1. Economical Dispatch Representation of the Problem
Cost curve:
- fuel cost of unit 1, a probability distribution.
- fuel cost of unit 2, a probability distribution.
Incremental cost curve:
The format of the problem:
Min F=F1+F2 = 
+
S.t P1+P2 = D (D = 400)
According to KKT condition:
(1)
(2)
(3)
1.1 If there is no overlap between the incremental cost curves of the two units, it is quite easy to determine the output level for each unit.
For the above case, according to equal incremental cost
criteria, unit always loads first until it is fully loaded, unit 2 loads the
left demand after unit 1. Output 
, 
, then the production cost is
F1+F2= 
+
=a*v1+b*v2
If the distribution of 
and 
are known, based on
multiple operands calculation, the distribution of production cost can be
obtained from P-bounds software with information on CDF and IDF.
1.2 If there is overlap between the incremental cost curves of the two units, the output level of each unit is a random variable decided by the distribution and correlation of the fuel price.
From (1) (2) and (3) 
(5)
(6)
If the distribution of 
and 
are known, based on
multiple operands calculation, the distribution of P1 and
P2 can be obtained from P-bounds software with information on
CDF and IDF. The production cost is
F1+F2= 
+ 
=f1(v1, v2)*v1+
f2(v1, v2)*v2
The same, the distribution of production cost can be obtained through multiple operands calculation.
2. Decision Tree Representation of the Problem.
Cost curve:
- Fuel cost of unit 1, a probability distribution.
- Fuel cost of unit 2, a probability distribution.
Incremental cost curve:
The format of the problem:
Min F=F1+F2 = 
+
S.t P1+P2 = D (D = 400)
Decision Tree Representation:
Node 1 represents the decision set of unit 1; node 2 represents the decision set of unit 2.
Since the decision set located in the range from minimal output to maximal output continuously, there is infinite number of leaves out of the node, which can’t be represented by enumeration of possible leaf.
If the fuel cost is independent to each other, the decision tree is symmetric from each leaf of node 1, otherwise the tree is not symmetric.
The decision tree can be solved by backward propagation: first assume a possible value of fuel cost v2, then choose the feasible value set of v1 according to v2, determine the optimal output level of each unit for all the possible value of v2 (the distribution of P1 and P2). The procedure is similar to the Monte Carlo method in solving the production cost problem.
3. Influence Diagram Representation of the Problem.
The above diagram is the influence diagram representation of the problem with correlation on fuel cost. Since the fuel cost is correlated, it is represented by a single chance event – oven in the influence diagram. Generally in order to solve the influence diagram, it is first converted to decision tree, then solve it like the above procedure.