Patent Publication Number: US-2017373331-A1

Title: State detection device and method for fuel cell

Description:
TECHNICAL FIELD 
     This invention relates to state detection device and method for fuel cell. 
     BACKGROUND ART 
     A state detection device for fuel cell is known which measures a voltage value and an impedance value of a fuel cell and detects an internal state of the fuel cell on the basis of these values. 
     For example, it is proposed in Japanese Patent No. 4640661 to calculate a first impedance in a first frequency region corresponding to an electrolyte membrane resistance and a second impedance in a second frequency region corresponding to the sum of the electrolyte membrane resistance and a catalyst layer resistance and lower than the first frequency region and calculate a water content of a catalyst layer on the basis of a differential impedance between the second and first impedances. 
     Further, it is described in JP2005-285614A to acquire complex impedances corresponding to a frequency F 1  at an intersection with a real axis of a complex impedance curve (Cole-Cole plot) of a fuel cell, a frequency F 2  in a first region expressing a reaction resistance (reaction resistance of a cathode electrode) when oxygen reacts and a frequency F 3  in a second region expressing a resistance concerning oxygen diffusion and obtain an internal resistance value from the obtained complex impedances. 
     SUMMARY OF INVENTION 
     However, it is not possible to grasp each of state quantities of an anode electrode and those of a cathode electrode in Japanese Patent No. 4640661. Further, it is also difficult in JP2005-285614A to individually grasp the state of the anode electrode and that of the cathode electrode since the state of the anode electrode and that of the cathode electrode are mixed in the impedance curve. 
     The present invention was developed, focusing on such a problem, and aims to provide a state detection device and method for fuel cell capable of individually detecting internal state quantities such as state quantities of an anode electrode and those of a cathode electrode in a fuel cell. 
     According to one aspect of the present invention, the present invention provides a state detection device for a fuel cell for generating power upon receiving a supply of anode gas and cathode gas. More specifically, the state detection device includes an impedance acquisition unit configured to acquire a high frequency impedance based on a frequency selected from a high frequency band and a low frequency impedance based on a frequency selected from a low frequency band, the high frequency band including a frequency band which shows responsiveness at least to a state quantity of an anode electrode, the low frequency band including a frequency band which shows responsiveness at least to a state quantity of a cathode electrode, and an internal state quantity estimation unit configured to estimate each of the state quantity of the anode electrode and the state quantity of the cathode electrode by combining the acquired high frequency impedance and low frequency impedance, the state quantity of the anode electrode and the state quantity of the cathode electrode serving as internal states of the fuel cell. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a perspective view of a fuel cell according to an embodiment of the present invention, 
         FIG. 2  is a sectional view along II-II of the fuel cell of  FIG. 1 , 
         FIG. 3  is a schematic configuration diagram of a fuel cell system according to the embodiment of the present invention. 
         FIG. 4A  is a diagram showing a path of a current flowing in a simplified equivalent circuit model of a fuel cell in the case of applying an alternating-current voltage in a low frequency band. 
         FIG. 4B  is a diagram showing a path of a current flowing in the simplified equivalent circuit model of the fuel cell in the case of applying an alternating-current voltage in a frequency band higher than in the case of  FIG. 4A . 
         FIG. 4C  is a diagram showing a path of a current flowing in the simplified equivalent circuit model of the fuel cell in the case of applying an alternating-current voltage in a frequency band higher than in the case of  FIG. 4B . 
         FIG. 4D  is a diagram showing a path of a current flowing in the simplified equivalent circuit model of the fuel cell in the case of inputting an alternating-current voltage in a high frequency band. 
         FIG. 5  is a flow chart showing the flow of state quantity estimation according to one embodiment. 
         FIG. 6  is a flow chart showing the flow of state quantity estimation according to one embodiment. 
         FIG. 7  is a graph showing I-V characteristic curves of the fuel cell respectively in steady time and in unsteady time. 
         FIG. 8  is a flow chart showing the flow of state quantity estimation according to one embodiment. 
         FIG. 9  shows frequency responses of candidates for an electrical double layer capacitance of a cathode electrode. 
         FIG. 10A  shows frequency responses of candidates for an electrical double layer capacitance of an anode electrode. 
         FIG. 10B  shows frequency responses of candidates for a reaction resistance value of the anode electrode  112 . 
         FIG. 11  is a flow chart showing the flow of state quantity estimation according to one embodiment. 
         FIG. 12  shows an I-V characteristic curve of the fuel cell  1  in steady time, 
         FIG. 13  is a graph showing an example of a method for setting a set of current and voltage for the calculation of a gradient ΔV/ΔI in the I-V characteristic curve, and 
         FIG. 14  is a block diagram schematically showing a main part relating to an impedance measurement in a fuel cell system according to one embodiment. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Hereinafter, embodiments of the present invention are described with reference to the drawings and the like. 
     A fuel cell is configured such that an electrolyte membrane is sandwiched by an anode electrode serving as a fuel electrode and a cathode electrode serving as an oxidant electrode. The fuel cell generates power using anode gas containing hydrogen and supplied to the anode electrode and cathode gas containing oxygen and supplied to the cathode electrode. Electrode reactions which proceed in both anode and cathode electrodes are as follows. 
       Anode electrode: 2H 2 →4H + +4e −    (1)
 
       Cathode electrode: 4H + +4e − +O 2 →2H 2 O   (2)
 
