Merit order

The following is based on Biggar and Hesamzadeh (2014)[8] and Kirschen (2010).[9] 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), suppose that S_k is the rate of generation, and D_k is the rate of consumption at bus k. Suppose, further, that C_k(S_k) is the cost function of producing power (i.e., the rate at which the generator incurs costs when producing at rate S_k), and V_k(D_k) is the rate at which the load receives value or benefits (expressed in currency units) when consuming at rate D_k. The total welfare is then

${\displaystyle W=\sum _{k=1}^{n}V_{k}(D_{k})-\sum _{k=1}^{n}C_{k}(S_{k})}$

The economic dispatch task is to find the combination of rates of production and consumption (S_k, D_k) which maximise this expression W subject to a number of constraints:

${\displaystyle \max _{S_{k},D_{k}}W}$

The first constraint, which is necessary to interpret the constraints that follow, is that the net injection at each bus is equal to the total production at that bus less the total consumption:

${\displaystyle \forall k,\;I_{k}=S_{k}-D_{k}}$

The power balance constraint requires that the sum of the net injections at all buses must be equal to the power losses in the branches of the network:

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

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:

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

where F_l is the flow on branch l, and F_l^max 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:

${\displaystyle {\mathcal {L}}=\sum _{k=1}^{n}C_{k}(I_{k})+\pi \left[L(I_{1},I_{2},\dots ,I_{n-1})-\sum _{k=1}^{n}I_{k}\right]+\sum _{l=1}^{m}\mu _{l}\left[F_{l}^{max}-F_{l}(I_{1},I_{2},\dots ,I_{n-1})\right]}$

where π and μ are the Lagrangian multipliers of the constraints. The conditions for optimality are then:

${\displaystyle {\partial {\mathcal {L}} \over \partial I_{k}}=0\qquad k=1,\dots ,n}$

${\displaystyle {\partial {\mathcal {L}} \over \partial \pi }=0}$

${\displaystyle {\partial {\mathcal {L}} \over \partial \mu _{l}}=0\qquad l=1,\dots ,m}$

${\displaystyle \mu _{l}\cdot \left[F_{l}^{max}-F_{l}(I_{1},I_{2},\dots ,I_{n-1})\right]=0\quad \mu _{l}\geq 0\quad k=1,\dots ,n}$

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.

There is a special case which is found in much of the literature. This is the case in which demand is assumed to be perfectly inelastic (i.e., unresponsive to price). This is equivalent to assuming that ${\displaystyle V_{k}(D_{k})=M\min(D_{k},{\bar {D}}_{k})}$ for some very large value of ${\displaystyle M}$ and inelastic demand ${\displaystyle {\bar {D}}_{k}}$. Under this assumption, the total economic welfare is maximised by choosing ${\displaystyle D_{k}={\bar {D}}_{k}}$. The economic dispatch task reduces to:

${\displaystyle \min _{S_{k}}\sum _{k=1}^{n}C_{k}(S_{k})}$

Subject to the constraint that ${\displaystyle \forall k,\;I_{k}=S_{k}-{\bar {D}}_{k}}$ and the other constraints set out above.

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.[1] 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.[1] Other methods used to address the economic emission dispatch problem include Particle Swarm Optimization (PSO)[10] and neural networks[11]

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.[12] 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).[12] 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.[13]