genform.m

MATPOWER 一款基于MATLAB的电力系统潮流计算及优化的程序

%GENFORM: Help file describing the generalized OPF formulation used by
%         the fmincopf and MINOPF solvers.
%
%==========
% CONTENTS
%==========
%
%1. General OPF Problem Formulation
%2. General Linear Constraints
%3. Generalized Cost Function
%4. Piecewise Linear Convex Cost Formulation Using Constrained Cost Variables
%5. Generator P-Q Capability Curves
%6. Dispatchable Loads
%7. Problem Data Transformation
%8. Example of Additional Linear Constraint
%9. Miscellaneous
%
%
%====================================
% 1. GENERAL OPF PROBLEM FORMULATION
%====================================
%
%The problem is formulated in terms of 3 groups of optimization variables,
%labeled x, y and z. The vector x = [ Theta; V; Pg; Qg ] contains the OPF
%variables, consisting of the voltage angles Theta and magnitudes V at each of
%the nb buses, and real and reactive generator injections Pg and Qg for each of
%the ng generators. The y variables are the helper variables used by the
%constrained cost variable (CCV) formulation of the piecewise linear generator
%cost functions. Additional user defined variables are grouped in z.
%
%The optimization problem can be expressed as follows:
%
%   min  sum( f1i(Pgi) + f2i(Qgi) ) + sum(y) + 0.5 * w' * H * w + Cw' * w
%  x,y,z
%
%subject to
%  g(x) <=> 0          (nonlinear constraints: bus power balance equations
%                                              & branch flow limits)
%  l <= A*[x;y;z] <= u (general linear constraints)
%  xmin <= x <= xmax   (variable bounds: voltage limits, generation limits)
%
%The most significant additions to the traditional, simple OPF formulation
%appear in the generalized cost terms containing w and in the general linear
%constraints involving the matrix A, described in the next two sections.
%
%
%===============================
% 2. GENERAL LINEAR CONSTRAINTS
%===============================
%
%In addition to the standard non-linear equality constraints for nodal power
%balance and non-linear inequality constraints for line flow limits, this
%formulation includes a framework for additional linear constraints involving
%the full set of optimization variables.
%
%  l <= A*[x;y;z] <= u (general linear constraints)
%
%Some portions of these linear constraints are supplied directly by the user,
%while others are generated automatically based on the case data. Automatically
%generated portions include:
%
%* rows for constraints that define generator P-Q capability curves
%* rows for constant power factor constraints for dispatchable loads
%* rows and columns for the y variables from the CCV implementation of
%  piecewise linear generator costs and their associated constraints
%
%In addition to these automatically generated constraints, the user can provide
%a matrix Au and vectors lu and uu to define further linear constraints. These
%user supplied constraints could be used, for example, to restrict voltage
%angle differences between specific buses. The matrix Au must have at least nx
%columns where nx is the number of x variables. If Au has more than nx columns,
%a corresponding z optimization variable is created for each additional column.
%These z variables also enter into the generalized cost terms described below,
%so Au and N must have the same number of columns.
%
%  lu <= Au*[x;z] <= uu (user supplied linear constraints)
%
%*Change from MATPOWER 3.0*: The Au matrix supplied by the user no longer
%includes the (all zero) col-umns corresponding to the y variables for
%piecewise linear generator costs. This should simplify signifi-cantly the
%creation of the desired Au matrix.
%
%
%==============================
% 3. GENERALIZED COST FUNCTION
%==============================
%
%The cost function consists of 3 parts. The first two are the polynomial and
%piecewise linear costs, respectively, of generation. A polynomial or piecewise
%linear cost is specified for each generator's active output and, optionally,
%reactive output in the appropriate row(s) of the gencost matrix. Any piecewise
%linear costs are implemented using the CCV formulation described below which
%introduces corresponding helper y variables. The general formulation allows
%generator costs of mixed type (polynomial and piece-wise linear) in the same
%problem.
%
%The third part of the cost function provides a general framework for imposing
%additional costs on the optimization variables, enabling things such as using
%penalty functions as soft limits on voltages, additional costs on variables
%involved in constraints handled by Langrangian relaxation, etc. This general
%cost term is specified through a set of parameters  H, Cw,  N and fparm,
%described below. It consists of a general quadratic function of an  nw x 1
%vector w of transformed optimization variables.
%
%  1/2 * w' * H * w + Cw' * w
%
%H is the nw x nw symmetric, sparse matrix of quadratic coefficients and Cw
%is the nw x 1 vector of linear coefficients. The sparse N matrix is nw x nxz,
%where the number of columns must match that of any user supplied Au matrix.
%And fparm is nw x 4, where the 4 columns are labeled as
%
%  fparm = [ d rhat h m ].
%
%The vector w is created from the x and z optimization variables by first
%applying a general linear transformation
%
%  r = N * [x; z],
%
%followed by a scaled function with a shifted "dead zone", defined by the
%remaining elements of fparm. Each element of r is transformed into the
%corresponding element of w as follows:
%
%        /  mi * fi(ri - rhati + hi),  for ri - rhati < -hi
%  wi = <   0,                         for -hi <= ri - rhati <= hi
%        \  mi * fi(ri - rhati - hi),  for ri - rhati > hi
%
%where the function fi is a predetermined function selected by the index in di.
%The current implementation includes linear and quadratic options.
%
%           /  t,    for di = 1
%  fi(t) = <
%           \  t^2,  for di = 2
%
%See the User's Manual for an illustration of the linear case. 
%
%
%==============================================================================
% 4. PIECEWISE LINEAR CONVEX COST FORMULATION USING CONSTRAINED COST VARIABLES
%==============================================================================
%
%The OPF formulations in MATPOWER allow for the specification of convex
%piecewise linear cost functions for active or reactive generator output. An
%example of such a cost curve is shown below.
%
% cost axis
%     ^
%c2   |                           *
%     |                        *
%     |                     *
%c1   |                  *
%     |          *
%c0   |   *
%     |------------------------------->   Pg axis
%         x0            x1         x2
%        Pmin                     Pmax
%This non-differentiable cost is modeled using an extra helper cost variable
%for each such cost curve and additional constraints on this variable and Pg,
%one for each segment of the curve. The constraints build a convex "basin"
%equivalent to requiring the cost variable to lie in the epigraph of the cost
%curve. When the cost is minimized, the cost variable will be pushed against
%this basin. If the helper cost variable is y, then the contribution of the
%generator's cost to the total cost is exactly y. In the above case, the two
%additional required constraints are
%
%  1)  y >= m1*(Pg - x0) + c0   (y must lie above the first segment)
%  2)  y >= m2*(Pg - x1) + c1   (y must lie above the second segment)
%
%where  m1 and m2 are the slopes of the two segments. Also needed, of course,
%are the box restrictions on Pg:  Pmin