       FIGS. 1 and 2  are views showing the configuration of a fuel cell  10  according to one embodiment of the present invention.  FIG. 1  is a perspective view of the fuel cell  10 .  FIG. 2  is a sectional view along II-II of the fuel cell  10  of  FIG. 1 . 
     As shown in  FIGS. 1 and 2 , the fuel cell  10  includes a membrane electrode assembly (MEA)  11 , and an anode separator  12  and a cathode separator  13  arranged to sandwich the MEA  11 . 
     The MEA  11  is composed of an electrolyte membrane  111 , an anode electrode  112  and a cathode electrode  113 . The MEA  11  includes the anode electrode  112  on one surface side of the electrolyte membrane  111  and the cathode electrode  113  on the other surface side. 
     The electrolyte membrane  11  is a proton conductive ion exchange membrane formed of fluororesin. The electrolyte membrane  111  exhibits good electrical conductivity in a wet state. It should be noted that another material such as a material having a phosphoric acid (H 3 PO 4 ) impregnated in a predetermined matrix may be used according to a possible response of a fuel cell. 
     The anode electrode  112  includes a catalyst layer  112 A and a gas diffusion layer  112 B. The catalyst layer  112 A is a member formed of platinum or carbon black particles carrying platinum or the like and provided in contact with the electrolyte membrane  111 . The gas diffusion layer  112 B is provided on an outer side of the catalyst layer  112 A. The gas diffusion layer  112 B is a member formed of carbon cloth having gas diffusion property and electrical conductivity and provided in contact with the catalyst layer  112 A and the anode separator  12 . 
     Similarly to the anode electrode  112 , the cathode electrode  13  also includes a catalyst layer  113 A and a gas diffusion layer  113 B. The catalyst layer  113 A is arranged between the electrolyte membrane  111  and the gas diffusion layer  113 B and the gas diffusion layer  113 B is arranged between the catalyst layer  113 A and the cathode separator  13 . 
     The anode separator  112  is arranged on an outer side of the gas diffusion layer  112 B. The anode separator  12  includes a plurality of anode gas flow passages  121  for supplying anode gas (hydrogen gas) to the anode electrode  112 . The anode gas flow passages  121  are formed as groove-like passages. 
     The cathode separator  13  is arranged on an outer side of the gas diffusion layer  113 B. The cathode separator  13  includes a plurality of cathode gas flow passages  131  for supplying cathode gas (air) to the cathode electrode  113 . The cathode gas flow passages  131  are formed as groove-like passages. 
     The anode separator  12  and the cathode separator  13  are so configured that the anode gas flowing in the anode gas flow passages  121  and the cathode gas flowing in the cathode gas flow passages  131  flow in directions opposite to each other. It should be noted that the anode separator  12  and the cathode separator  13  may be so configured that these gases flow in the same direction. 
     In the case of using such a fuel cell  10  as a power source for an automotive vehicle, a fuel cell stack in which several hundreds of fuel cells  10  are laminated is used since required power is large. Power for driving the vehicle is taken out by configuring a fuel cell system for supplying anode gas and cathode gas to the fuel cell stack. It should be noted that although an impedance measurement to be described later is conducted for each fuel cell stack in which the fuel cells  10  are laminated in the present embodiment, the impedance measurement may be conducted for each fuel cell  10  or for each part (e.g., several tens of cells) of the fuel cell stack. 
     Further, in the fuel cell stack, an anode electrode, a cathode electrode and an electrolyte membrane serving as sums are configured by arranging the anode electrodes  112 , the cathode electrodes  113  and the electrolyte membranes  111  of a plurality of the fuel cells  10  in series. However, for the convenience of description, these anode electrode, cathode electrode and electrolyte membrane serving as the sums are also denoted by the same reference signs as the anode electrode  112 , the cathode electrode  113  and the electrolyte membrane  111  of the single cell. 
       FIG. 3  is a schematic diagram of a fuel cell system  100  according to one embodiment of the present invention. 
     The fuel cell system  100  includes a fuel cell  1 , a cathode gas supplying/discharging device  2 , an anode gas supplying/discharging device  3 , a power system  5  and a controller  6 . 
     The fuel cell  1  is a laminated battery formed by laminating a plurality of fuel cells  10  (unit cells) as described above. The fuel cell  1  generates power necessary to drive a vehicle upon receiving the supply of the anode gas and the cathode gas. The fuel cell  1  includes an anode electrode side terminal  1 A and a cathode electrode side terminal  1 B as output terminals for taking out power. 
     The cathode gas supplying/discharging device  2  supplies the cathode gas to the fuel cell  1  and discharges cathode off-gas discharged from the fuel cell  1  to outside. The cathode gas supplying/discharging device  2  includes a cathode gas supply passage  21 , a cathode gas discharge passage  22 , a filter  23 , an air flow sensor  24 , a cathode compressor  25 , a cathode pressure sensor  26 , a water recovery device (WRD)  27  and a cathode pressure control valve  28 . 
     The cathode gas supply passage  21  is a passage in which the cathode gas to be supplied to the fuel cell  1  flows. One end of the cathode gas supply passage  21  is connected to the filter  23  and the other end is connected to a cathode gas inlet part of the fuel cell  1 . 
     The cathode gas discharge passage  22  is a passage in which the cathode off-gas discharged from the fuel cell  1  flows. One end of the cathode gas discharge passage  22  is connected to a cathode gas outlet part of the fuel cell  1  and the other end is formed as an opening end. The cathode off-gas is mixture gas containing the cathode gas, steam produced by the electrode reaction and the like. 
     The filter  23  is a member for removing dust, dirt and the like contained in the cathode gas to be taken into the cathode gas supply passage  21 . 
     The cathode compressor  25  is provided downstream of the filter  23  in the cathode gas supply passage  21 . The cathode compressor  25  supplies the cathode gas in the cathode gas supply passage  21  to the fuel cell  1  by feeding the cathode gas under pressure. 
     The air flow sensor  24  is provided between the filter  23  and the cathode compressor  25  in the cathode gas supply passage  21 . The air flow sensor  24  detects a flow rate of the cathode gas to be supplied to the fuel cell  1 . 
     The cathode pressure sensor  26  is provided between the cathode compressor  25  and the WRD  27  in the cathode gas supply passage  21 . The cathode pressure sensor  26  detects a pressure of the cathode gas to be supplied to the fuel cell  1 . The cathode gas pressure detected by the cathode pressure sensor  26  represents a pressure of an entire cathode system including the cathode gas flow passages of the fuel cell  1  and the like. 
     The WRD  27  is connected over the cathode gas supply passage  21  and the cathode gas discharge passage  22 . The WRD  27  is a device for recovering moisture in the cathode off-gas flowing in the cathode gas discharge passage  22  and humidifying the cathode gas flowing in the cathode gas supply passage  21  with that recovered moisture. 
     The cathode pressure control valve  28  is provided downstream of the WRD  27  in the cathode gas discharge passage  22 . The cathode pressure control valve  28  is controlled to open and close by the controller  6  and adjusts the pressure of the cathode gas to be supplied to the fuel cell  1 . 
     Next, the anode gas supplying/discharging device  3  is described. 
     The anode gas supplying/discharging device  3  supplies the anode gas to the fuel cell  1  and discharges anode off-gas discharged from the fuel cell to the cathode gas discharge passage  22 . The anode gas supplying/discharging device  3  includes a high-pressure tank  31 , an anode gas supply passage  32 , an anode pressure control valve  33 , an anode pressure sensor  34 , an anode gas discharge passage  35 , a buffer tank  36 , a purge passage  37  and a purge valve  38 . 
     The high-pressure tank  31  is a container for storing the anode gas to be supplied to the fuel cell  1  in a high-pressure state. 
     The anode gas supply passage  32  is a passage for supplying the anode gas discharged from the high-pressure tank  31  to the fuel cell  1 . One end of the anode gas supply passage  32  is connected to the high-pressure tank  31  and the other end is connected to an anode gas inlet part of the fuel cell  1 . 
     The anode pressure control valve  33  is provided downstream of the high-pressure tank  31  in the anode gas supply passage  32 . The anode pressure control valve  33  is controlled to open and close by the controller  6  and adjusts the pressure of the anode gas to be supplied to the fuel cell  1 . 
     The anode pressure sensor  34  is provided downstream of the anode pressure control valve  33  in the anode gas supply passage  32 . The anode pressure sensor  34  detects a pressure of the anode gas to be supplied to the fuel cell  1 . The anode gas pressure detected by the anode pressure sensor  34  represents a pressure of an entire anode system including the buffer tank  36 , the anode gas flow passages of the fuel cell  1  and the like. 
     The anode gas discharge passage  35  is a passage in which the anode off-gas discharged from the fuel cell  1  flows. One end of the anode gas discharge passage  35  is connected to an anode gas outlet part of the fuel cell  1  and the other end is connected to the buffer tank  36 . The anode off-gas contains the anode gas not used in the electrode reaction, impurity gas such as nitrogen having leaked from the cathode gas flow passages  131  to the anode gas flow passages  121 , moisture and the like. 
     The buffer tank  36  is a container for temporarily storing the anode off-gas flowing from the anode gas discharge passage  35 . The anode off-gas pooled in the buffer tank  36  is discharged to the cathode gas discharge passage  22  through the purge passage  37  when the purge valve  38  is opened. 
     The purge passage  37  is a passage for discharging the anode off-gas. One end of the purge passage  37  is connected to the anode gas discharge passage  35  and the other end is connected to a part of the cathode gas discharge passage  22  downstream of the cathode pressure control valve  28 . 
     The purge valve  38  is provided in the purge passage  37 . The purge valve  38  is controlled to open and close by the controller  6  and controls a purge flow rate of the anode off-gas discharged from the anode gas discharge passage  35  to the cathode gas discharge passage  22 . 
     When a purge control is executed to open the purge valve  38 , the anode off-gas is discharged to outside through the purge passage  37  and the cathode gas discharge passage  22 . At this time, the anode off-gas is mixed with the cathode off-gas in the cathode gas discharge passage  22 . By mixing the anode off-gas and the cathode off-gas and discharging the mixture gas to outside in this way, an anode gas concentration (hydrogen concentration) in the mixture gas is set at a value not larger than a discharge allowable concentration. 
     The power system  5  includes a current sensor  51 , a voltage sensor  52 , a travel motor  53 , an inverter  54 , a battery  55  and a DC/DC converter  56 . 
     The current sensor  51  detects an output current extracted from the fuel cell  1 . The voltage sensor  52  detects an output voltage of the fuel cell  1 , i.e., an inter-terminal voltage between the anode electrode side terminal  1 A and the cathode electrode side terminal  1 B. The voltage sensor  52  may be configured to detect a voltage of each fuel cell  10  or may be configured to detect a voltage of each group composed of a plurality of the fuel cells  10 . 
     The travel motor  53  is a three-phase alternating-current synchronous motor and a drive source for driving wheels. The travel motor  53  has a function serving as a motor to be rotationally driven upon receiving the supply of power from the fuel cell  1  and the battery  55  and a function serving as a generator for generating power by being rotationally driven by an external force. 
     The inverter  54  is composed of a plurality of semiconductor switches such as IGBTs. The semiconductor switches of the inverter  54  are switching-controlled by the controller  6 , thereby converting direct-current power into alternating-current power or alternating-current power into direct-current power. The inverter  54  converts composite direct-current power of output power of the fuel cell  1  and output power of the battery  55  into three-phase alternating-current power and supplies this power to the travel motor  53  when the travel motor  53  is caused to function as the motor. In contrast, the inverter  54  converts regenerative power (three-phase alternating-current power) of the travel motor  53  into direct-current power and supplies this power to the battery  55  when the travel motor  53  is caused to function as the generator. 
     The battery  55  is configured to be charged with a surplus of the output power of the fuel cell  1  and the regenerative power of the travel motor  53 . The power charged into the battery  55  is supplied to the travel motor  53  and auxiliary machines such as the cathode compressor  25  if necessary. 
     The DC/DC converter  56  is a bidirectional voltage converter for increasing and decreasing the output voltage of the fuel cell  1 . By controlling the output voltage of the fuel cell  1  by the DC/DC converter  56 , the output current of the fuel cell  1  and the like are adjusted. 
     The controller  6  is configured by a microcomputer including a central processing unit (CPU), a read-only memory (ROM), a random access memory (RAM) and an input/output interface (I/O interface). To the controller  6  are input signals from sensors such as an accelerator stroke sensor (not shown) for detecting a depressed amount of an accelerator pedal besides signals from various sensors such as the current sensor  51  and the voltage sensor  52 . 
     The controller  6  adjusts the pressures and flow rates of the anode gas and the cathode gas to be supplied to the fuel cell  1  by controlling the anode pressure control valve  33 , the cathode pressure control valve  28 , the cathode compressor  25  and the like according to an operating state of the fuel cell system  100 . 
     Further, the controller  6  calculates target output power on the basis of power required by the travel motor  53 , power required by the auxiliary machines such as the cathode compressor  25 , charge/discharge requests of the battery  55  and the like. The controller  6  calculates a target output current of the fuel cell  1  on the basis of the target output power by referring to an IV characteristic (current-voltage characteristic) of the fuel cell  1  determined in advance. Then, the controller  6  controls the output voltage of the fuel cell  1  by the DC/DC converter  56  such that the output current of the fuel cell  1  reaches the target output current, and executes a control to supply a necessary current to the travel motor  53  and the auxiliary machines. 
     Further, the controller  6  controls the cathode compressor  25  and the like such that a degree of wetness (water content) of each electrolyte membrane  111  of the fuel cell  1  is in a state suitable for power generation. 
     Further, the controller calculates an impedance Z of the fuel cell  1  at a predetermined frequency by dividing an amplitude value of a voltage value, in which an alternating-current signal of the predetermined frequency is superimposed on an output voltage of the fuel cell  1 , by an amplitude value of a current value likewise superimposed with an alternating-current signal in first to sixth embodiments described later. 
     In the fuel cell system  100  described as above, a state detection device for the fuel cell  1  is configured by the controller  6 , the current sensor  51 , the voltage sensor  52  and the DC/DC converter  56 . 
     In the present embodiment, a simplified equivalent circuit model taking into account of a reaction resistance R a  and an electrical double layer capacitance C a , which are state quantities of the anode electrode  112  in the fuel cell  1 , a reaction resistance R c  and an electrical double layer capacitance C c , which are state quantities of the cathode electrode  113 , and an electrolyte membrane resistance value R m , which is a state quantity of the electrolyte membrane  111 , is set and a state of the fuel cell  1  is estimated on the basis of this simplified equivalent circuit model. 
     It should be noted that the electrolyte membrane resistance value R m  is a state quantity whose value is determined according to a degree of wetness of the electrolyte membrane  111 . Normally, as the electrolyte membrane  111  becomes drier, the electrolyte membrane resistance value R m  tends to increase. 
     Further, the reaction resistance value R a  of the anode electrode  112  increases and decreases according to the reaction of the anode gas in the anode electrode  112 . For example, if there is a factor due to which the reaction does not smoothly proceed such as a shortage of the anode gas, the reaction resistance value R a  increases according to this. 
     Furthermore, the electrical double layer capacitance C a  of the anode electrode  112  is modeled to represent an electrical capacitance of the anode electrode  112  in the fuel cell  1 . Thus, the electrical double layer capacitance C a  is determined on the basis of various elements such as a constituting material, the size and the like of the anode electrode  112 . 
     Further, the reaction resistance value R c  of the cathode electrode  113  increases and decreases according to the reaction of the cathode gas in the cathode electrode  113 . For example, if there is a factor due to which the reaction does not smoothly proceed such as a shortage of the cathode gas, the reaction resistance value R c  increases according to this. 
     Furthermore, the electrical double layer capacitance C c  of the cathode electrode  113  is modeled to represent an electrical capacitance of the cathode electrode  113 . Thus, the electrical double layer capacitance value C c  is determined on the basis of various elements such as a constituting material, the size and the like of the cathode electrode  113 . 
     Here, the present inventors found out that there was a frequency dependent characteristic in a path, along which an alternating-current signal (alternating current) superimposed on an output current of the fuel cell  1  flowed in the fuel cell, in the simplified equivalent circuit model of the fuel cell  1 . The frequency dependent characteristic in the path along which the alternating current flows in the fuel cell is described below. 
       FIGS. 4A to 4D  are diagrams schematically showing a path, along which an alternating current superimposed on an output current of the fuel cell  1  flows, in the equivalent circuit model of the fuel cell  1  according to the present embodiment for each frequency band of the alternating current. 
       FIG. 4A  shows a path of an alternating current of a frequency belonging to a low frequency band, for example, near 0 Hz (hereinafter, also written as a first frequency band). Further,  FIG. 4B  shows a path of an alternating current of a frequency belonging to a frequency band slightly higher than the first frequency band by about several Hz (hereinafter, also written as a second frequency band). Furthermore,  FIG. 4C  shows a path of an alternating current of a frequency belonging to a frequency band slightly higher than the second frequency band by about several tens of Hz to several KHz (hereinafter, also written as a third frequency band). Further,  FIG. 4D  shows a path of an alternating current of a frequency belonging to a highest frequency band of several tens of KHz or higher (hereinafter, also written as a fourth frequency band). Note that the path of the alternating current is shown by a thick line in  FIGS. 4A to 4D . 
     First, the value of the alternating current of the frequency belonging to the first frequency band shown in  FIG. 4A  moderately varies since the frequency is low, and properties of the alternating current are close to those of a direct current having a constant current value. Thus, the alternating current having the properties close to those of the direct current does not flow to the electrical double layer capacitance of the anode electrode  112  and the electrical double layer capacitance of the cathode electrode  113  or, even if the alternating current flows, the magnitude thereof is small to a negligible extent. Specifically, as shown in  FIG. 4A , the alternating current substantially flows only to the reaction resistance of the anode electrode  112 , the electrolyte membrane resistance and the reaction resistance of the cathode electrode  113 . 
     Next, the value of the alternating current of the frequency belonging to the second frequency band shown in  FIG. 4B  more largely varies as compared to the alternating current of the frequency belonging to the first frequency band, and properties as the alternating current are intensified. Thus, as shown in  FIG. 4B , the alternating current is thought to start flowing also toward the electrical double layer capacitance of the cathode electrode  113 . 
     On the other hand, since the reaction resistance value R a  of the anode electrode  112  is known to have a much smaller value than the reaction resistance value R c  of the cathode electrode  113 , the current relatively easily flows toward the reaction resistance of the anode electrode  112 . Thus, it is thought that the alternating current of the frequency in the second frequency band still does not flow toward the electrical double layer capacitance part of the anode electrode  112  or, even if the alternating current flows, the magnitude thereof is small to a negligible extent. 
     Further, the value of the alternating current of the frequency belonging to the third frequency band shown in  FIG. 4C  more largely varies as compared to the alternating current of the frequency belonging to the second frequency band, and properties as the alternating current are further intensified. Thus, the influence of the electrical double layer capacitance of the anode electrode  112  can be no longer ignored and the current is thought to flow also to the electrical double layer capacitance of the anode electrode  112 . 
     On the other hand, in this third frequency band, an oxidation/reduction reaction in the cathode electrode  13  cannot follow a variation speed of the value of the above alternating current and a state occurs in which this oxidation/reduction reaction does not apparently occur. 
     Accordingly, the cathode gas substantially does not react in the cathode electrode  113 , wherefore the influence of the reaction resistance of the cathode electrode  113  due to the above oxidation/reduction reaction can be ignored. 
     Specifically, in the third frequency band, the alternating current does not flow to the reaction resistance of the cathode electrode  113  or, even if the alternating current flows, the magnitude thereof is small to a negligible extent. Thus, the alternating current is thought to substantially flow only to the electrical double layer capacitance component. 
     It should be noted that performance of the oxidation/reduction reaction to follow a variation of the value of the alternating current is relatively high in the anode electrode  112  and this oxidation/reduction reaction can still follow the variation of the value of the alternating current in the third frequency band. Thus, as shown in  FIG. 4C , the alternating current of the frequency belonging to the third frequency band is thought to still flow through the reaction resistance of the anode electrode  112 . 
     The value of the alternating current of the frequency belonging to the fourth frequency band shown in  FIG. 4D  even more largely varies as compared to the alternating current of the frequency belonging to the third frequency band, wherefore not only the oxidation/reduction reaction in the cathode electrode  113 , but also the oxidation/reduction reaction in the anode electrode  112  can no longer follow the variation of the value of this alternating current. 
     Accordingly, the reaction substantially does not occur in the anode electrode  112  in addition to in the cathode electrode  113 , and the influence of both the reaction resistance of the cathode electrode  113  and that of the anode electrode  112  can be ignored. 
     Specifically, in the fourth frequency band, the alternating current does not flow to the reaction resistances of both the cathode electrode  113  and the anode electrode  112  or, even if the alternating current flows, the magnitude thereof is small to a negligible extent. Thus, as shown in  FIG. 4D , the alternating current of the frequency belonging to the fourth frequency band is thought to flow only toward the electrical double layer capacitance of each of the cathode electrode  113  and the anode electrode  112 . 
     As is understood from the above description, the paths along which the alternating current of the frequency selected from the aforementioned first frequency band, the alternating current of the frequency selected from the aforementioned second frequency band, the alternating current of the frequency selected from the aforementioned third frequency band and the alternating current of the frequency selected from the aforementioned fourth frequency band flow to each element in the simplified equivalent circuit of the fuel cell differ. 
     Accordingly, the present inventors arrived at individual estimation of various state quantities from impedances based on frequencies belonging to each frequency band with reference to the following equation for impedance obtained on the basis of the simplified equivalent circuit utilizing differences of the paths of the alternating currents corresponding to the frequencies as just described: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Equation 
                        
