 
                                 
                               
                                
                               k 
                                
                               
                                   
                               
                                
                               
                                 ln 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       
                                         x 
                                         ~ 
                                       
                                       
                                         x 
                                         ^ 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 x 
                                 ~ 
                               
                             
                             - 
                             
                               
                                 L 
                                 0 
                               
                                
                               
                                  
                                 u 
                                  
                               
                                
                               
                                 M 
                                 1 
                               
                                
                               
                                 
                                   x 
                                   ~ 
                                 
                                 2 
                               
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         &lt; 
                           
                          
                         0 
                       
                     
                   
                 
               
               
                 25 
               
             
           
         
       
     
         [0044]    The last step follows since for {tilde over (x)}&gt;0, 
         [0000]    
       
         
           
             
               
                 ln 
                  
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         x 
                         ~ 
                       
                       x 
                     
                   
                   ) 
                 
               
                
               
                 x 
                 ~ 
               
             
             &gt; 
             0. 
           
         
       
     
         [0000]    Suppose {tilde over (x)}&lt;0. Then 
         [0000]    
       
         
           
             
                
               
                 
                   x 
                   ~ 
                 
                 
                   x 
                   ^ 
                 
               
                
             
             &lt; 
             1. 
           
         
       
     
       Hence, 
       [0045]    
       
         
           
             
               
                 
                   
                     
                       ln 
                        
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               x 
                               ~ 
                             
                             
                               x 
                               ^ 
                             
                           
                         
                         ) 
                       
                     
                      
                     
                       x 
                       ~ 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 x 
                                 ~ 
                               
                               
                                 x 
                                 ^ 
                               
                             
                             ) 
                           
                           2 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               
                                 1 
                                 2 
                               
                                
                               
                                 
                                   x 
                                   ~ 
                                 
                                 
                                   x 
                                   ^ 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           x 
                           ^ 
                         
                       
                       + 
                       
                         
                           
                             ( 
                             
                               
                                 x 
                                 ~ 
                               
                               
                                 x 
                                 ^ 
                               
                             
                             ) 
                           
                           4 
                         
                          
                         
                           ( 
                           
                             
                               1 
                               3 
                             
                             - 
                             
                               
                                 1 
                                 4 
                               
                                
                               
                                 
                                   x 
                                   ~ 
                                 
                                 
                                   x 
                                   ^ 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           x 
                           ^ 
                         
                       
                       + 
                       
                         
                           
                             ( 
                             
                               
                                 x 
                                 ~ 
                               
                               
                                 x 
                                 ^ 
                               
                             
                             ) 
                           
                           6 
                         
                          
                         
                           ( 
                           
                             
                               1 
                               5 
                             
                             - 
                             
                               
                                 1 
                                 6 
                               
                                
                               
                                 
                                   x 
                                   ~ 
                                 
                                 
                                   x 
                                   ^ 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           x 
                           ^ 
                         
                       
                       + 
                       … 
                     
                     &gt; 
                     0. 
                   
                 
               
               
                 26 
               
             
           
         
       
     
         [0046]    Hence the strategy/algorithm given by (21) and (22) converges for a large class of generic flow batteries. 
         [0047]      FIG. 6  depicts a flow chart for application of the desired model depending upon the class of flow battery. 
         [0048]    While the disclosure has been illustrated and described in detail in the drawings and foregoing description, the same should be considered as illustrative and not restrictive in character. It is understood that only the preferred embodiments have been presented and that all changes, modifications and further applications that come within the spirit of the disclosure are desired to be protected.