Wednesday, January 19, 2011

Comparison between the Supply-Demand Curves and a Fuel Cell V-I Curves

One of the main topics in any microeconomics course is the supply-demand curve, and how the two curves meet together at an "equilibrium" point.
Note: The term "equilibrium" stems from the fact that the economists in the 1800's borrowed from the language of mechanics & physics at the time (as is often done, sometimes for good and sometimes not.)

We need a new formulation of economics: one that does not assume that the universe is in equilibrium. Instead, we need a new economics for open, far-from-equilibrium systems trying to maximize the production of entropy via the maximization of the rate of return on investment on work invested.

So, I was thinking today about the supply-demand curve and it seems to me that there are some similarities between supply-demand curves of economics and the V-I curves of fuel cells & electrolyzer. In this blog, I hope to point out the similarities once you convert currency [$] into units of work [kW-hr].

This is the summary of what will be presented below:
If one makes the substitution of work for currency, then one can convert the demand curves of economics into a voltage-current plot of a fuel cell and convert the supply curves of economics into a voltage-current plot of an electrolysis unit. The operating voltage of an electrolysis unit is the marginal work required to move an additional mol of charged species, and is analogous to the marginal cost of producing one more product. The operating voltage of a fuel cell is the marginal work generated in moving an additional mol of charged species, and is analogous to the marginal value of demanding one more product. (See the following post for the difference between price, cost and value.)



The demand curve of economics graphs the number of items that will be bought per unit time as a function of the price per unit. If the price is low, then there is large demand, and as the price increases, the demand decreases. This type of curve is very similar to the V-I curve of fuel cell (or a photovoltaic cell). Here Voltage (V) corresponds to the $'s per unit of a demand curve, and the current (I) corresponds to the # of objects that are demanded at a given price, i.e. the current is the # of charges that can be generated for a given voltage. For high voltage [work/charge] ($ per unit), then there are only a few charges (units) demanded per time.  (The comparison here is:  kJ with  $,  Charge  with  units, &  time with time) This comparison does not assume equilibrium, and this is clear because rates (per time) show up in the variables.

When you multiply current times voltage, you get the power output from the fuel cell or photovoltaic cell. Similarly, when you multiple $ per unit by #units/time, you get $/s. And since I showed earlier that money ($) is equivalent to work (kW-hr) in a previous blog, you can see that one also obtains the "power" from a supply-demand curve. Note that there is a direct correspondence between voltage and $/unit as well as current and #of units/time.
Electricity Backed Currency

To be correct, a supply curve should relate the # of units per time as a function of the price that people are willing to pay a given price per unit, as opposed to the typical curve which just relates the # of units as function of price per unit.

For a fuel cell, the reason that the voltage decreases as you increase the current (current density to be precise) is that the irreversibility of the system increases. Irreversibility destroys the capability of generating work (voltage) and hence destroys the capability to make money. Does the concept of irreversibility have an analogous term in the demand curve?

Yes!

The people who are willing to buy the unit at a high price are willing to do so because they will use the product efficiently and can derive more value from the product than people. What this means is that they will use the product with minimal irreversibility. However, the people who can only afford the product at a lower price will not use the product as efficiently, and this is means that they will cause more irreversibility.


So, let's continue with the supply curve of economics. What's the equivalent for a fuel cell? I believe that the supply curve of economics is equivalent to the V-I curve of an electrolyzer (which is a fuel cell run backwards. For example, an electrolyzer generates hydrogen and oxygen from water, whereas a fuel cell generates water from hydrogen and oxygen.) The supply curve of economics graphs the rate of units that can be produced as a function of the price that is willing to be paid. As the price increases, the rate of produced units increases. This is quite similar to what happens in the V-I curve of an electrolyzer. (As you increase the voltage of the electrolyzer, you increase the current that you can generate. So, once again, the analogy is between voltage and $/unit  as well as current and units/time.)

Once again, the concept of irreversibility is crucial in understanding both the electrolyzer V-I curve and the supply curve. As you increase the current (or unit/time), you increase the irreversibility at the margin. To draw more current, the paths remaining for charged particles are more resistive than the paths already used. The same holds for the supply curve. To produce more units/time, you have to use factories that are less efficient (more irreversible) than the factories that are already being used. You start building at the cheapest factories, and then you move to more expensive factories only if you can sell units at a higher price. This is even easier to see when looking at what's called the Incremental Price Curve of an electricity market.
This Dispatch Curve comes from the PJM training courses on Locating the Marginal Electricity Price. I suggest taking a few of PJM's free online courses if you are interested in understanding how an exceptional electricity market works. PJM has become the world's largest electricity market...and there's a good reason...because they seem to know what they're doing and they've been in the business for awhile. The PJM market (as opposed to the California deregulation debacle) is a good example of how an electricity market should work.
The shape of the V-I curve of an electrolysis unit (or a battery being charged) will be a monotonically increasing shape for the same basic reason for the monotonically increasing shape of a supply curve in economics, or termed the Incremental Price Curve in certain electricity markets. As more products (or charges) are required to be supplied, processes with more irreversibility are required. The operating voltage is the marginal work required to move one more mol of charged species. The marginal voltage is higher when more current is required because all of the low irreversibility routes to move charged spcies are already filled. Similarly, the marginal cost of producing one more MW of electricity increases as the amount of required MW increases because all of the low irreversibility (i.e. low cost) routes of generation/transmission have already been filled. Note that in an electricity-backed currency, the units from multiplying the operating value (x,y) are power [MW] for both (a) the electrolysis/electrodialysis V-I curve and (b)  the Incremental Price Curve. If you multiple the y-axis of the Incremental Cost Curve by the grid voltage and divide the x-axis by the grid voltage, then you obtain the same units for the y-axis (voltage) and x-axis (current) as you would obtain for any electrochemical device, either a solar cell, a battery, a fuel cell or an electrodialysis unit. The analogy is:  (price, cost, value) = Work [kW-Hrs], (product) = Charge, (time) = time. So, (cost / product) = Voltage,  and (product / time) = Current.


In summary, currency is equal to ‘work,’ which has units of energy [kW-hrs], in any rational theory of economics based solely on physical units. I have hopefully shown you that there is a strong analogy between 'voltage' and 'price/unit' because both are the marginal work per unit as well as a strong analogy between 'current' and '# of units/time.' And this is the start of making a theory of economics that does not assume equilibrium, and instead recognizes that the world is far-from-equilibrium.

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