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Economic dispatch is the short-term determination of the optimal output of a number of electricity generation facilities, to meet the system load, at the lowest possible cost, while serving power to the public in a robust and reliable manner. The Economic Dispatch Problem is solved by specialised computer software which should honour the operational and system constraints of the available resources and corresponding transmission capabilities. In the US Energy Policy Act of 2005 the term is defined as "the operation of generation facilities to produce energy at the lowest cost to reliably serve consumers, recognising any operational limits of generation and transmission facilities"
The main idea is that at minimum total cost, the marginal costs of all generators are the same. That is the cost of producing one additional MWh is the same for every generator. The historic methodology for economic dispatch was developed to manage fossil fuel burning power plants, relying on calculations involving the input/output characteristics of power stations. As new renewable generation has come online over the last few decades, new methods have had to be developed to deal with the fact that renewable generation does not have a fuel cost in the same sense as fossil generation, and that renewable generation is intermittent, and thus not as easily scheduled as fossil generation.
The economic dispatch problem can be thought of as maximising the economic welfare W of a power network whilst meeting system constraints. For a network with n buses (nodes), where Ik represents the net power injection at bus k, and Ck(Ik) is the cost function of producing power at bus k, the unconstrained problem is formulated as:
Constraints imposed on the optimisation are the need to maintain a power balance, and that the flow on any line must not exceed its capacity. For the power balance, the sum of the net injections at all buses must be equal to the power losses in the branches of the network:
The power losses L depend on the flows in the branches and thus on the net injections as shown in the above equation. However it cannot depend on the injections on all the buses as this would give an over-determined system. Thus one bus is chosen as the Slack bus and is omitted from the variables of the function L. The choice of Slack bus is entirely arbitrary, here bus n is chosen.
The second constraint involves capacity constraints on the flow on network lines. For a system with m lines this constraint is modelled as:
where Fl is the flow on branch l, and Flmax is the maximum value that this flow is allowed to take. Note that the net injection at the slack bus is not included in this equation for the same reasons as above.
These equations can now be combined to build the Lagrangian of the optimization problem:
where π and μ are the Lagrangian multipliers of the constraints. The conditions for optimality are then:
where the last condition is needed to handle the inequality constraint on line capacity.
Solving these equations is computationally difficult as they are nonlinear and implicitly involve the solution of the power flow equations. The analysis can be simplified using a linearised model called a DC power flow.
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