                       
                           
                       
                        
                       1 
                     
                     ] 
                   
                    
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   Z 
                   = 
                   
                     
                       R 
                       m 
                     
                     + 
                     
                       
                         
                           R 
                           a 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               j 
                                
                               
                                   
                               
                                
                               ω 
                                
                               
                                   
                               
                                
                               
                                 C 
                                 a 
                               
                                
                               
                                 R 
                                 a 
                               
                             
                           
                           ) 
                         
                       
                       
                         1 
                         + 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             C 
                             a 
                             2 
                           
                            
                           
                             R 
                             a 
                             2 
                           
                         
                       
                     
                     + 
                     
                       
                         
                           R 
                           c 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               j 
                                
                               
                                   
                               
                                
                               ω 
                                
                               
                                   
                               
                                
                               
                                 C 
                                 c 
                               
                                
                               
                                 R 
                                 c 
                               
                             
                           
                           ) 
                         
                       
                       
                         1 
                         + 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             C 
                             c 
                             2 
                           
                            
                           
                             R 
                             c 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     (where j denotes an imaginary unit). 
     For example, the alternating current of the frequency selected from the above fourth frequency band (hereinafter, also written as an “electrolyte membrane response frequency band”) flows to the electrolyte membrane resistance, the electrical double layer capacitance of the anode electrode  112  and the electrical double layer capacitance of the cathode electrode  113 . Thus, the impedance based on the frequency selected from this electrolyte membrane response frequency band (hereinafter, also written as an “electrolyte membrane response impedance”) includes information of the electrolyte membrane resistance value R m . 
     It should be noted that this electrolyte membrane response frequency band is a frequency band used in so-called HFR (High Frequency Resistance) measurement. Thus, if ω→∞ is assumed in Equation (1) for impedance, the impedance Z can be regarded to substantially match the electrolyte membrane resistance value R m . 
     Further, the alternating current of the frequency selected from the third frequency band (hereinafter, also written as an “anode electrode response frequency band”) flows to the electrolyte membrane resistance, the reaction resistance of the anode electrode  112 , the electrical double layer capacitance of the anode electrode  112  and the electrical double layer capacitance of the cathode electrode  113 . Thus, the impedance based on the frequency selected from this anode electrode response frequency band (hereinafter, also written as an “anode electrode response impedance”) includes information of at least the reaction resistance value R a  of the anode electrode  112  and the electrical double layer capacitance value C a  of the anode electrode  112 . 
     Particularly, since the reaction resistance of the cathode electrode  113  can be ignored in the equivalent circuit shown in  FIG. 4C  in this case, the following equation for impedance is given. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   Z 
                   = 
                   
                     
                       R 
                       m 
                     
                     + 
                     
                       
                         
                           R 
                           a 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               j 
                                
                               
                                   
                               
                                
                               ω 
                                
                               
                                   
                               
                                
                               
                                 C 
                                 a 
                               
                                
                               
                                 R 
                                 a 
                               
                             
                           
                           ) 
                         
                       
                       
                         1 
                         + 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             C 
                             a 
                             2 
                           
                            
                           
                             R 
                             a 
                             2 
                           
                         
                       
                     
                     - 
                     
                       j 
                        
                       
                         1 
                         
                           ω 
                            
                           
                               
                           
                            
                           
                             C 
                             c 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Further, the alternating current of the frequency selected from the second frequency band flows to the electrolyte membrane resistance, the reaction resistance of the anode electrode  112 , the reaction resistance of the cathode electrode  113  and the electrical double layer capacitance of the cathode electrode  113 . Thus, the impedance based on the frequency selected from this second frequency band includes information of the electrolyte membrane resistance value, the reaction resistance value of the anode electrode  112 , the reaction resistance value R c  of the cathode electrode  113  and the electrical double layer capacitance value C c  of the cathode electrode  113  as state quantities. 
     Furthermore, the alternating current of the frequency selected from the first frequency band (hereinafter, also written as a “low frequency band”), which is a lowest frequency band, flows to the electrolyte membrane resistance, the reaction resistance of the anode electrode  112  and the reaction resistance of the cathode electrode  113 . Thus, the impedance based on the frequency selected from this low frequency band (hereinafter, also written as an “low frequency response impedance”) includes information of at least the reaction resistance value R c  of the cathode electrode  113 . 
     The estimation of each state quantity using at least two of the above electrolyte membrane response frequency band, anode electrode response frequency band and low frequency band is described in detail in each embodiment below. 
     It should be noted that it is generally known that there is a relationship of ω=2πf between a “frequency f” and an “angular frequency ω”, and there is only a difference multiplied by a dimensionless constant 2π between these. Thus, the “frequency” and the “angular frequency” are identified with each other and a symbol “ω” is used in expressing the both to facilitate description in each embodiment. 
     First Embodiment 
     A first embodiment is described below. 
       FIG. 5  is a flow chart showing the flow of state quantity estimation according to the present embodiment. 
     As shown, first in Step S 101 , a frequency ω H  at one point in the electrolyte membrane response frequency band is selected and an impedance Z (ω H ) based on the frequency ω H  is obtained. 
     Specifically, the controller  6  controls the DC/DC converter  56  such that an alternating-current signal of the frequency ω H  in the electrolyte membrane response frequency band is superimposed on an output voltage and an output current output from the fuel cell  1  at an impedance measurement timing. 
     Further, the controller  6  applies a Fourier transform processing on a value V of the output voltage measured by the voltage sensor  52  to obtain a voltage amplitude value V(ω H ), applies a Fourier transform processing on a value I of the output current measured by the current sensor  51  to obtain a current amplitude value I(ω H ) and obtains a ratio V(ω H )/I(ω H ) of these as the impedance Z(ω H ). It should be noted that since a method for measuring the impedance Z(ω H ) is similar also in the case of measurement for the frequency selected from the anode electrode response frequency band or the low frequency band other than the electrolyte membrane response frequency band, detailed description is omitted hereinafter. 
     Subsequently, in Step S 102 , the controller  6  estimates the electrolyte membrane resistance value R m  from the obtained impedance Z(ω H ). Specifically, since the electrolyte membrane response frequency band is a frequency band used in the so-called HFR measurement as described above, the impedance Z(ω H ) based on the frequency ω H  selected from this high frequency band or a real component Z r (ω H ) thereof substantially matches the electrolyte membrane resistance value R m . Specifically, the value of the impedance Z(ω H ) or the real component Z r (ω H ) thereof is directly estimated as the electrolyte membrane resistance value R m . 
     In Step S 103 , the controller  6  selects frequencies ω 1 , ω 2  at two points in the anode electrode response frequency band and obtains anode electrode response impedances Z(ω 1 ), Z(ω 2 ) based on these frequencies ω 1 , ω 2 . 
     In Step S 104 , the controller  6  estimates the reaction resistance value R a  of the anode electrode  112  and the electrical double layer capacitance value C a  of the anode electrode  112  from the estimated electrolyte membrane resistance value R m  and the obtained two impedances Z(ω 1 ), Z(ω 2 ). 
     A mode of this estimation is specifically described. First, in the case of selecting the frequencies ω 1 , ω 2  at the two points in the anode electrode response frequency band, the reaction resistance of the cathode electrode  113  can be ignored as described above. Thus, Equation (2) obtained by removing the reaction resistance value R c  of the cathode electrode  113  from Equation (1) for impedance based on the simplified equivalent circuit can be used as an equation for impedance. 
     Here, the frequencies ω 1 , ω 2  at the two points, which are known values, and a combination of the impedances Z(ω 1 ) and Z(ω 2 ) based on these are substituted into in Equation (2) and real components Z r (ω 1 ) and Z r (ω 2 ) of the impedances Z(ω 1 ) and Z(ω 2 ) are taken. Considering that the estimated electrolyte membrane resistance value R m  is known, two equations with R a  and C a  serving as unknowns are obtained. Thus, R a  and C a  can be obtained if the obtained two equations are solved. 
     An example of a method for obtaining the unknowns R a  and C a  is described. First, if the real component of Equation (2) is taken and changed, the following equation is obtained. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     3 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     1 
                     
