# Economic dispatch

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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, subject to transmission and operational constraints. The Economic Dispatch Problem is solved by specialized computer software which should satisfy 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, in order to satisfy the load at a minimum total cost, the set of generators with the lowest marginal costs must be used first, with the marginal cost of the final generator needed to meet load setting the system marginal cost. This is the cost of delivering one additional MWh of energy onto the system. 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.

## Basic mathematical formulation

The following is based on Kirschen (2010). 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:

$\min _{I_{k}}\;(-W)=\min _{I_{k}}\;\left\{\sum _{k=1}^{n}C_{k}(I_{k})\right\}$ Constraints imposed on the model 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:

$\sum _{k=1}^{n}I_{k}=L(I_{1},I_{2},\dots ,I_{n-1})$ 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 modeled as:

$F_{l}(I_{1},I_{2},\dots ,I_{n-1})\leq F_{l}^{max}\qquad l=1,\dots ,m$ 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:

${\mathcal {L}}=\sum _{k=1}^{n}C_{k}(I_{k})+\pi \left[L(I_{1},I_{2},\dots ,I_{n-1})-\sum _{k=1}^{n}I_{k}\right]+\sum _{l=1}^{m}\mu _{l}\left[F_{l}^{max}-F_{l}(I_{1},I_{2},\dots ,I_{n-1})\right]$ where π and μ are the Lagrangian multipliers of the constraints. The conditions for optimality are then:

${\partial {\mathcal {L}} \over \partial I_{k}}=0\qquad k=1,\dots ,n$ ${\partial {\mathcal {L}} \over \partial \pi }=0$ ${\partial {\mathcal {L}} \over \partial \mu _{l}}=0\qquad l=1,\dots ,m$ $\mu _{l}\cdot \left[F_{l}^{max}-F_{l}(I_{1},I_{2},\dots ,I_{n-1})\right]=0\quad \mu _{l}\geq 0\quad k=1,\dots ,n$ 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.

## Environmental dispatch

In environmental dispatch, additional considerations concerning reduction of pollution further complicate the power dispatch problem. The basic constraints of the economic dispatch problem remain in place but the model is optimized to minimize pollutant emission in addition to minimizing fuel costs and total power loss. Due to the added complexity, a number of algorithms have been employed to optimize this environmental/economic dispatch problem. Notably, a modified bees algorithm implementing chaotic modeling principles was successfully applied not only in silico, but also on a physical model system of generators.. Other methods used to address the economic emission dispatch problem include Particle Swarm Optimization (PSO)  and neural networks 

Another notable algorithm combination is used in a real-time emissions tool called Locational Emissions Estimation Methodology (LEEM) that links electric power consumption and the resulting pollutant emissions. The LEEM estimates changes in emissions associated with incremental changes in power demand derived from the locational marginal price (LMP) information from the independent system operators (ISOs) and emissions data from the US Environmental Protection Agency (EPA). LEEM was developed at Wayne State University as part of a project aimed at optimizing water transmission systems in Detroit, MI starting in 2010 and has since found a wider application as a load profile management tool that can help reduce generation costs and emissions.