Can anyone please explain what is meant by penalty factor in generator dispatching (power systems)? I know this is used when we define the Lagrangian equation for non linear variable costs, but don't really understand how it fits in.
A problem is that the term "penalty factor" is used in quit a few ways in the power generation and distribution industry. A number of these are relevant enough to Lagrangian analysis that one can't be sure that they are what you want.As Kortuk says , it would be a good idea to quote a specific example.
(0) This paper (abstract only here) -
A dual augmented Lagrangian approach for optimal power flow
seems to use the term exactly as you intended (I think :-) ).
- The authors present a method for solving the active and reactive optimal power-flow problem. The method is based on nonlinear programming techniques combining dual and penalty approaches. The classical Langrangian is defined for both equality and inequality constraints. The function is then augmented by penalty terms for all constraints and it is minimized by Newton's method. An intrinsic multiplier updating rule eliminates the necessity of increasing the penalty factors to very large values. The method unifies the treatment equality and inequality constraints. It also avoids the critical aspects present in the determination of the binding constraints set
What's not to like ? :-)
(Runs screaming from room).
(1) At the most general level the term is used to describe what are effectively 'fiddle factors' added in order to ensure that hard or soft constraints are met when analysis is carried out.
This paper uses the term very much in the context that you mention.
- βγ Penalty factor associated with emission constraint
- βλ Penalty factor associated with power balance
- βη Penalty factor associated with transmission constraint
Here the key point is that they have an iterative solution system which can become unstable during solution and they introduce factors to limit instabilities.
It sounds extremely like magic empirical hand waving. eg on page 80 they say:
- Selection of parameters: In the ALHN model, the parameters have to be predetermined including sigmoid function slope, updating step sizes for neurons and penalty factors for augmented Lagrange function. A proper parameter selection will guarantee rapid convergence to ALHN. The parameters of ALHN are selected via tuning. By experiments, the values of sigmoid function slope and penalty factors are fixed at 100 and 0.001, respectively. The values of the others will vary depending on the data of test systems.
(2) A slightly more orthodox use is to provide weighting factors for power stations when performing routing and optimum cost analyses for grids of energy sources. These seem to linearly represent energy losses due to transmission "costs" between nodes. In some instances they seem to use sparate parameters for reactive and resistive components.
This abstract for the paper Transmission loss penalty factors for area energy interchange
- The concept of transmission loss penalty factors for area energy interchange is introduced... The algorithm for penalty factors is based upon the Jacobian method ...electrical distances between energy sending and receiving points and the area inter-tie locations are considered in the calculation. ... an area penalty factor ... reflects only the portion of the internal area loss induced by the interchange with the area of interest.
This sounds relevant in the same context. For $.
This 5 page 1995 Chilean paper
Penalty Factor calculations for marginal pricing of transmission systems in a hydroelectrical system looks like it may be especially useful in this context. Not a searchable PDF unfortunately.
(3) Some papers use the term "penalty" factor asa means of providing feasibility weighting
eg see the formula at the bottom of the 3rd page here
Short-Term Hydrothermal Generation Scheduling Model Using a Genetic Algorithm where panalty is effectovely a linear cost factor.
Worth a look.
- This paper with abstract given at the end ("Economic disptach with emission ...") proposes an augmented Lagrange Hopfield network (ALHN) for solving economic dispatch (ED) problem with ramp rate, emission and transmission constraints. The proposed ALHN method is the continuous Hopfield neural network with its energy function based on augmented Lagrangian function. In ALHN, the energy function is augmented by Hopfield terms from Hopfield neural network and penalty factors from augmented Lagrangian function to damp out oscillation of the Hopfield network during its convergence.