                       Z 
                       r 
                     
                   
                   = 
                   
                     
                       
                         ω 
                         2 
                       
                        
                       
                         C 
                         a 
                         2 
                       
                        
                       
                         R 
                         a 
                       
                     
                     + 
                     
                       1 
                       
                         R 
                         a 
                       
                     
                     - 
                     
                       
                         R 
                         m 
                       
                       
                         
                           Z 
                           r 
                         
                          
                         
                           ( 
                           
                             
                               Z 
                               r 
                             
                             - 
                             
                               R 
                               m 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Considering a plane with ω 2  represented on a horizontal axis and 1/Z r  represented on a vertical axis, a straight line is represented by Equation (3) on this plane and a gradient m r  thereof is given by the following equation. 
     [Equation 4] 
       m r =C a   2 R a    (4)
 
     Here, the frequencies ω 1 , ω 2  at the two points are known. Thus, if these frequencies ω 1 , ω 2  at the two points and the real components Z r (ω 1 ) and Z r (ω 2 ) of the impedance measurement values corresponding to these frequencies are plotted on the above plane, a straight line connecting these points is determined and the value of the gradient m r  is determined. Specifically, unknowns of Equation (4) are R a  and C a . 
     Subsequently, an intercept a of the straight line represented by Equation (3) is given by the following equation. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     5 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   a 
                   = 
                   
                     
                       1 
                       
                         R 
                         a 
                       
                     
                     - 
                     
                       
                         R 
                         m 
                       
                       
                         
                           Z 
                           r 
                         
                          
                         
                           ( 
                           
                             
                               Z 
                               r 
                             
                             - 
                             
                               R 
                               m 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Here, the value of the intercept a is determined by the frequencies ω 1 , ω 2  at the points and the real components Z r1  and Z r2  of the impedance measurement values corresponding to these frequencies similarly to the value of the gradient m r . Since Z r  is equivalent to the real components Z r1  and Z r2  of the impedance measurement values, only R a  is unknown in Equation (5). 
     Thus, according to Equation (5), the reaction resistance value R a  of the anode electrode  112  can be obtained as follows. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Equation 
                        
                       
                           
                       
                        
                       6 
                     
                     ] 
                   
                    
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     R 
                     a 
                   
                   = 
                   
                     
                       
                         Z 
                         r 
                       
                        
                       
                         ( 
                         
                           
                             Z 
                             r 
                           
                           - 
                           
                             R 
                             m 
                           
                         
                         ) 
                       
                     
                     
                       
                         
                           Z 
                           r 
                         
                          
                         
                           a 
                            
                           
                             ( 
                             
                               
                                 Z 
                                 r 
                               
                               - 
                               
                                 R 
                                 m 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         R 
                         m 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Further, by substituting R a  determined by Equation (6) into Equation (4), the electrical double layer capacitance value C a  of the anode electrode  112  can be obtained as follows. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Equation 
                        
                       
                           
                       
                        
                       7 
                     
                     ] 
                   
                    
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     C 
                     a 
                   
                   = 
                   
                     
                       
                         m 
                         r 
                       
                       
                         R 
                         a 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     It should be noted that a method for calculating R a  and C a  is not limited to the above calculation method and various suitable calculation methods can be used. 
     Subsequently, in Step S 105 , the controller  6  selects a frequency ω L  at one point in the low frequency band and measures an impedance Z(ω L ) based on this frequency ω L . 
     In Step S 106 , the controller  6  estimates the electrical double layer capacitance value C c  of the cathode electrode  113  using the already estimated electrolyte membrane resistance value R m , reaction resistance value R a  of the anode electrode  112  and electrical double layer capacitance value C a  of the anode electrode  112  and the measured impedance Z(ω L ). 
     A mode of this estimation is specifically described. An alternating current of the frequency ω L  in the low frequency band flows to all the circuit elements in the simplified equivalent circuit of the fuel cell  1 , i.e., the reaction resistance and the electrical double layer capacitance of the anode electrode  112 , the electrolyte membrane resistance and the reaction resistance and the electrical double layer capacitance of the cathode electrode  113  as described above. Thus, the low frequency impedance Z(ω L ) obtained on the basis of the frequency ω L  includes information of the reaction resistance R a  and the electrical double layer capacitance C a  of the anode electrode  112 , the electrolyte membrane resistance R m  and the reaction resistance R c  and the electrical double layer capacitance C c  of the cathode electrode  113 . Thus, Equation (1) taking into account of all the above circuit elements needs to be used as the equation for impedance. 
     The frequency ω L , which is a known value, and the impedance Z(ω L ) based on this frequency are substituted into Equation (1), and a real component Z r (ω L ) and an imaginary component Z i  (ω L ) are taken. Considering that the estimated electrolyte membrane resistance value R m , reaction resistance value R a  of the anode electrode  112  and electrical double layer capacitance C a  of the anode electrode  112  are known, two equations with R c  and C c  serving as unknowns are obtained. Thus, the unknowns R c  and C c  can be obtained if these two equations are solved. 
     An example of a method for obtaining the unknowns R c  and C c  is described. First, if the real component of Equation (1) is taken and changed, the following equation is obtained. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Equation 
                        
                       
                           
                       
                        
                       8 
                     
                     ] 
                   
                    
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     Z 
                     r 
                   
                   = 
                   
                     
                       R 
                       m 
                     
                     + 
                     
                       
                         R 
                         a 
                       
                       
                         1 
                         + 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             C 
                             a 
                             2 
                           
                            
                           
                             R 
                             a 
                             2 
                           
                         
                       
                     
                     + 
                     
                       
                         R 
                         c 
                       
                       
                         1 
                         + 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             C 
                             c 
                             2 
                           
                            
                           
                             R 
                             c 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Further, if the imaginary component of Equation (1) is taken and changed, the following equation is obtained. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Equation 
                        
                       
                           
                       
                        
                       9 
                     
                     ] 
                   
                    
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     Z 
                     i 
                   
                   = 
                   
                     
                       
                         
                           - 
                           ω 
                         
                          
                         
                             
                         
                          
                         
                           C 
                           a 
                         
                          
                         
                           R 
                           a 
                         
                       
                       
                         1 
                         + 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             C 
                             a 
                             2 
                           
                            
                           
                             R 
                             a 
                             2 
                           
                         
                       
                     
                     + 
                     
                       
                         
                           - 
                           ω 
                         
                          
                         
                             
                         
                          
                         
                           C 
                           c 
                         
                          
                         
                           R 
                           c 
                         
                       
                       
                         1 
                         + 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             C 
                             c 
                             2 
                           
                            
                           
                             R 
                             c 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Here, the frequency ω L , the real component Zr(ω L ) and the imaginary component Z i (ω L ) of the impedance measurement value corresponding to the frequency ω L  and R a  and C a  are known. If these are substituted into Equations (8) and (9) and Equations are changed, the electrical double layer capacitance value C c  of the cathode electrode  113  is as follows. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     10 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     C 
                     c 
                   
                   = 
                   
                     
                       1 
                       
                         ω 
                          
                         
                             
                         
                          
                         
                           R 
                           c 
                         
                       
                     
                      
                     
                       
                         
                           
                             R 
                             c 
                           
                           - 
                           A 
                         
                         A 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     In Equation (10), ω is ω L  and A is defined as in the following Equation (11). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     11 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   A 
                   = 
                   
                     
                       Z 
                       r 
                     
                     - 
                     
                       R 
                       m 
                     
                     - 
                     
                       
                         R 
                         a 
                       
                       
                         1 
                         + 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             C 
                             a 
                             2 
                           
                            
                           
                             R 
                             a 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Further, the reaction resistance value R c  of the cathode electrode  113  is obtained as follows. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     12 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     R 
                     c 
                   
                   = 
                   
                     
                       
                         
                           1 
                           - 
                           
                             
                               2 
                                
                               
                                 B 
                                 2 
                               
                             
                             ± 
                             
                               
                                 1 
                                 - 
                                 
                                   4 
                                    
                                   
                                     B 
                                     2 
                                   
                                 
                               
                             
                           
                         
                         
                           2 
                            
                           B 
                         
                       
                        
                       A 
                     
                     + 
                     A 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     A in Equation (12) is defined as in the above Equation (11) and B in Equation (12) is defined as in the following Equation (13). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     13 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   B 
                   = 
                   
                     
                       Z 
                       i 
                     
                     + 
                     
                       
                         ω 
                          
                         
                             
                         
                          
                         
                           C 
                           a 
                         
                          
                         
                           R 
                           a 
                         
                       
                       
                         1 
                         + 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             C 
                             a 
                             2 
                           
                            
                           
