1. Introduction
The problem of optimal and well-diversified energy distribution in a given country is central to the well-functioning of economic and social infrastructures. There are two aspects to the problem. On the one hand, we have to consider the problem of satisfying the demand without unnecessary surplus (unless there is a market for the surplus). On the other hand, we must ensure that the resulting distribution is well-diversified.
The first problem is a typical exercise of demand-supply with given cost constraints. Costs include economic, ecological, and social costs. Any cost that can be translated into a constraint on the energy flow between consumer and supplier (or source and sink) should be taken into account. This problem has received considerable attention in the applied mathematics literature when the data about the amounts produced are exact. A review of the literature is available in [
1].
In addition, as there might be uncertainties in the demand and/or the supply, it is realistic to suppose that the data are specified up to intervals. Of course, there are obvious interval constraints of this type: a given supplier cannot produce more than a certain amount at a given site. However, the amount produced may fluctuate within a range within the maximum capacity. Similar limitations can be observed on the demand side.
Linear programming techniques are one of the standard tools to deal with the problem, especially when there is a monetary cost that must be minimized. The main goal of this paper is to propose the method of maximum entropy on the mean to deal with the transportation problem with cost constraints and range constraints on the solutions. The method that we propose is based on an extension of the standard method of maximum entropy proposed in [
2]. Attempts to solve the problem using maximum entropy based methods are long-dated (see Chapter 13 in Kapur’s [
3] and the references in that chapter for earlier work, in particular to the work by Wilson [
4]). An interesting application can be found in [
5]. Our approach to the use of the method of maximum entropy is completely different.
The problem of diversification is reviewed in [
6,
7], where more references are listed, and we bring in the entropy-based measure proposed there to measure the diversification of the solution to the problem considered above.
To state the problems treated here, consider M sources of electricity and denote by the amount supplied by each source. The labeling addresses the possibility that in the same geographical area there may exist two different types of electricity sources, which would be labeled by two different subscripts.
Consider N different electricity consumers, each of whom requiring units of energy per unit time. These may describe geographical areas and/or different types of consumer within a given area. For example, within a geographical area, there may exist large chemical industries or mineral processing plants that may need large amounts of electricity, besides what is needed for domestic use by the local population.
The first group of constraints is imposed by the nature of the generation process or by the local demand of electricity: We suppose that the demand at each site and the supply at site are both known. Demand at site i represents the demand by a type of energy consumer which might be distributed geographically, but which shares some type of collective characteristic, such as being a domestic (household) consumer in a geographic area.
1.1. Problem Statements
If there were no other constraints, the simplest problem of distributing the electricity could be stated as follows.
Problem 1 (First electricity supply-demand problem). Denote by the amount of electricity required at site i and produced at site j per unit time. We suppose that the network is such that the every electricity-producing site is connected to every consuming site. Otherwise, we set the corresponding without further ado.
The electricity matching problem consists of determining values that satisfy the following demand-supply constraints: Comments:
- (i)
Production per unit time refers to an average produced or required during some standardized time interval (one hour, for example).
- (ii)
These constraints can be replaced by intervals to allow for uncertainty in the demand or uncertainty in the supply. We describe this further below.
- (iii)
Note that we are taking into account possible nonlinear constraints, resulting from the actual physical transport of energy through the network, in which the losses could depend on the amount of energy being transported.
In practice, cost constraints may also exist. Here, we suppose that costs are regulated and fixed (by competition or government agencies) and fairly passed on to consumers, but these constraints have to be taken into account.
There might also exist regulatory environmental constraints. Each mode of electricity generation has an environmental impact, measured by
grams of contaminant per unit of electricity generated by the
ith supplier per unit time. Therefore, the total amount of contaminants generated by the supplier is
With this, instead of Problem 1, we consider now the following:
Problem 2 (Second electricity supply-demand problem).
With the notation introduced above, the electricity matching problem consists of determining that satisfy the demand-supply constraints (1) as well as cost constraints solving: Here, is a maximum cost (environmental impact) generated (or incurred) by the electricity-producing system that regulators allow. Most of the time the constraint will not depend on the connection (edge) but only on the source (type of energy that is produced at the jth source). In this case, does not change as i changes.
For a combination of technological and economical reasons, we might be forced to consider flexible constraints. For example, to cover for possible downward or upward movement in demand or supply, we might consider those values to fall in a range. Similarly, we could include a tolerance for sudden fluctuations in the cost of production, or the demand at a certain node. Instead of point-valued constraints, we might replace both Problems 1 and 2 by:
Problem 3 (Third electricity supply-demand problem).
To allow for different types of constraints, we extend our previous notation. Denote by a -matrix, where and denote its elements by , where labels the cost constraints (or cost restrictions). Now, instead of a point value K, we consider a range dataset given by:The problem to solve is to find in some given range such that:When reduces to a point, that is, when with we have a problem with point constraints.
Problems 1–3 are ill-posed linear inverse problems subject to convex constraints. As such, each might have infinitely many possible solutions because the number of unknowns is usually significantly larger than that of the dataset. The convex constraints include positivity constraints as well as those in (3).
The method of maximum entropy in the mean is especially designed to deal with this type of problems. The representation of the solution is such that it allows for explicit sensitivity analysis. The standard maximum entropy method (SME) is the stepping stone towards the method of maximum entropy in the mean (MEM).
1.2. Contents of the Paper
Building on the aforecited notation, we briefly describe the solution to Problems 1–3 in
Section 2. The solution of the three problems, given by (21), looks the same, although what changes is the specification of the problem data. The actual derivation of the solutions of Problems 1 and 2 follows the same pattern, and Problem 3 uses the solution of Problem 2 as a stepping stone.
In
Section 3, we recall the mathematical details of the procedure to arrive at the results listed in
Section 2. We include a short digression on using the concept of entropy to quantify how diversified is a solution to the supply-demand problem. Further, we explicitly compute it for each of the examples.
In
Section 4, we work out two illustrating toy examples which take into account all the essentials. In the first one, we consider only point data, while, in the second example, we show the flexibility of the approach with some of the data in intervals. The natural production constraints are specified up to an interval to incorporate the possibility of fluctuation in the energy generation output.