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caseDistributionSystemMeasure.m
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classdef caseDistributionSystemMeasure < caseDistributionSystem
% This is the class of distribution system. We assume all the
% evaluations are conducted under practical measurements
properties
dataE % the estimated data
dataO % the process data in the optimization iterations
boundA % the approximated bound
sigmaReal % the deviation of the real state variables
prior % the prior assumptions of the G and B matrix
A_FIM % the approximated fisher information matrix
A_FIMP % the (sparse) FIM of active power injection
A_FIMQ % the (sparse) FIM of reactive power injection
initPar % the initial estimation of parameters and state variables
truePar % the ground truth of the parameters
grad % the gradient vector
gradChain % the chain of the gradients
gradP % the gradient vector from the measurement of P
gradQ % the gradient vector from the measurement of Q
gradVm % the gradient vector from the measurement of Vm
gradVa % the gradient vector from the measurement of Va
numGrad % the number of the gradient elements
loss % the sum-of-squares loss function
lossChain % the chain of the loss functions
parChain % the chain of the parameters
kZero % the ratio that we set nondiagonal elements to zero
maxIter % the maximum iteration in the gradient-based methods
step % the step length of the iterations
stepInit % the initial step
stepMin % the minimum step length
stepMax % the maximum step length
stepChain % the chain of the step length
iter % the current iteration step
updateStepFreq % the frequency to update the step length
momentRatio % the part we maintain from the past gradient
momentRatioMax % the maximum momentRatio
momentRatioMin % the minimum momentRatio
vmvaWeight % the additional weight to the vm and va
isConverge % if the iteration concerges
isGB % whether to iterate the GB part
H % the Hessian matrix
HP % the P part
HQ % the Q part
HVm % the Vm part
HVa % the Va part
J % the Jacobian matrix
Topo % the topology matrix
Tvec % the topology vector
thsTopo % the threshold of Topology
lambda % the damping ratio of LM algorithm
lambdaMax % the maximum value
lambdaMin % the minimum value
lambdaChain % the chain of lambda ratio, the first order ratio
gradOrigin % the original gradient in LM algorithm
gradPast % the gradient of the last step
lossMin % the theoretical minimum loss
momentLoss % the moment of loss
ratioMax % the ratio of second order / first order (final value)
ratioMaxMax % the maximum of ratioMax
ratioMaxMin % the minimum of ratioMin
ratioMaxChain % the chain of ratioMax
ratioChain % the chain of ratio
lambdaCompen % the additional compensate when second order is too large
% idGB % the address of G and B matrix
% idVmVa % the id of Vm and Va
isBoundChain % if attain the bound that Hessian is too large
deRatio % ratio of step and lambda when loss decreases
inRatio % ratio of step and lambda when loss increases
isSecond % the mode of second order
isFirst % whether we force the mode to be the first order
lastState % we temporally save the last state for the second order mode
regretRatio % the regret ratio in the second mode
second % the absolute proportion of second order
secondChain % the chain of second
secondMax % the maximum value of second
secondMin % the minimum value of second
tuneGrad % whether tune grad
boundTuned % the tuned bound
ratioMaxConst % the constant of ratio max
err % the evaluation errors
vaPseudoWeight % the enlarge sigma of the va pseudo measurement
vaPseudoWeightInit % the initial value of vaPseudoWeight
vaPseudoMax % the maximum value of vaPseudoWeight
vaPseudoMin % the minimum value of vaPseudoWeight
startPF % the loss value to start PF calculation log10(loss/lossMin)
maxD2Chain % the chain of maxD2
maxD2Upper % the upper bound of maxD2
maxD2Lower % the lower bound of maxD2
D2D1Chain % the chain of D2/D1
updateStart % the start iteration number to update the topology
updateStep % the number of steps we calculate the judge whether stop iteration
updateRatio % the long term ratio and the short term ratio to stop iteration
updateLast % the last update step
updateLastLoss % the last loss function
updateRatioLast % the last topology update ratio
isLBFGS % whether we use the LBFGS method or the newton method
numStore % the number of parameters we want to store in our memory
numEstH % the number of history we use to estimate the H
sChain % the Chain of s_k = x_k+1 - x_k
yChain % the Chain of y_k = g_k+1 - g_k
rhoChain % the Chain of rho_k = 1/(y_k^T * s_k)
isLHinv % whether we use the low memory version to get the inverse
isPHinv % whether we use the pseudo inverse
isIll % whether it is ill-conditioned
ls_c % the c value of line search c*alpha*g'*d
ls_alpha % the alpha ratio of line search c*alpha*g'*d
ls_maxTry % the maximum try numbers of the line search
end
methods
function obj = caseDistributionSystemMeasure(caseName, numSnap, range)
% the construction function
obj = obj@caseDistributionSystem(caseName, numSnap, range);
end
function obj = preEvaluation(obj, varargin)
% This method evaluate the parameters before approximating the
% FIM. The evaluated value has low accuracy. We only use one
% snapshot for the Vm and Va.
