Topics: Unit commitment, Problems solving on the priority list, Dynamic programming method, optimal scheduling of hydrothermal system.
Before discussing unit commitment problem, let us have a brief introduction of Economic load dispatch problem.
Economic Load Dispatch: The figure below shows the configuration of N thermal generating units connected to a single bus bar serving a load of P-Load. P1, P2, P3 ......PN is power generation outputs. Letus consider the production cost of each unit is Fi i.e (F1, F2, F3.......FN).
The ELD problem is economically scheduling the operation of generating units to meet the load demand such that the total production cost (Ftotal=F1+F2+F3+.......FN) should be minimum.
Mathematically the problem can be represented as follows
---------------(1)
---------------(2)
---------------(3)
This is a constrained optimization problem that may be attacked formally using advanced calculus methods that involve the Lagrange function. In order to establish the necessary conditions for an extreme value of the objective function, add the constraint function to the objective function after the constraint function has been multiplied by an undetermined multiplier. This is known as the Lagrange function and is
------------------------------------(4)
--------------------------------------------(5)
That is, the necessary condition for the existence of a minimum cost operating condition for the thermal power system is that the incremental cost rates of all the units be equal to some undetermined value, l. Of course, to this necessary condition, we must add the constraint equation that the sum of the power outputs must be equal to the power demanded by the load. In addition, there are two inequalities that must be satisfied for each of the units. That is, the power output of each unit must be
greater than or equal to the minimum power permitted and must also be less than or equal to the maximum power permitted on that particular unit. These conditions and inequalities may be summarized as shown in the set of equations.
When we recognize the inequality constraints, then the necessary conditions may be expanded slightly as shown in the set of equations.
From this discussion, the economic load dispatch problem can be solved by equating the first differentiation of F and equating it to λ, the final objective is to find the power generation levels from the generating units that satisfy the given load demand and to verify whether the power outputs should be within the minimum and maximum power limits so that the total production cost can be minimized.
Economic Load Dispatch: The figure below shows the configuration of N thermal generating units connected to a single bus bar serving a load of P-Load. P1, P2, P3 ......PN is power generation outputs. Letus consider the production cost of each unit is Fi i.e (F1, F2, F3.......FN).
The ELD problem is economically scheduling the operation of generating units to meet the load demand such that the total production cost (Ftotal=F1+F2+F3+.......FN) should be minimum.
Mathematically the problem can be represented as follows
---------------(1)
---------------(2)
---------------(3)
This is a constrained optimization problem that may be attacked formally using advanced calculus methods that involve the Lagrange function. In order to establish the necessary conditions for an extreme value of the objective function, add the constraint function to the objective function after the constraint function has been multiplied by an undetermined multiplier. This is known as the Lagrange function and is
------------------------------------(4)
where L is a Lagrangian function and λ isLagrangianngian operator. The necessary conditions for an extreme value of the objective function result when we take the first derivative of the Lagrange function with respect to each of the independent variables and set the derivatives equal to 0.
--------------------------------------------(5)That is, the necessary condition for the existence of a minimum cost operating condition for the thermal power system is that the incremental cost rates of all the units be equal to some undetermined value, l. Of course, to this necessary condition, we must add the constraint equation that the sum of the power outputs must be equal to the power demanded by the load. In addition, there are two inequalities that must be satisfied for each of the units. That is, the power output of each unit must be
greater than or equal to the minimum power permitted and must also be less than or equal to the maximum power permitted on that particular unit. These conditions and inequalities may be summarized as shown in the set of equations.
When we recognize the inequality constraints, then the necessary conditions may be expanded slightly as shown in the set of equations.
From this discussion, the economic load dispatch problem can be solved by equating the first differentiation of F and equating it to λ, the final objective is to find the power generation levels from the generating units that satisfy the given load demand and to verify whether the power outputs should be within the minimum and maximum power limits so that the total production cost can be minimized.
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