                             R 
                             a 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     As described above, the electrolyte membrane resistance value R m , the reaction resistance value R a  of the anode electrode  112 , the electrical double layer capacitance value C a  of the anode electrode  112 , the reaction resistance value R c  of the cathode electrode  113  and the electrical double layer capacitance value C c  of the cathode electrode  113  are estimated as the state quantities of the fuel cell  1  by Steps S 101  to S 106 . 
     According to the present embodiment described above, the following effects can be obtained. In the present embodiment, the state detection device is configured by the controller  6 , the current sensor  51 , the voltage sensor  52  and the DC/DC converter  6 . Further, impedance acquisition unit and internal state quantity estimation unit are configured by the controller  6 . 
     According to the present embodiment, the impedance acquisition unit of the state detection device for the fuel cell  1  for generating power upon receiving the supply of the anode gas and the cathode gas acquires the high frequency impedances Z(ω H ), Z(ω 1 ) and Z(ω 2 ) based on the frequencies ω H , ω 1  and ω 2  selected from the high frequency band (anode electrode response frequency band and electrolyte membrane response frequency band) including a frequency band which shows responsiveness at least to the state quantities R a , C a  of the anode electrode  112  and the low frequency impedance Z(ω L ) based on the frequency ω L  selected from the low frequency band including a frequency band which shows responsiveness at least to the state quantities R c , C c  of the cathode electrode (Step S 101 , Step S 103 , Step S 105 ). 
     The internal state quantity estimation unit of the state detection device for the fuel cell  1  estimates each of the state quantities R a , C a  of the anode electrode  112  and the state quantities R c , C c  of the cathode electrode  113  serving as the internal states of the fuel cell  1  by combining the obtained high frequency impedances Z(ω H ), Z(ω 1 ) and Z(ω 2 ) and low frequency impedance Z(ω L ). 
     According to this, at least each of the state quantities R a , C a  of the anode electrode  112  and the state quantities R c , C c  of the cathode electrode  113  can be individually detected on the basis of the obtained high frequency impedances Z(ω H ), Z(ω 1 ) and Z(ω 2 ) and low frequency impedance Z(ω L ), i.e., impedance information obtained from the different frequency bands, utilizing a following speed difference of the reaction of the anode electrode  112  and the reaction of the cathode electrode  113  in response to a current variation according to the magnitude of the frequency. Thus, highly accurate information of the state quantities R a , C a  of the anode electrode  112  and the state quantities (R c , C c ) of the cathode electrode  113  can be obtained, with the result that an operation control of the fuel cell  1  executed utilizing these state quantities can be made more proper. 
     Further, according to the present embodiment, the internal state quantity estimation unit estimates the internal state quantities R m , R a  and C a  on the basis of the high frequency impedances Z(ω H ), Z(ω 1 ) and Z(ω 2 ) and estimates the other internal state quantities R c  and C c  on the basis of the estimated internal state quantities R m , R a  and C a  and the low frequency impedance Z(ω L ). 
     In this way, the internal state quantities R c , C c  that cannot be determined only from the low frequency impedance Z(ω L ) in the low frequency band, which is one frequency band, can be determined on the basis of the internal state quantities R m , R a  and C a  estimated from the high frequency impedances Z(ω H ), Z(ω 1 ) and Z(ω 2 ) in the high frequency band, which is another frequency band. Specifically, each of a plurality of types of internal state quantities R m , R a , C a , R c  and C c  can be more reliably distinguished. 
     It should be noted that the internal state quantity estimation unit may, conversely, estimate a certain internal state quantity on the basis of the low frequency impedance Z(ω L ) and estimate another internal state quantity on the basis of the estimated internal state quantity and the high frequency impedances Z(ω H ), Z(ω 1 ) and Z(ω 2 ). 
     Further, according to the present embodiment, the above high frequency band (anode electrode response frequency band and electrolyte membrane response frequency band) includes the anode electrode response frequency band, which is a frequency band which shows responsiveness to the state quantities R a , C a  of the anode electrode  112  of the fuel cell  1 , and the electrolyte membrane response frequency band, which is a frequency band higher than the anode electrode response frequency band and which shows responsiveness to the state quantity R m  of the electrolyte membrane of the fuel cell  1 . The impedance acquisition unit acquires both the anode electrode response impedances Z(ω 1 ), Z(ω 2 ) based on the frequencies selected from the anode electrode response frequency band and the electrolyte membrane response impedance Z(ω H ) based on the frequency selected from the electrolyte membrane response frequency band as the high frequency impedances Z(ω H ), Z(ω 1 ) and Z(ω 2 ) (Step S 101 , Step S 103 ). 
     In this way, each of the state quantity R m  of the electrolyte membrane  111  of the fuel cell  1  and the state quantities R a , C a  of the anode electrode  112  can be estimated on the basis of the electrolyte membrane response impedance Z(ω H ) and the anode electrode response impedances Z(ω 1 ), Z(ω 2 ). 
     Further, according to the present embodiment, the internal state quantity estimation unit estimates the state quantity R m  of the electrolyte membrane  111  on the basis of the electrolyte membrane response impedance Z(ω H ) (Step S 102 ) and estimates the state quantities R a , C a  of the anode electrode  112  on the basis of the estimated electrolyte membrane resistance R m  and the anode electrode response impedances Z(ω 1 ), Z(ω 2 ) (Step S 104 ). 
     In this way, the state quantities R a , C a  of the anode electrode  112  can be estimated in clearer distinction from the other state quantities on the basis of the estimated state quantity R m  of the electrolyte membrane  111  and the anode electrode response impedances Z(ω 1 ), Z(ω 2 ). 
     Particularly, in the present embodiment, the state quantities R a , C a  of the anode electrode  112  include the reaction resistance value R a  and the electrical double layer capacitance value C a  of the anode electrode  112 , and the state quantities R c , C c  of the cathode electrode  113  include the reaction resistance value R c  and the electrical double layer capacitance value C c  of the cathode electrode  113 . The internal state quantity estimation unit estimates the reaction resistance value R a  of the anode electrode  112  and the electrical double layer capacitance value C a  of the anode electrode  112  on the basis of the anode electrode response impedance Z(ω 1 ), Z(ω 2 ) (Step S 104 ). Further, the internal state quantity estimation unit estimates the reaction resistance value R c  of the cathode electrode  113  on the basis of the estimated state quantity R m  of the electrolyte membrane  111 , reaction resistance value R a  of the anode electrode  112 , electrical double layer capacitance value C a  of the anode electrode  112  and the low frequency impedance Z(ω L ) (Step S 106 ). 
     According to this, the reaction resistance value R a  and the electrical double layer capacitance value C a  of the anode electrode  112  estimated on the basis of the anode electrode response impedances (Z(ω 1 ), Z(ω 2 )) and the state quantity R m  of the electrolyte membrane  111  estimated on the basis of the electrolyte membrane response impedance Z(ω H ) can be applied to the low frequency impedance Z(ω L ) in the low frequency band including all pieces of information other than the reaction resistance value R c  of the cathode electrode  113 . 
     Accordingly, the targeted state quantity R c  can be suitably distinguished and estimated from the low frequency impedance Z(ω L ) in the low frequency band including information other than the targeted state quantity R c . 
     Second Embodiment 
     A second embodiment is described below. It should be noted that elements similar to those of the already described first embodiment are denoted by the same reference signs. 
       FIG. 6  is a flow chart showing the flow of state quantity estimation according to the second embodiment. Since Steps S 101  to S 104  in  FIG. 6  are similar to Steps S 101  to S 104  in  FIG. 5 , no detailed description is given. In the second embodiment, a gradient of a straight part of a characteristic curve in an I-V characteristic curve diagram (I-V characteristic diagram) of a fuel cell  1  set in advance is regarded and acquired as a low frequency impedance instead of measuring a low frequency impedance at a frequency in a low frequency band. 
     As shown, after Steps S 101  to S 104 , i.e., estimation values of the reaction resistance value R a  and the electrical double layer capacitance value C a  of the anode electrode  112  are acquired, the gradient ΔV/ΔI of the straight part of the characteristic curve in the I-V characteristic diagram of the fuel cell  1  is regarded and acquired as the low frequency impedance Z(ω L ) in Step S 205 . 
       FIG. 7  shows I-V characteristic curves of the fuel cell  1  respectively in steady time and in unsteady time. It should be noted that these I-V characteristic curves of the fuel cell  1  are determined in advance on the basis of an experiment or the like. A characteristic curve Cv 1  shows an I-V characteristic in steady time and a characteristic curve Cv 2  shows an I-V characteristic in unsteady time. Here, the I-V characteristic in steady time means an output characteristic of the fuel cell  1  during stable travel not in a sudden accelerating state such as during vehicle startup or during vehicle stop. 
     Particularly, as understood from  FIG. 7 , a variation of the gradient ΔV/ΔI is small, has a substantially constant value and is linear in a steady region P of the characteristic curve Cv 1  in steady time. Thus, in the steady region P, the gradient ΔV/ΔI can be regarded as a constant value regardless of an output current I. 
     As just described, the steady region P where the value of ΔV/ΔI is constant is a section of a horizontal axis (output current I) in which the value of ΔV/ΔI of the characteristic curve Cv 1  in steady time is not larger than a predetermined value. 
     In the present embodiment, the controller  6  stores the value of ΔV/ΔI in this steady region P in an unillustrated memory or the like in advance, reads the value of ΔV/ΔI from this memory at an acquisition timing of the low frequency impedance Z(ω L ) and regards this value as the low frequency impedance Z(ω L ). The low frequency impedance Z(ω L ) obtained in this way matches well an actual value. 
     In Step S 206 , the reaction resistance value R c  of the cathode electrode  113  is estimated using the value of ΔV/ΔI acquired as the low frequency impedance Z(ω L ). 
     This is specifically described. If ω is assumed to be a low frequency (ω→0) in Equation (1) described above, the following equation is thought to hold. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     14 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       lim 
                       
                         ω 
                         -&gt; 
                         0 
                       
                     
                      
                     Z 
                   
                   = 
                   
                     
                       R 
                       m 
                     
                     + 
                     
                       R 
                       a 
                     
                     + 
                     
                       R 
                       c 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     Thus, if the impedance Z is substituted by ΔV/ΔI in Equation (14), the following equation is obtained. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     15 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     R 
                     c 
                   
                   = 
                   
                     
                       
                         Δ 
                          
                         
                             
                         
                          
                         V 
                       
                       
                         Δ 
                          
                         
                             
                         
                          