if nargin == 2
obj.prior = varargin{1};
elseif nargin == 1
obj.prior.Gmin = 0.1;
obj.prior.Bmin = 0.1;
obj.prior.ratio = 0.05;
obj.prior.Gmax = 1000;
obj.prior.Bmax = 1000;
end
% we first evaluate the vm and the va
% obj.dataE.Vm = obj.data.Vm;%_noised;
% obj.dataE.Va = obj.data.Va;%_noised;
obj.sigmaReal.Vm = cov(obj.data.Vm');
mu = mean(obj.data.Vm, 2);
rng(5);
obj.dataE.Vm = mvnrnd(mu, obj.sigmaReal.Vm, obj.numSnap)';
obj.sigmaReal.Va = cov(obj.data.Va');
mu = mean(obj.data.Va, 2);
rng(6);
obj.dataE.Va = mvnrnd(mu, obj.sigmaReal.Va, obj.numSnap)';
% obj.dataE.Vm = obj.data.Vm;%_noised;
% We then evaluate the G and B.
% obj.dataE.G = obj.data.G;
% obj.dataE.B = obj.data.B;
obj = approximateY(obj);
obj.dataE.Va = zeros(obj.numBus, obj.numSnap);
obj.dataE.Va(2:end, :) = - pinv(obj.dataE.G(2:end, 2:end)) * obj.data.P_noised(2:end, :);
if any(any(isnan(obj.dataE.Va)))
obj.dataE.Va = zeros(obj.numBus, obj.numSnap);
end
% mu = mean(obj.data.Va, 2);
% obj.sigmaReal.P = cov(obj.data.P');
% obj.sigmaReal.Va = zeros(obj.numBus, obj.numBus);
% obj.sigmaReal.Va(2:end, 2:end) = ...
% ((1.5*obj.dataE.G(2:end, 2:end)) \ obj.sigmaReal.P(2:end, 2:end)) / (1.5*obj.dataE.G(2:end, 2:end));
% rng(7);
% obj.dataE.Va = mvnrnd(mu, obj.sigmaReal.Va, obj.numSnap)';
% obj.dataE.Va(2:end, :) = -obj.dataE.G(2:end, 2:end) \ obj.data.P_noised(2:end, :);
end
function obj = approximateFIM(obj, varargin)
% This method approximate the fisher information matrix based
% on the pre-evaluation results of the parameters.
if nargin == 2
obj.k = varargin{1};
elseif nargin == 1
obj.k.G = 5;
obj.k.B = 10;
obj.k.vm = 10;
obj.k.va = 1000;
end
% initialize the A_FIM matrix
obj.numFIM.G = (obj.numBus - 1) * obj.numBus / 2;
obj.numFIM.B = (obj.numBus - 1) * obj.numBus / 2;
obj.numFIM.Vm = obj.numSnap * (obj.numBus - 1); % exclude the source bus
obj.numFIM.Va = obj.numSnap * (obj.numBus - 1);
obj.numFIM.Sum = obj.numFIM.G + obj.numFIM.B + obj.numFIM.Vm + obj.numFIM.Va;
%initialize the sparsify measurement matrix
numVector = obj.numSnap * obj.numBus * ((obj.numBus-1)*4*2 + 2);
obj.mRow = zeros(1, numVector);
obj.mCol = zeros(1, numVector);
obj.mVal = zeros(1, numVector);
obj.spt = 1;
% Initialize the idGB
obj.idGB = zeros(obj.numBus, obj.numBus);
id = 1;
for i = 1:obj.numBus
obj.idGB(i, i+1:end) = id:id+obj.numBus-i-1;
obj.idGB(i+1:end, i) = id:id+obj.numBus-i-1;
id = id+obj.numBus-i;
end
obj.numMeasure = obj.numSnap *...
