Method for maximum net power calculation for fuel cell system based on online polarization curve estimation
View Patent ↗An algorithm for determining the maximum net power available from a fuel cell stack as the stack degrades over time using an online adaptive estimation of a polarization curve of the stack. The algorithm separates the current density range of the stack into sample regions, and selects a first sample region from the far left of the estimated polarization curve. The algorithm then calculates the cell voltage for that current density sample region, and determines whether the calculated cell voltage is less than or equal to a predetermined cell voltage limit. If the calculated cell voltage is not less than the cell voltage limit, then the algorithm selects the next sample region along the polarization curve. When the calculated cell voltage does reach the cell voltage limit, then the algorithm uses that current density for the sample region being analyzed to calculate the maximum power of the fuel cell stack.
1. A method for estimating the maximum stack power from a fuel cell stack during operation of the fuel cell stack, said method comprising:
executing at any time during operation of the fuel cell stack, using at least one computing device coupled to a memory, instructions for:
selecting an average cell voltage limit;
selecting a minimum cell voltage limit;
estimating a separate polarization curve of the fuel cell stack for both an average cell voltage and a minimum cell voltage;
executing an algorithm to determine a maximum current density for each of the average cell voltage polarization curve and the minimum cell voltage polarization curve, the algorithm including:
separating a current density range of the fuel cell stack into a predetermined number of sample regions in ascending order;
selecting a first sample region and a current density corresponding to the first sample region;
calculating an average cell voltage and a minimum cell voltage at the current density selected for the first sample region being examined;
determining whether the average cell voltage is less than the average cell voltage limit;
selecting a next sample region in the current density range and a current density corresponding to the next sample region if the average cell voltage is not less than the average cell voltage limit;
setting the current density selected for the sample region being examined as the maximum current density for the average cell voltage polarization curve when the average cell voltage is less than the average cell voltage limit;
calculating a maximum net power using the maximum current density estimated at the average cell voltage limit;
determining whether the minimum cell voltage is less than the minimum cell voltage limit;
selecting a next sample region in the current density range and a current density corresponding to the next sample region if the minimum cell voltage is not less than the minimum cell voltage limit;
setting the current density selected for the sample region being examined as the maximum current density for the minimum cell voltage polarization curve when the minimum cell voltage is less than the minimum cell voltage limit;
calculating a maximum net power using the maximum current density estimated at the minimum voltage limit; and
selecting as the maximum stack power, the lesser of the maximum net power using the maximum current density estimated at the average cell voltage limit and the maximum net power using the maximum current density estimated at the minimum voltage limit.
2. The method according to claim 1 wherein calculating the net power includes calculating the net power as a maximum cell voltage times an overall current density times a number of fuel cells in the fuel cell stack times an area of the fuel cells of the fuel cell stack minus a predetermined parasitic power that are used while running the fuel cell stack.
3. The method according to claim 2 further comprising adding a correction to the calculated maximum power.
4. The method according to claim 1 wherein determining an average cell voltage and a minimum cell voltage includes using the equation:
E
cell
=
E
rev
-
(
j
+
a
)
*
R
HFR
-
(
0.07
*
log
10
(
j
+
a
j
0
)
+
c
log
10
(
1
-
j
j
∞
)
)
where E call is the cell voltage, j is the current density, R HFR is the cell HFR resistance, E rev is the thermodynamic reversible cell potential, α is the background current density from cell shorting/cell crossover, j D is the exchange current density, j ∞ is the limiting current density and c is the mass transfer coefficient.
5. The method according to claim 1 wherein the average cell voltage limit is about 0.525 V and the minimum cell voltage limit is about 0.3 V.
6. The method according to claim 1 wherein separating a current density range of the fuel cell stack into a predetermined number of sample regions includes separating a current density range of 0.1-2.0 A/cm 2 into sample regions of 0.1 A/cm 2 .
7. The method according to claim 1 wherein selecting a next sample region in the current density range includes selecting the next sample region in order from a low current density in the range to a high current density in the range.
8. The method according to claim 1 wherein the fuel cell stack is a split fuel cell stack where the maximum stack power is determined for both spilt stacks.
