genform.m

以上所传的内容为电力系统比较全面的分析程序,才用的是matlab编写m的文件

% GENFORM: Help file on the formulation used in MINOPF and adopted
% as a generalized formulation for MATPOWER.
%
% *-*-*-*-*
% CONTENTS
% *-*-*-*-*
%
% 1: General OPF problem formulation
% 2: Input transformations for data & general linear restrictions
% 3: Piecewise linear convex cost formulation
% 4: Constant power factor constraints for curtailable/price sensitive loads
% 5: Additional structure information.
% 
% *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
% 1. GENERAL OPF PROBLEM FORMULATION
% *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
%
% The generalized formulation expects problems of the following type:
%
% min  sum(f_i(p_gi)) + c* [x; y; z]
%
% subject to
%
%  g(x) <=> 0     (nonlinear constraints: power flow balance and branch limits)
%  a <= x <= b    (box bound constraints on variables: voltage limits, 
%                  generation limits)
%  l <= A*[x;y;z] <= u  (general linear constraints)
%
% where x = [Theta; V; Pg; Qg ], with (Pg,Qg) being the ng active
% and ng reactive injections into the network and (V,Theta) being
% the voltage magnitudes and voltage phase angles at each of the
% nb buses in the network.  The objective is a generator-wise additive 
% function f(Pg) (perhaps f(Pg,Qg)). The nonlinear constraints g()
% are grouped as
%
% g(x) = [gp(x)  ;  % the nb bus real power mismatch constraints
%         gq(x)  ;  % the nb bus reactive power mismatch constraints
%         gsf(x) ;  % the nl upper branch MVA injection limits at the "from" end
%         gst(x) ;. % the nl upper branch MVA injection limits at the "to" end
%              ];
% The additional variables y and z can optionally augment the
% problem to allow casting more general constraints and is also
% the mechanism though which piecewise linear convex costs are
% modeled (more later on). The linear cost vector allows the addition
% of a general linear cost on the whole set [x;y;z] of variables.
% With this setup, mixed polynomial/piecewise linear costs can
% be dealt with, as well as curtailable or price-sensitive loads
% requiring a constant power factor.
%
%
% *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
% 2. INPUT DATA TRANSFORMATIONS AND GENERAL LINEAR RESTRICTIONS
% *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
%
% If the user wants to add general linear constraints of his or her own,
% it is necessary to understand the standard transformations performed
% on the input data (bus, gen, branch, area and gencost tables) before
% the problem is actually solved in order to know where the optimization
% variables end up in the x vector. All of these transformations are
% reversed after solving the problem so that output data is in the
% right place in the tables.
%
% The first step filters out inactive generators and branches;
% original tables are saved for data output.
%
%    comgen = find(gen(:,GEN_STATUS) > 0);        % Find online generators
%    onbranch  = find(branch(:,BR_STATUS) ~= 0);  % Find online branches
%    gen   = gen(comgen, :);
%    branch = branch(onbranch, :);
%
% The second step is a renumbering of the bus numbers in the bus table 
% so that the resulting table contains consecutively-numbered buses starting 
% from 1:
%
%    [i2e, bus, gen, branch, areas] = ext2int(bus, gen, branch, areas);
%
% where i2e is saved for inverse reordering at the end.  Finally,
% generators are further reordered by bus number:
%
%    ng = size(gen,1);                 % number of generators or injections
%    [tmp, igen] = sort(gen(:, GEN_BUS));
%    [tmp, inv_gen_ord] = sort(igen);  % save for inverse reordering at the end
%    gen  = gen(igen, :);
%    if ng == size(gencost,1)          % This is because gencost might have
%      gencost = gencost(igen, :);     % twice as many rows as gen if there
%    else                              % are reactive injection costs.
