I have a real-life problem that I would like to express in maths, especially to use an algorithm to solve it over time. Both because I want to learn something new, both because, as a wise consumer, I would like to have a solution to a more general problem than a one-month specific case, comparing different companies by performance too.
Let's start from a generic problem. I am customer of Shinra Power Corp who provides electricity to my house. They offer me to choose between two tariffs, where the change from one to another has no administrative fee. I have option to change my tariff, but not to change my habits or use energy-saving home equipment in the scope of this question. So the obvious answer "Get the variable rate and consume during night, run the dishwasher weekly and super full" is not viable. And here we are talking only about electricity, not water or gas.
- Variable rate: the electricity charge is computed by coefficients $\alpha$, $\beta$ and $\gamma$ in different time slots, as explained later. Each variable is currency-over-kilowatt (${€ \over kW}$)
- Constant rate: the electricity has a single 24/7 rate (${€ \over kW}$), named $\delta$
The variable rate consists in 3 different figures read by the power metre, F1 (daylight, associated with alpha, most expensive), F2 (commute and dinner hours, associated with beta, cheaper) and F3 (nights and weekends, associated to gamma, cheapest). The constant fare, for which I'll tell later the exact figure, is delta and is in some between alpha and gamma. Following disequations always apply in general.
$$ \alpha > \beta > \gamma ; \alpha > \gamma, \alpha > \delta > \gamma $$
The time slots are designed to make it more convenient for consumers/families to use more power in the nights and weekends when factories/offices are closed. This is not true in the era of remote working. I.e. if a family can afford to run all the appliances at night and none in the working hours they achieve best saving, by design.
My question is
I want to express a mathematical problem that can be solved with the help of computer tools (e.g. Excel, Wolfram, C#, etc). A visual tool, or visually-interactive, is preferred. I want to find the the break-even point between the two tariffs, given that the coefficients $\alpha, \beta, \gamma, \delta$ are known. The incognitos are the percents of power distributed during the three time slots. The total power usage (kilowatts), which makes the ultimate price, is irrelevant in this context. In particular, different sets of coefficients will change the break-even point, which is what I want to compare.
Suppose in the future either Shinra changes the 4 power rates, or I want to sign up electricity from AVALANCHE (who provides greener power? ), I'd like to check how much nightly power (washers, dryer, etc) is required for the variable tariff.
One more constraint: I know that the results depend in the value of the coefficients, which can be found on the electricity contract and may vary during time. And are at the bottom of the question.
Here is what I did so far
I have tried to express the constraints as equations first. The following is how the power part of the bill (i.e. the taxes and general fees will be added later)
$$ \left\{ \begin{array}{c} \alpha F1 + \beta F2 + \gamma F3 = \chi F \\ \delta (F1+F2+F3) = \delta F \\ (F1+F2+F3)=F \end{array} \right. $$
Where now $F_i$ is the kilowatt power usage for each time slot, and the new $\chi$ variable is the average cost of electricity per kW, and $F$ is just the total billed power.
If I introduce incognitos $x, y, z$, the problem gets closer to my percentage formulation. By dividing every $F_i$ by $F$ itself.
$$ \left\{ \begin{array}{c} \alpha x + \beta y + \gamma z = \chi \\ 0<x,y,z<1 \\ x+y+z=1 \\ \alpha > \delta > \beta, \gamma \end{array} \right. $$
In this scope, $x,y,z$ are percetanges of power usage respectively in expensive (x) and cheap (y,z) time slots.
My question becomes
Once I know the exact values of the tariff coefficients, as the below example, how can I determine possible $(x, y, z)$ as percentages that reduce the average cost of electricity to $\chi < \delta $?
$$ \left\{ \begin{array}{c} \alpha = 0.0510 \\ \beta = 0.0480 \\ \gamma = 0.0370 \\ \delta = 0.0449 \\ \end{array} \right. $$
Experimental finding
I did a retrospective looking at my real data December 2020 ($F1=78kW, F2=79kW, F3=72kW$), thus with an equal distribution Wolfram said the constant tariff shown above is more convenient but the price impact is negligible for the record.