Memo: Economical Dispatch with Correlation on Fuel Price
From: Rujun Hu
To: Dr. G. Sheble, D. Berleant
CC: Jianzhong Zhang
Date: Feb. 26 2001.
Introduction: Economical dispatch is the most traditional and basic issue in the steady state operation of power system. It is the base and indispensable part of other application such as unit commitment and production cost. Although with the deregulation and restructuring, some new trend, decentralized unit commitment and self economical dispatch, are under discussing, economical dispatch will continue to play an important role in the Energy Management System(EMS) recently. Even when central economical dispatch is substituted by bid and auction, the theory of economical dispatch will still be applied in other application such as production cost.
In the common known model, the correlation between the fuel price of different units is neglected, but in actual, there are only two main source of fuel for most of the generation units, gas and coal, not only the fuel prices of units using same fuel are correlated, but also there is correlation exists among the units employing different fuel.
So consider the correlation between the fuel prices of different units will provide a more accurate dispatch, which results in less generation cost. The model will also be more reasonable to guide the operation of power system.
The first part of the memo elaborates the modified economical dispatch model with correlation on the fuel price and its difference between traditional the model is discussed. How to apply a new mathematic tool P-bounds, which can accommodate all the known and unknown correlation between the random variable, to solve the problem and give out the most information under the uncertain condition is also discussed in the first part. Decision tree representation and influence diagram representation of the problem are presented in part 2 and part 3 to provide a investigation of the problem from other prospects.
Key Words: Economical Dispatch, Fuel Price, Correlation, P-bounds, Decision Tree, Influence Diagram.
1. Economical Dispatch Model with Correlation on Fuel Price
Cost curve:
- fuel Price of unit 1, a probability distribution.
- fuel Price of unit 2, a probability distribution.
Incremental cost curve:
The format of the problem:
Min F=F1+F2 = 
+
(1)
S.t P1+P2 = D (D = 400) (2)
According to KKT condition:
(3)
(4)
(5)
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 since unit 1 always loads first, until it is fully loaded, unit 2 bears the left demand after unit 1. (Note that the incremental cost curve a group of curves depend on the distribution of fuel price, not just a single curve.)
Output 
(6)
(7)
Then the production cost is
F1+F2= 
+ 
=a*v1+b*v2 (8)
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 (3) (4) and (5) 
(9)
(10)
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= 
+ 
(11)
=f1(v1, v2)*v1+ f2(v1, v2)*v2
The same, the distribution of production cost can be
obtained through multiple operands calculation no matter 
and 
are independent
or unknown dependent (positive, negative, partially or perfect
correlated).
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.
Conclusion: With the new model that simulates the real system more accurately, more information obtained for the really dispatch result that leads to least generation cost. The result is no longer exact value for each unit, they are distribution at a certain range depends on fuel price. With P-bounds, no matter the fuel prices are independent or unknown dependent (positive, negative, partially or perfect correlated) the most information available is provided to the decision making to support it as much as possible.