%      gencost = gencost( [igen; igen+ng], :);
%    end
%
% Having done this, the variables inside the x vector now have the same
% ordering as in the bus, gen tables:
%
% x = [  Theta ;    %  nb bus voltage angles
%          V   ;    %  nb bus voltage magnitudes
%          Pg  ;    %  ng active power injections (p.u.) (ascending bus order)
%          Qg ];    %  ng reactive power injections (p.u.)(ascending bus order)
%
% and the nonlinear constraints have the same order as in the bus, branch
% tables
%
% g = [  gp ;       % nb real power flow mismatches (p.u.)
%        gq;        % nb reactive power flow mismatches (p.u.)
%        gsf;       % nl "From" end apparent power injection limits (p.u.)
%        gst ];     % nl "To" end apparent power injection limits (p.u.)
%
% With this setup, box bounds on the variables are applied as follows:
% the reference angle is upper and lower bounded with the value that
% came for it in the original bus() table.  The V section of x is
% lower and upper-bounded with the corresponding values for VMIN
% and VMAX in the bus() table.  The Pg and Qg sections of x are
% lower and upper-bounded with the corresponding values for
% PMAX, PMIN, QMAX and QMIN in the gen() table.
% The nonlinear constraints are similarly setup so that gp and gq
% are equality constraints (zero RHS) and the limits for gsf, gst
% are taken from the RATE_A column in the branch() table.
%
% Example: in the standard solution to case9.m, the voltage angle
% for bus 7 lags the voltage angle in bus 2 by 6.09 degrees.
% We want to limit that lag to 5 degrees at the most.  A linear
% restriction of the form
%
%    Theta(2) - Theta(7) <=  5 degrees
%
% would do the trick.  We have nb = 9 buses, ng = 3 generators
% and nl = 9 branches.  Therefore the first 9 elements of
% x are bus voltage angles, elements 10:18 are bus voltage magnitudes,
% elements 19:21 are active injections corresponding to the generators
% in buses 1, 2 and 3 (in that order) and elements 22:24 are the
% corresponding reactive injections.  Note that in this case the
% generators in the original data already appear in ascending bus
% order, so no permutation with respect to the original data is
% necessary.  Going back to the angle restriction, we see that
% it can be cast as
%
%  [ 0 1 0 0 0 0 -1 0 0 zeros(1,nb+ng+ng) ] * x  <= 5 degrees
%
% We can set up the problem as follows:
%
%    A = sparse([1;1], [2;7], [1;-1], 1, 24);
%    l = 0;
%    u = 5 * pi/180;
%    mpopt = mpoption;
%    mpopt(11) = 520; % use fmincon with generalized formulation
%    opf('case9', A, l, u, mpopt)
%
% which indeed restricts the angular separation to 5 degrees.
% NOTE: in this example, the total number of variables is 24,
% but if there are any piece wise-linear cost functions, there
% may be additional "helper" variables used by the solver and
% in that case the number of columns in A may need to be
% larger.  Read the next section to understand how this is done.
% If all costs are polynomial, however, no extra variables
% are needed.
%
% *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
% 3. PIECEWISE LINEAR CONVEX COST FORMULATION
% *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
%
% The generalized formulation allows an easy way to model
% piecewise linear convex generation costs.  A typical cost vs. power curve
% looks like
%
%  cost axis
%      ^
% c2   |                           *
%      |                        *
%      |                     *
% c1   |                  *
%      |          *
% c0   |   *
%      |------------------------------->   Pg axis
%          x0            x1         x2
%         Pmin                     Pmax
%
% This nondifferentiable cost can be modeled using one helper cost
% variable for each such cost curve and additional restrictions
% on this variable and Pg, one restriction for each segment of the
% curve.  The restrictions build a convex "basin" and they are
% equivalent to saying that the cost variable must 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 generators' cost to the total cost is
% exactly y, and in the above case the two restrictions needed 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)
%
% (here, m1 and m2 are the slopes of the two segments)
% and of course, the box restrictions on Pg:  Pmin <= Pg <= Pmax .
% The additive part of the cost contributed by this generator is y.
%