                         I 
                       
                     
                     - 
                     
                       R 
                       m 
                     
                     - 
                     
                       R 
                       a 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     In this way, the reaction resistance value R c  of the cathode electrode  113  can be calculated by substituting the electrolyte membrane resistance value R m  estimated in the process of Steps S 101  to S 104  and the reaction resistance value R a  of the anode electrode  112  into Equation (15). 
     According to the state detection device for the fuel cell  1  according to the present embodiment described above, the controller  6  serving as the impedance acquisition unit acquires the gradient ΔV/ΔI of the I-V characteristic curve of the fuel cell  1  as the low frequency impedance Z(ω 1 ). Specifically, the low frequency impedance Z(ω 1 ) can be acquired without being directly measured. 
     It should be noted that the low frequency impedances Z(ω 1 ) may be acquired by both methods for acquiring the low frequency impedance Z(ω 1 ) as the value of the gradient ΔV/ΔI of the I-V characteristic curve and acquiring the low frequency impedance Z(ω 1 ) by measurement and the highly accurate low frequency impedance Z(ω 1 ) acquired such as by comparing/correcting the low frequency impedances Z(ω 1 ) obtained by these two methods may be used for the estimation of the reaction resistance value R c  of the cathode electrode  113 . 
     Further, in the present embodiment, the controller  6  serving as the impedance acquisition unit acquires the gradient ΔV/ΔI as the low frequency impedance Z(ω 1 ) in the steady region P where the variation of the value of the gradient in the I-V characteristic curve Cv 1  of the fuel cell  1  is not larger than the predetermined value. 
     As just described, in the steady region P where the variation of the gradient ΔV/ΔI is relatively small, there is no problem in regarding the value of the gradient ΔV/ΔI as constant regardless of a measurement value of the output current I. Thus, it is not necessary to calculate the value of the gradient ΔV/ΔI for each of the measurement values of the output voltage V and the output current I and the amount of calculation can be reduced. 
     Third Embodiment 
     A third embodiment is described below. It should be noted that elements similar to those of the already described embodiments are denoted by the same reference signs. 
       FIG. 8  is a flow chart showing the flow of state quantity estimation according to the present embodiment. As shown, the estimation of the electrolyte membrane resistance value R m  using the frequency in the electrolyte membrane response frequency band equivalent to Steps S 101  and S 102  shown in  FIG. 5  is omitted. 
     Particularly, in the present embodiment, the reaction resistance value R a  of the anode electrode  112 , the electrical double layer capacitance value C a  of the anode electrode  112 , the electrical double layer capacitance value C c  of the cathode electrode  113  and the electrolyte membrane resistance value R m  serving as state quantities are estimated, using anode electrode response impedances Z(ω 1 ), Z(ω 2 ) acquired at two frequencies ω 1 , ω 2  in the anode electrode response frequency band in specific Step S 304  (Step S 304 ). 
     A mode of the state quantity estimation in Step S 304  is described below. 
     Also in the present embodiment, calculation is performed on the basis of Equation (2) for impedance described above. A step of obtaining Equation (3) by taking a real component of Equation (2) and obtaining Equation (4) on the basis of Equation (3) is as in the case of estimating the reaction resistance value R a  of the anode electrode  112  and the electrical double layer capacitance value C a  of the anode electrode  112  according to the first embodiment. 
     If Equation (4) is changed, the following equation is obtained. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     16 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     R 
                     a 
                   
                   = 
                   
                     
                       m 
                       r 
                     
                     
                       C 
                       a 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     It should be noted that m r  is a gradient of a straight line connecting two impedances Z(ω 1 ) and Z(ω 2 ) and a known value as described above. 
     On the other hand, if an imaginary component of Equation (2) is taken, the following equation is obtained. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     17 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     Z 
                     i 
                   
                   = 
                   
                     
                       - 
                       
                         
                           ω 
                            
                           
                               
                           
                            
                           
                             C 
                             a 
                           
                            
                           
                             R 
                             a 
                             2 
                           
                         
                         
                           1 
                           + 
                           
                             
                               ω 
                               2 
                             
                              
                             
                               C 
                               a 
                               2 
                             
                              
                             
                               R 
                               a 
                               2 
                             
                           
                         
                       
                     
                     - 
                     
                       1 
                       
                         ω 
                          
                         
                             
                         
                          
                         
                           C 
                           c 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Here, if R a  of Equation (16) is substituted into the above Equation (17) and both sides are multiplied by ω, the following equation is obtained. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     18 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ω 
                      
                     
                         
                     
                      
                     
                       Z 
                       i 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           
                             ω 
                             2 
                           
                            
                           
                             m 
                             r 
                             2 
                           
                         
                         
                           
                             C 
                             a 
                             3 
                           
                           + 
                           
                             
                               ω 
                               2 
                             
                              
                             
                               m 
                               r 
                               2 
                             
                              
                             
                               C 
                               a 
                             
                           
                         
                       
                     
                     - 
                     
                       1 
                       
                         C 
                         c 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     If the above known frequencies ω 1  and ω 2  and imaginary components Z i1  and Z i2  of impedance measurement values corresponding to these frequencies are respectively substituted into Equation (18) to obtain two equations and the electrical double layer capacitance C c  of the cathode is erased by taking a difference between these two equations, the following quartic equation for the unknown electrical double layer capacitance C a  of the anode is obtained. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     19 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       C 
                       a 
                       4 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             ω 
                             1 
                             2 
                           
                           + 
                           
                             ω 
                             2 
                             2 
                           
                         
                         ) 
                       
                        
                       
                         m 
                         r 
                         2 
                       
                        
                       
                         C 
                         a 
                         2 
                       
                     
                     + 
                     
                       
                         
                           
                             ω 
                             1 
                             2 
                           
                           - 
                           
                             ω 
                             2 
                             2 
                           
                         
                         
                           
                             
                               ω 
                               1 
                             
                              
                             
                               Z 
                               
                                 i 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                           - 
                           
                             
                               ω 
                               2 
                             
                              
                             
                               Z 
                               
                                 i 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                        
                       
                         m 
                         r 
                         2 
                       
                        
                       
                         C 
                         a 
                       
                     
                     + 
                     
                       
                         ω 
                         1 
                         2 
                       
                        
                       
                         ω 
                         2 
                         2 
                       
                        
                       
                         m 
                         r 
                         4 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     When the quartic equation of Equation (19) is solved and it is considered that C a  cannot be an imaginary value, the following two solutions are obtained as candidates for the electrical double layer capacitance C a  of the anode. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Equation 
                        
                       
                           
                       
                        
                       20 
                     
                     ] 
                   
                    
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     C 
                     
                       a 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   = 
                   
                     
                       
                         
                           t 
                           1 
                         
                       
                       + 
                       
                         
                           
                             - 
                             
                               t 
                               1 
                             
                           
                           - 
                           
                             2 
                              
                             
                               
                                 m 
                                 r 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ω 
                                     1 
                                     2 
                                   
                                   + 
                                   
                                     ω 
                                     2 
                                     2 
                                   
                                   + 
                                   
                                     
                                       
                                         ω 
                                         1 
                                         2 
                                       
                                       - 
                                       
                                         ω 
                                         2 
                                         2 
                                       
                                     
                                     
                                       
                                         
                                           t 
                                           1 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               ω 
                                               1 
                                             
                                              
                                             
                                               Z 
                                               
                                                 i 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 1 
                                               
                                             
                                           
                                           - 
                                           
                                             
                                               ω 
                                               2 
                                             
                                              
                                             
                                               Z 
                                               
                                                 i 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       Equation 
                        
                       
                           
                       
                        
                       21 
                     
                     ] 
                   
                    
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     C 
                     
                       a 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   = 
                   
                     
                       
                         
                           t 
                           1 
                         
                       
                       - 
                       
                         
                           
                             - 
                             
                               t 
                               1 
                             
                           
                           - 
                           
                             2 
                              
                             
                               
                                 m 
                                 r 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ω 
                                     1 
                                     2 
                                   
                                   + 
                                   
                                     ω 
                                     2 
                                     2 
                                   
                                   + 
                                   
                                     
                                       
                                         ω 
                                         1 
                                         2 
                                       
                                       - 
                                       
                                         ω 
                                         2 
                                         2 
                                       
                                     
                                     
                                       
                                         
                                           t 
                                           1 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               ω 
                                               1 
                                             
                                              
                                             
                                               Z 
                                               
                                                 i 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 1 
                                               
                                             
                                           
                                           - 
                                           
                                             
                                               ω 
                                               2 
                                             
                                              
                                             
                                               Z 
                                               
                                                 i 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 2 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     It should be noted that the quartic equation of Equation (19) can be solved by various methods known to a person skilled in the art. 
     Here, t 1  is a constant defined as follows. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Equation 
                        
                       
                           
                       
                        
                       22 
                     
                     ] 
                   
                    
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     t 
                     1 
                   
                   = 
                   
                     
                       
                         
                           - 
                           
                             
                               
                                 27 
                                  
                                 
                                   A 
                                   0 
                                 
                               
                               + 
                               
                                 2 
                                  
                                 
                                   A 
                                   2 
                                   3 
                                 
                               
                               - 
                               
                                 9 
                                  
                                 
                                   A 
                                   2 
                                 
                                  
                                 
                                   A 
                                   1 
                                 
                               
                             
                             54 
                           
                         
                         + 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       27 
                                        
                                       
                                         A 
                                         0 
                                       
                                     
                                     + 
                                     
                                       2 
                                        
                                       
                                         A 
                                         2 
                                         3 
                                       
                                     
                                     - 
                                     
                                       9 
                                        
                                       
                                         A 
                                         2 
                                       
                                        
                                       
                                         A 
                                         1 
                                       
                                     
                                   
                                   54 
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       3 
                                        
                                       
                                         A 
                                         1 
                                       
                                     
                                     - 
                                     
                                       A 
                                       2 
                                       2 
                                     
                                   
                                   9 
                                 
                                 ) 
                               
                               3 
                             
                           
                         
                       
                       3 
                     
                     + 
                     
                       
                         
                           - 
                           
                             
                               
                                 27 
                                  
                                 
                                   A 
                                   0 
                                 
                               
                               + 
                               
                                 2 
                                  
                                 
                                   A 
                                   2 
                                   3 
                                 
                               
                               - 
                               
                                 9 
                                  
                                 
                                   A 
                                   2 
                                 
                                  
                                 
                                   A 
                                   1 
                                 
                               
                             
                             54 
                           
                         
                         - 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       27 
                                        
                                       
                                         A 
                                         0 
                                       
                                     
                                     + 
                                     
                                       2 
                                        
                                       
                                         A 
                                         2 
                                         3 
                                       
                                     
                                     - 
                                     
                                       9 
                                        
                                       
                                         A 
                                         2 
                                       
                                        
                                       
                                         A 
                                         1 
                                       
                                     
                                   
                                   54 
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       3 
                                        
                                       
                                         A 
                                         1 
                                       
                                     
                                     - 
                                     
                                       A 
                                       2 
                                       2 
                                     
                                   
                                   9 
                                 
                                 ) 
                               
                               3 
                             
                           
                         
                       
                       3 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     Although the embodiments of the present invention have been described above, the above embodiments are merely an illustration of some application examples of the present invention and not intended to limit the technical scope of the present invention to the specific configurations of the above embodiments. 
     Further, A 2 , A 1  and A 0  in Equation are respectively as follows. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     23 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       A 
                       2 
                     
                     = 
                     
                       2 
                        
                       
                         ( 
                         
                           
                             ω 
                             1 
                             2 
                           
                           + 
                           
                             ω 
                             2 
                             2 
                           
                         
                         ) 
                       
                        
                       
                         m 
                         r 
                         2 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       A 
                       1 
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               
                                 ω 
                                 1 
                                 2 
                               
                               + 
                               
                                 ω 
                                 2 
                                 2 
                               
                             
                             ) 
                           
                           2 
                         
                          
                         
                           m 
                           r 
                           4 
                         
                       
                       - 
                       
                         4 
                          
                         
                           ω 
                           1 
                           2 
                         
                          
                         
                           ω 
                           2 
                           2 
                         
                          
                         
                           m 
                           r 
                           4 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       A 
                       0 
                     
                     = 
                     
                       
                         - 
                         
                           
                             ( 
                             
                               
                                 
                                   ω 
                                   1 
                                   2 
                                 
                                 - 
                                 
                                   ω 
                                   2 
                                   2 
                                 
                               
                               
                                 
                                   
                                     ω 
                                     1 
                                   
                                    
                                   
                                     Z 
                                     
                                       i 
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     ω 
                                     2 
                                   
                                    
                                   
                                     Z 
                                     
                                       i 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                        
                       
                         m 
                         r 
                         4 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     Further, by substituting each of C a1  and C a2  into the above Equation (16), R a1  and R a2  are determined as candidates for the estimation value of the reaction resistance in correspondence with C a1  and C a2 . The candidates R a1  and R a2  for the estimation value are as follows. 
     
