Economic Dispatch is the process of allocating the required load demand between the available generation units such that the cost of operation is minimized. There have been many algorithms proposed for economic dispatch: Merit Order Loading, Range Elimination, Binary Section, Secant Section, Graphical/Table Look-Up, Convex Simplex, Dantzig-Wolf Decomposition, Separable Convex Linear Programming, Reduced Gradient with Linear Constraints, Steepest Descent Gradient, First Order Gradient, Merit Order Reduced Gradient, etc. The close similarity of the above techniques can be shown if the solution steps are compared. These algorithms are well documented in the literature. We will use only the graphical (LaGrangian Relaxation) techniques.

Generation Models

 The electric power system representation for Economic Dispatch consists of models for the generating units and can also include models for the transmission system. The generation model represents the cost of producing electricity as a function of power generated and the generation capability of each unit. We can specify it as:

1. Unit cost function:

(1)

where F_i = production cost

F_i(.) = energy to cost conversion curve

P_i = production power

2. Unit capacity limits:

(2)

3. System constraints (demand – supply balance)

(3)

Formulation of the LaGrangian

We are now in a position to formulate our optimization problem. Stated in words, we desire to minimize the total cost of generation subject to the inequality constraints on individual units (2) and the power balance constraint (3). Stated analytically, we have:

Minimize:

Subject to: (1)~(3)

 The equality constraint h(x) = b for the general case was allowed to contain multiple constraints. Here, in the EDC problem, we see that there is only one equality constraint, i.e., h and b are both scalars. This implies that lambda is a scalar also. The Lagrangian function, then, is:

(4)

KKT Conditions

Application of the KKT conditions to the LaGrangian function of (4) results in:

i=1,2…N (5)

(6)

The KKT conditions provide us with a set of equations that can be solved. The unknowns in these equations include the generation levels P₁, P₂, …, P_n and the LaGrange multipliers , a total of (n+1) unknowns. We note that (5) provides n equations, (6) provides one equation. Thus, we have a total of (n+1) equations.

KKT Conditions for a 2-unit system

To illustrate more concretely, let’s consider a simple system having only two generating units. The LaGrangian function, from (4), is:

The KKT conditions, from (5) and (6) become:

(7)

(8)

Graphical Solution

Recall the first KKT condition when applied to the generation system, if we assume that all binding inequality constraints have been converted to equality, then the equations becomes

(9)

This equation implies that for all regulating generators (i.e. units not at their limits.) each generators incremental costs are the same and are equal to lambda.

This very important principle provides the basis on which to apply the graphical solution method. The graphical solution is illustrated in Figure 1 (note that "ICC" means incremental-cost-curve). The unit's data are simply plotted adjacent to each other. Then, a value for lambda is chosen (judiciously) and the generations are added. If the total generation is equal to the total demand "P_T" then the optimal solution has been found. Otherwise, a new value for lambda is chosen and the process repeated. The limitations of each unit are included as vertical lines since the rulers must not include generation beyond unit capabilities. The unit is simply fixed at the value crossed.

Figure 1. Graphical Solution of EDC

Example:

There is a simple two units system including two very similar units that have the following input-output cost function and incremental cost function:

(10)

(11)

(12)

Discretize the I/O curve and incremental cost curve with 10MW space each, 9.92 is the mean value among the discrete points of the incremental cost curve for unit 1, 10.2 is the mean value for unit 2.

Suppose the system lambda value is known, then apply the equal lambda criteria (graphical solution) to the two units, corresponding to each specific value of lambda within the minimal and maximal of incremental cost, there is optimal output level for each unit. From the spreadsheet, we can see that the correlation coefficient between the optimal output levels is ; for the lambda value outside of the minimal or maximal incremental cost of any unit, the output level for that unit is at minimal or maximal, another unit bears the left demand, their correlation coefficient is also 1 (perfect correlation). According to the correlation defined in pearson, the whole correlation is 1 in the whole data set.

Suppose the system demand level is known, then apply the graphical method to the economic dispatch problem of the two units, and determine the corresponding lambda for the demand level. If value of lambda is within the minimal and maximal of incremental cost, the specific output level of each unit is determined, from the spreadsheet, we can see that the correlation between the optimal output levels is 1, if the lambda value outside of the minimal or maximal incremental cost of any unit, the output level for that unit is at minimal or maximal, another unit bears the left demand, their correlation coefficient is also 1 (perfect correlation). According to the correlation defined in pearson, the whole correlation is 1 in the whole data set.

The result is coincident with the fact that in the competitive market, the optimal bid for identical or very similar units are strongly correlated, which is an important aspect that should be considered carefully in making decision on optimal bidding strategy.