9. A method for estimating the maximum stack power from a fuel cell stack during operation of the fuel cell stack, said method comprising:
executing during operation of the fuel cell stack, using at least one computing device coupled to a memory, instructions for:
selecting at least one cell voltage limit for a fuel cell in the fuel cell stack;
estimating a polarization curve of the fuel cell stack for at least one cell;
separating a current density range of the fuel cell stack into a predetermined number of sample regions;
selecting a first sample region;
determining a cell voltage at a current density for the selected sample region;
determining whether the cell voltage is less than the cell voltage limit;
selecting a next sample region in the current density range if the cell voltage is not less than the cell voltage limit; and
calculating a maximum net power for the cell voltage if the cell voltage is less than the cell voltage limit.
10. The method according to claim 9 wherein selecting at least one cell voltage limit includes selecting an average cell voltage limit corresponding to an average cell voltage polarization curve and a minimum cell voltage limit corresponding to a minimum cell voltage polarization curve, and wherein estimating a polarization curve of the fuel cell stack includes estimating a polarization curve for both the average cell voltage and the minimum cell voltage, and wherein determining a cell voltage includes determining an average cell voltage and a minimum cell voltage at the current density for the selected sample region.
11. The method according to claim 9 wherein calculating the net power includes calculating the net power as a maximum cell voltage times an overall current density times a number of fuel cells in the fuel cell stack times an area of the fuel cells of the fuel cell stack minus a predetermined parasitic power that are used while running the fuel cell stack.
12. The method according to claim 11 further comprising adding a correction to the calculated maximum power.
13. The method according to claim 9 wherein determining the cell voltage includes using the equation:
E
cell
=
E
rev
-
(
j
+
a
)
*
R
HFR
-
(
0.07
*
log
10
(
j
+
a
j
0
)
+
c
log
10
(
1
-
j
j
∞
)
)
where E cell is the cell voltage, j is the current density, R HFR is the cell HFR resistance, E rev is the thermodynamic reversible cell potential, α is the background current density from cell shorting/cell crossover, j 0 is the exchange current density, j ∞ is the limiting current density and c is the mass transfer coefficient.
14. The method according to claim 9 wherein separating a current density range of the fuel cell stack into a predetermined number of sample regions includes separating a current density range of 0.1-2.0 A/cm 2 into sample regions of 0.1 A/cm 2 .
15. A system for estimating the maximum stack power from a fuel cell stack during operation of a fuel cell stack, said system comprising;
means for selecting an average cell voltage limit;
means for selecting a minimum cell voltage limit;
means for estimating a polarization curve of the fuel cell stack for both an average fuel cell and a minimum fuel cell voltage;
means for separating a current density range of the fuel cell stack into a predetermined number of sample regions;
means for selecting a first sample region;
means for determining an average cell voltage and a minimum cell voltage at the current density for the selected sample region;
means for determining whether the average cell voltage and the minimum cell voltage are less than the average cell voltage limit and the minimum cell voltage limit, respectively;
means for selecting a next sample region in the current density range if the average cell voltage is not less than the average cell voltage limit;
means for selecting a next sample region in the current density range if the minimum cell voltage is not less than the minimum cell voltage limit;
means for calculating a maximum net power for the average cell voltage when the average cell voltage is less than the average cell voltage limit and for calculating a maximum net power for the minimum cell voltage when the minimum cell voltage is less than the minimum cell voltage limit; and
means for selecting the less of the average cell voltage maximum net power and the minimum cell voltage maximum net power as a maximum stack power.
16. The system according to claim 15 wherein the means for calculating the net power includes means for calculating the net power as the maximum cell voltage times the overall current density times the number of fuel cells times the area of the fuel cells of the fuel cell stack minus predetermined parasitic power that are used while running the fuel cell system.
17. The system according to claim 15 wherein the average cell voltage limit is about 0.525 V and the minimum cell voltage limit is about 0.3 V.
18. The system according to claim 15 wherein separating a current density range of the fuel cell stack into a predetermined number of sample regions includes separating a current density range of 0.1-2.0 A/cm 2 into sample regions of 0.1 A/cm 2 .
19. The system according to claim 15 wherein the fuel cell stack is a split fuel cell stack where the maximum stack power is determined for both spilt stacks.
20. The system according to claim 15 wherein the means for selecting a next sample region in the current density range includes means for selecting the next sample region in order from a low current density in the range to a high current density in the range.