       
         
           
             [ 
             
               Equation 
                
               
                   
               
                
               24 
             
             ] 
           
         
       
       
         
           
             
               
                 
                   
                     R 
                     
                       a 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   = 
                   
                     
                       4 
                        
                       
                         m 
                         r 
                       
                     
                     
                       
                         { 
                         
                           
                             
                               t 
                               1 
                             
                           
                           + 
                           
                             
                               
                                 - 
                                 
                                   t 
                                   1 
                                 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   
                                     m 
                                     r 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         1 
                                         2 
                                       
                                       + 
                                       
                                         ω 
                                         2 
                                         2 
                                       
                                       + 
                                       
                                         
                                           
                                             ω 
                                             1 
                                             2 
                                           
                                           - 
                                           
                                             ω 
                                             2 
                                             2 
                                           
                                         
                                         
                                           
                                             
                                               t 
                                               1 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   ω 
                                                   1 
                                                 
                                                  
                                                 
                                                   Z 
                                                   
                                                     i 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     1 
                                                   
                                                 
                                               
                                               - 
                                               
                                                 
                                                   ω 
                                                   2 
                                                 
                                                  
                                                 
                                                   Z 
                                                   
                                                     i 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     2 
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         } 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       Equation 
                        
                       
                           
                       
                        
                       25 
                     
                     ] 
                   
                    
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     R 
                     
                       a 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   = 
                   
                     
                       4 
                        
                       
                         m 
                         r 
                       
                     
                     
                       
                         { 
                         
                           
                             
                               t 
                               1 
                             
                           
                           - 
                           
                             
                               
                                 - 
                                 
                                   t 
                                   1 
                                 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   
                                     m 
                                     r 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         1 
                                         2 
                                       
                                       + 
                                       
                                         ω 
                                         2 
                                         2 
                                       
                                       + 
                                       
                                         
                                           
                                             ω 
                                             1 
                                             2 
                                           
                                           - 
                                           
                                             ω 
                                             2 
                                             2 
                                           
                                         
                                         
                                           
                                             
                                               t 
                                               1 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   ω 
                                                   1 
                                                 
                                                  
                                                 
                                                   Z 
                                                   
                                                     i 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     1 
                                                   
                                                 
                                               
                                               - 
                                               
                                                 
                                                   ω 
                                                   2 
                                                 
                                                  
                                                 
                                                   Z 
                                                   
                                                     i 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     2 
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         } 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     Here, it is necessary to determine a true estimation value conforming to an actual characteristic from the aforementioned candidates C a1  and C a2  for the electrical double layer capacitance value of the anode electrode  112  and candidates R a1  and R a2  for the reaction resistance value. An example of that method is described. 
     In the present embodiment, the determination of this true estimation value is judged not only from the values of C a1 , R a1 , C a2  and R a2 , but also by the following equation for the electrical double layer capacitance value C c  of the cathode electrode  113  obtained by changing the equation for the impedance imaginary component in the above Equation (17). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     26 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     C 
                     c 
                   
                   = 
                   
                     - 
                     
                       
                         1 
                         + 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             C 
                             a 
                             2 
                           
                            
                           
                             R 
                             a 
                             2 
                           
                         
                       
                       
                         
                           
                             ω 
                             2 
                           
                            
                           
                             C 
                             a 
                           
                            
                           
                             R 
                             a 
                             2 
                           
                         
                         + 
                         
                           ω 
                            
                           
                               
                           
                            
                           
                             
                               Z 
                               i 
                             
                              
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     ω 
                                     2 
                                   
                                    
                                   
                                     C 
                                     a 
                                     2 
                                   
                                    
                                   
                                     R 
                                     a 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
       FIG. 9  shows frequency responses of the candidates C c1  and C c2  for the electrical double layer capacitance value of the cathode electrode  113 . It should be noted that this graph is based on data of the candidates C a1  and C a2  for the electrical double layer capacitance value obtained by continuously changing the frequencies ω 1  and ω 2  calculated by an experiment or the like in advance in a range of the anode electrode response frequency band. 
     It should be noted that a line of C c1  is represented by a broken line and a line of C c2  is represented by a solid line. Further, a frequency ω d  is a frequency at which (C a1 , R a1 ) =(C a2 , R a2 ) for sets (C a1 , R a1 ), (C a2 , R a2 ) of the candidates for the reaction resistance value and the electrical double layer capacitance value of the anode electrode  112 . Specifically, the inside of the radical sign in the above Equations (20), (21), (24) and (25) expressing C a1  and R a1  C a2  and R a2  is 0. 
     As shown, in a region where the frequency ω&lt;ω d , the estimation value candidate C c2  for the electrical double layer capacitance value is basically 0 or smaller and the value of C c2  is extremely sensitive to a change of the frequency immediately before ω d . Thus, in the region where the frequency ω&lt;ω d , C c1  is a true estimation value which should be actually employed. 
     Accordingly, also for the electrical double layer capacitance value and the reaction resistance value of the cathode electrode  113 , C a1  and R a1  corresponding to C c1  are respectively employed in the region where the frequency ω&lt;ω d . 
     On the other hand, in a region where ω&gt;ω d , it is difficult to judge which of C c1  and C c2  should be employed only by looking at changes of the candidates (C c1 , C c2 ) for the electrical double layer capacitance value of the cathode electrode  113 . Accordingly, this judgment is made by directly studying the sets (C a1 , R a1 ), (C a2 , R a2 ) of the candidates for the reaction resistance value and the electrical double layer capacitance value of the anode electrode  112 . 
       FIG. 10A  shows frequency responses of the candidates C a1 , C a2  for the electrical double layer capacitance value of the anode electrode  112 . Further,  FIG. 10B  shows frequency responses of the candidates R a1 , R a2  for the reaction resistance value of the anode electrode  112 . It should be noted that these graphs are also based on data of the sets (C a1 , R a1 ), (C a2 , R a2 ) of the candidates obtained by continuously changing the frequencies ω l  and ω 2  calculated by an experiment or the like in advance in the range of the anode electrode response frequency band. 
     With reference to  FIG. 10A , in a region where the frequency ω&gt;ω d , the candidate C a1  for the electrical double layer capacitance value of the anode electrode  112  is extremely sensitive to the frequency. Thus, in the region where ω&gt;ω d , C a2  is a value which should be actually employed as a true estimation value of the electrical double layer capacitance value of the anode electrode  112 . Therefore, in the region where the frequency ω&gt;ω d , C a2  and R a2  corresponding thereto should be respectively employed. 
     It should be noted that, as understood with reference to  FIG. 10B , the candidate R a2  for the reaction resistance value is extremely sensitive to a frequency change in a region of ω&lt;ω d  where the frequency ω d  is smaller. Thus, the candidate R a1  for the reaction resistance value is judged to be a true estimation value which should be actually employed. Thus, in the region where the frequency ω&lt;ω d , C a1  corresponding to R a1  and R a1  should be respectively employed. This point is found to match considerations based on the frequency response of the electrical double layer capacitance value of the cathode electrode  113 . 
     Further, when the frequency ω=ω d , (C a1 , R a1 )=(C a2 , R a2 ). Thus, it does not matter which of these sets of the candidates is employed as the set of the true candidates. 
     Based on the above considerations, it is found that values to be determined from the sets (C a1 , R a1 ) and (C a2 , R a2 ) of the candidates change according to the frequency in determining the true estimation values. 
     Specifically, the appropriate one of the sets (C a1 , R a1 ) and (C a2 , R a2 ) of the candidates is determined according to the frequencies ω 1 , ω 2  at two points in the anode electrode response frequency band and the magnitude of the frequency ω d . Further, if the determined estimation values of the electrical double layer capacitance value C a  and reaction resistance value R a  of the anode electrode  112  are substituted into Equation (3), the electrolyte membrane resistance value R m  is obtained since the frequency ω and the real component Z r  of the impedance measurement value are known. 
     Subsequent Steps S 105  and S 106  are performed as in the first embodiment to also estimate the reaction resistance value R c  of the cathode electrode  113 , using the estimation values of the electrical double layer capacitance value C a  and reaction resistance value R a  of the anode electrode  112  and the electrolyte membrane resistance value R m  obtained in this way. 
     According to the state detection for the fuel cell  1  according to the present embodiment described above, only the anode electrode response impedances Z(ω 1 ) and Z(ω 2 ) are acquired as the high frequency impedances and the state quantities C a  and R a  of the anode electrode  112  are estimated on the basis of the anode electrode response impedances Z(ω 1 ) and Z(ω 2 ) by the controller  6  serving as the impedance acquisition unit and the internal state quantity estimation unit. 
     In this way, the state quantities C a  and R a  of the anode electrode  112  can be estimated while reducing a load to the controller  6  by omitting the estimation of the electrolyte membrane resistance value R m  on the basis of the measurement of the electrolyte membrane response impedance and, finally, the reaction resistance value R c , which is the state quantity of the cathode electrode  113 , can be estimated. 
     Fourth Embodiment 
     A fourth embodiment is described. It should be noted that elements similar to those of the already described embodiments are denoted by the same reference signs. 
       FIG. 11  is a flow chart showing the flow of state quantity estimation according to the present embodiment. As shown, in the present embodiment, the anode electrode response impedances Z(ω 1 ) and Z(ω 2 ) are obtained in Step S 103  and the estimation values of the reaction resistance value R a  and electrical double layer capacitance value C a  of the anode electrode  112 , the electrical double layer capacitance value C c  of the cathode electrode  113  and the electrolyte membrane resistance value R m  are obtained in Step S 304  as in the third embodiment. 
     Thus, as in the second embodiment, the low frequency impedance ΔV/ΔI is acquired on the basis of the I-V characteristic of the fuel cell  1  in Step S 205  and the reaction resistance value R c  of the cathode electrode  113  is estimated from the estimation values of the low frequency impedance ΔV/ΔI and the electrolyte membrane resistance value R m  in Step S 206 . 
     Accordingly, according to the state detection of the fuel cell  1  according to the present embodiment, the low frequency impedance Z(ω L ) can be estimated without being directly measured and the estimation of the electrolyte membrane resistance value R m  on the basis of the measurement of the electrolyte membrane response impedance can be omitted. Thus, a load to the controller  6  can be further reduced. 
     Fifth Embodiment 
     A fifth embodiment is described. It should be noted that elements similar to those of the already described embodiments are denoted by the same reference signs. 
     In the present embodiment, measurement values of actual output voltage V and output current I are used to calculate the value of ΔV/ΔI instead of a mode of storing the value of ΔV/ΔI in the steady region P of the characteristic curve Cv 1  in steady time of  FIG. 7  in Step S 205  according to the second and fourth embodiments. 
       FIG. 12  shows an I-V characteristic curve of the fuel cell  1  in steady time. Particularly, in the present embodiment, the gradient ΔV/ΔI is calculated by calculating −(V 1 -V 2 )/(I 1 -I 2 ) for output currents I 1 , I 2  measured by the current sensor  51  at predetermined measurement timings and output voltages V 1 , V 2  measured by the voltage sensor  52  at the same predetermined measurement timings. 
     Specifically, the gradient ΔV/ΔI regarded as the low frequency impedance is determined according to the measurement values of the output currents and the output voltages. 
     In the present embodiment, the gradient ΔV/ΔI in the I-V characteristic curve of the fuel cell  1  is calculated on the basis of two sets (I 1 , V 1 ), (I 2 , V 2 ) of the measurement values of the current and the voltage as just described. In this way, the value of ΔV/ΔI more accurately reflecting an actual characteristic than in the case of using the gradient ΔV/ΔI regarded and determined as a constant value in the steady region P can be obtained. As a result, the accuracy of the estimation value of the reaction resistance value R c  of the cathode electrode  113  calculated assuming this value of ΔV/ΔI as the low frequency impedance is also improved. 
     Sixth Embodiment 
     A sixth embodiment is described. It should be noted that elements similar to those of the already described embodiments are denoted by the same reference signs. 
     In the present embodiment, in order to obtain the gradient ΔV/ΔI in the I-V characteristic curve, the gradient ΔV/ΔI in the I-V characteristic curve is calculated using one set (I 3 , V 3 ) of measurement values of the output current and the output voltage and one set (I set , V set ) set beforehand instead of measuring two sets (I 1 , V 1 ), (I 2 , V 2 ) of the measurement values of the output current and the output voltage as in the fifth embodiment. 
       FIG. 13  is a graph showing an example of a method for setting one set of current and voltage for the calculation of the gradient ΔV/ΔI in the I-V characteristic curve. It should be noted that, in this graph, the characteristic curve Cv 1  in steady time is shown by a broken line to clarify the drawing. As shown, a point shown by a black square of  FIG. 13  is equivalent to (I set , V set ) described above in the present embodiment. Particularly, I set =0. 
     Accordingly, the value of the gradient ΔV/ΔI is calculated by calculating −(V set -V 3 )/(I set -I 3 ) on the basis of the above measurement values (I 3 , V 3 ) and the preset values (I set , V set ). 
     As described above, according to the present embodiment, the value of the gradient ΔV/ΔI in the I-V characteristic curve is calculated on the basis of one set (1 3 , V 3 ) of the measurement values of the current and the voltage and one set (I set , V set ) of the values of the current and the voltage set beforehand. 
     Accordingly, in calculating the gradient ΔV/ΔI in the I-V characteristic curve of the fuel cell  1 , it is possible to ensure calculation accuracy of a specified level or higher by using the measurement values (I 3 , V 3 ) at one point while suppressing the amount of calculation using (I set , V set ) set beforehand at another point out of two points on the I-V characteristic curve used to calculate the value of the gradient. 
     Seventh Embodiment 
     A seventh embodiment is described. It should be noted that elements similar to those of the already described embodiments are denoted by the same reference signs. 
     In the present embodiment, in the measurement of the impedance of the fuel cell  1  performed in the first embodiment and the like, an excitation current application method in which a current I is supplied from a predetermined current source for measurement to the fuel cell  1  and an impedance Z=V/I is calculated on the basis of this supplied current I and an output voltage V is employed instead of the configuration for measuring the output current I and the output voltage V superimposed with the alternating-current signal. 
       FIG. 14  is a block diagram schematically showing a main part relating to an impedance measurement in a fuel cell system  100  according to the present embodiment. 
     As shown, the fuel cell system  100  according to the present embodiment includes an applied alternating current adjustment unit  200  configured to apply an alternating current to a fuel cell  1  while adjusting the alternating current. 
     The applied alternating current adjustment unit  200  is connected to an intermediate terminal  1 C besides a positive electrode terminal (cathode electrode side terminal)  1 B and a negative electrode terminal (anode electrode side terminal)  1 A of a fuel cell  1  configured as a stack. It should be noted that a part connected to the intermediate terminal  1 C is grounded as shown. 
     The applied alternating current adjustment unit  200  includes a positive electrode side voltage measurement sensor  210  configured to measure a positive electrode side alternating-current potential difference V 1  of the positive electrode terminal  1 B with respect to the intermediate terminal  1 C and a negative electrode side voltage measurement sensor  212  configured to measure a negative electrode side alternating-current potential difference V 2  of the negative electrode terminal  1 A with respect to the intermediate terminal  1 C. 
     Further, the applied alternating current adjustment unit  200  includes a positive electrode side alternating-current power supply unit  214  configured to apply an alternating current I 1  to a circuit composed of the positive electrode terminal  1 B and the intermediate terminal  1 C, a negative electrode side alternating-current power supply unit  216  configured to apply an alternating current  12  to a circuit composed of the negative electrode terminal  1 A and the intermediate terminal  1 C, a controller  218  configured to adjust amplitudes and phases of these alternating currents I 1  and I 2 , and a calculation unit 220 configured to calculate an impedance Z of the fuel cell  1  on the basis of the electrode side alternating-current potential differences V 1 , V 2  and the alternating currents I 1 , I 2 . 
     In the present embodiment, the controller  218  adjusts the amplitudes and phases of the alternating currents I 1  and I 2  such that the positive electrode side alternating-current potential difference V 1  and the negative electrode side alternating-current potential difference V 2  become equal. It should be noted that this controller  218  may be configured by the controller  6  shown in  FIG. 3 . 
     Further, the calculation unit  220  includes hardware such as an unillustrated AD converter and a microcomputer chip and software configuration such as a program for calculating the impedance, calculates an impedance Z 1  from the intermediate terminal  1 C to the positive electrode terminal  1 B by dividing the positive electrode side alternating-current potential difference V 1  by the alternating current I 1  and calculates an impedance Z 2  from the intermediate terminal  1 C to the negative electrode terminal  1 A by dividing the negative electrode side alternating-current potential difference V 2  by the alternating current I 2 . Furthermore, the calculation unit  220  calculates the total impedance Z of the fuel cell  1  by taking the sum of the impedances Z 1  and Z 2 . 
     According to a state detection device for fuel cell according to the above embodiment, the following effects can be obtained. 
     The state detection device for fuel cell according to the present embodiment includes the alternating-current power supply units  214 ,  216  connected to the fuel cell  1  and configured to output the alternating currents I 1 , I 2  to the fuel cell  1 , the controller  218  serving as an alternating current adjustment unit configured to adjust the alternating currents I 1 , I 2  on the basis of the positive electrode side alternating-current potential difference V 1 , which is a potential difference obtained by subtracting the potential of the intermediate terminal  1 C from the potential of the positive electrode terminal  1 B of the fuel cell  1 , and the negative electrode side alternating-current potential difference V 2 , which is a potential difference obtained by subtracting the potential of the intermediate terminal  1 C from the potential of the negative electrode terminal  1 A of the fuel cell  1 , and the impedance calculation unit  220  configured to calculate the impedance Z of the fuel cell  1  on the basis of the adjusted alternating currents I 1 , I 2  and the positive electrodes alternating-current potential difference V 1  and the negative electrode side alternating-current potential difference V 2 . 
     The controller  218  adjusts the amplitudes and phases of the alternating current I 1  applied by the positive electrode side alternating-current power supply unit  214  and the alternating current I 2  applied by the negative electrode side alternating-current power supply unit  216  such that the positive electrode side alternating-current potential difference V 1  on the positive electrode side of the fuel cell  1  and the negative electrode side alternating-current potential difference V 2  on the negative electrode side substantially match. Since the amplitude of the positive electrode side alternating-current potential difference V 1  and that of the negative electrode side alternating-current potential difference V 2  become equal in this way, the positive electrode terminal  1 B and the negative electrode terminal  1 A are substantially at an equal potential. Thus, the alternating currents I 1 , I 2  for the impedance measurement are prevented from flowing to a load  53 , wherefore the influence of the fuel cell  1  on power generation is prevented. 
     Further, in the case of carrying out the above impedance measurement when the fuel cell  1  is in a power generation state, an alternating-current potential for measurement is superimposed on a voltage generated by this power generation. Thus, the values of the positive electrode side alternating-current potential difference V 1  and the negative electrode side alternating-current potential difference V 2  themselves become larger. However, since the phases and amplitudes of the positive electrode side alternating-current potential difference V 1  and the negative electrode side alternating-current potential difference V 2  themselves do not change, a highly accurate impedance measurement can be carried out as in the case where the fuel cell  1  is not in the power generation state. 
     Although the embodiments of the present invention have been described above, the above embodiments are merely an illustration of some application examples of the present invention and not intended to limit the technical scope of the present invention to the specific configurations of the above embodiments. For example, the steps of acquiring the anode electrode response impedance, the electrolyte membrane response impedance and the low frequency impedance (Steps, S 101 , S 103  and S 105 ) and the like in each embodiment can be arbitrarily changed without being limited to the sequence of the steps described in each embodiment. 
     For example, each state quantity may be estimated after all the steps of acquiring the anode electrode response impedance, the electrolyte membrane response impedance and the low frequency impedance are performed. 
     Further, the modes of estimating a plurality of internal state quantities in the fuel cell  1  are not limited to the modes described in each of the above embodiments. 
     For example, instead of the mode of selecting one frequency ω L  from the low frequency band in Step S 105  in the first or third embodiment, two frequencies ω L1 , ω L2  may be selected in the low frequency band and low frequency impedances Z(ω L1 ) and Z(ω L2 ) may be obtained. In this way, not only the estimation value of the reaction resistance R c  of the cathode electrode  113 , but also that of the electrical double layer capacitance C c  of the cathode electrode  113  can be finally obtained. 
     Further, the mode of the simplified equivalent circuit of the fuel cell  1  is also not limited to that used in each of the above embodiments. For example, an equivalent circuit including other elements such as a diffusion resistance, an electron transport resistance and an ionomer resistance besides the circuit elements such as the reaction resistance and the electrical double layer capacitance of each electrode described in each of the above embodiments may be set, and a diffusion resistance value, an electron transport resistance value, an ionomer resistance value and the like serving as internal state quantities based on these other elements may be estimated.