sum([obj.isMeasure.P;obj.isMeasure.Q;obj.isMeasure.Vm;obj.isMeasure.Va]);
% obj.M = zeros(obj.numFIM.Sum, obj.numMeasure);
% obj.A_FIM = zeros(obj.numFIM.Sum, obj.numFIM.Sum);
% obj.A_FIMP = sparse(obj.numFIM.Sum, obj.numFIM.Sum);
% obj.A_FIMQ = sparse(obj.numFIM.Sum, obj.numFIM.Sum);
% obj.FIMVm = sparse(obj.numFIM.Sum, obj.numFIM.Sum);
% obj.FIMVa = sparse(obj.numFIM.Sum, obj.numFIM.Sum);
pt = 1;
% calculate the sub-matrix of P of all snapshots and all buses
for j = 1:obj.numSnap
% the id of Vm and Va
obj.idVm = 2*(obj.numBus-1)*(j-1)+1 : 2*(obj.numBus-1)*(j-1)+obj.numBus-1;
obj.idVa = 2*(obj.numBus-1)*(j-1)+obj.numBus : 2*(obj.numBus-1)*(j-1)+2*obj.numBus-2;
for i = 1:obj.numBus
if obj.isMeasure.P(i)
% profile on
obj = approximateFIMP(obj, i, j, pt);
pt = pt + 1;
% profile off
% profile viewer
end
end
end
% obj.A_FIM = obj.A_FIM + full(obj.A_FIMP);
% calculate the sub-matrix of Q of all snapshots and all buses
for j = 1:obj.numSnap
% the id of Vm and Va
obj.idVm = 2*(obj.numBus-1)*(j-1)+1 : 2*(obj.numBus-1)*(j-1)+obj.numBus-1;
obj.idVa = 2*(obj.numBus-1)*(j-1)+obj.numBus : 2*(obj.numBus-1)*(j-1)+2*obj.numBus-2;
for i = 1:obj.numBus
if obj.isMeasure.Q(i)
obj = approximateFIMQ(obj, i, j, pt);
pt = pt + 1;
end
end
end
% obj.A_FIM = obj.A_FIM + full(obj.A_FIMQ);
% calculate the sub-matrix of Vm of all snapshots and all buses
for j = 1:obj.numSnap
% the id of Vm and Va
obj.idVm = 2*(obj.numBus-1)*(j-1)+1 : 2*(obj.numBus-1)*(j-1)+obj.numBus-1;
for i = 1:obj.numBus
if obj.isMeasure.Vm(i)
obj = buildFIMVm(obj, i, j, pt);
pt = pt + 1;
end
end
end
% obj.A_FIM = obj.A_FIM + full(obj.FIMVm);
% calculate the sub-matrix of Va of all snapshots and all buses
for j = 1:obj.numSnap
% the id of Vm and Va
obj.idVa = 2*(obj.numBus-1)*(j-1)+obj.numBus : 2*(obj.numBus-1)*(j-1)+2*obj.numBus-2;
for i = 1:obj.numBus
if obj.isMeasure.Va(i)
obj = buildFIMVa(obj, i, j, pt);
pt = pt + 1;
end
end
end
% obj.A_FIM = obj.A_FIM + full(obj.FIMVa);
obj.mRow(obj.spt:end) = [];
obj.mCol(obj.spt:end) = [];
obj.mVal(obj.spt:end) = [];
obj.mVal(isnan(obj.mVal)) = 0;
Ms = sparse(obj.mRow, obj.mCol, obj.mVal, obj.numFIM.Sum, obj.numMeasure);
% Ms = sparse(obj.M);
obj.A_FIM = Ms * Ms';
end
function obj = approximateFIMP(obj, bus, snap, pt)
% This method approximate the P part of FIM. We ignore the sin
% part of the power flow equations.
h = zeros(obj.numFIM.Sum, 1);
theta_ij = obj.dataE.Va(bus, snap) - obj.dataE.Va(:, snap);
% Theta_ij = repmat(obj.dataE.Va(:, snap), 1, obj.numBus) - repmat(obj.dataE.Va(:, snap)', obj.numBus, 1);
% % G_ij\cos(\Theta_ij)+B_ij\sin(\Theta_ij)
% GBThetaP = obj.dataE.G .* cos(Theta_ij) + obj.dataE.B .* sin(Theta_ij);
% % G_ij\sin(\Theta_ij)-B_ij\cos(\Theta_ij)
% GBThetaQ = obj.dataE.G .* sin(Theta_ij) - obj.dataE.B .* cos(Theta_ij);
% G matrix
% H_G = zeros(obj.numBus, obj.numBus);
h_GG = obj.dataE.Vm(bus, snap) * obj.dataE.Vm(:, snap)' / obj.k.G;
h_GG = h_GG - obj.dataE.Vm(bus, snap)^2 / obj.k.G;
h(obj.idGB(bus, [1:bus-1 bus+1:end])) = h_GG([1:bus-1 bus+1:end]);
% H_G(bus, :) = h_GG; % .* cos(theta_ij')
% h_G = obj.matToColDE(H_G);
% h(1:obj.numFIM.G) = h_G;
% B matrix
% H_B = zeros(obj.numBus, obj.numBus);
h_BB = obj.dataE.Vm(bus, snap) * obj.dataE.Vm(:, snap)' .* sin(theta_ij') / obj.k.B;
h(obj.numFIM.G+obj.idGB(bus, [1:bus-1 bus+1:end])) = h_BB([1:bus-1 bus+1:end]);
% H_B(bus, :) = h_BB;
% h_B = obj.matToColDE(H_B);
% h(obj.numFIM.G+1:obj.numFIM.G+obj.numFIM.B) = h_B;
% Vm
% the first order term of other Vm
% H_Vm = zeros(obj.numBus, obj.numSnap);
h_Vm = obj.dataE.Vm(bus, snap) * obj.dataE.G(:, bus) / obj.k.vm; % obj.dataE.G(:, bus)
% the second order term of Vm(bus)
h_Vm(bus) = 2*obj.dataE.Vm(bus, snap) * obj.dataE.G(bus, bus) / obj.k.vm; % obj.dataE.G(bus, bus)
% the first order term of Vm(bus)
fOrderVm = obj.dataE.Vm(:, snap) .* obj.dataE.G(:, bus) / obj.k.vm; % obj.dataE.G(:, bus)
fOrderVm(bus) = 0;
h_Vm(bus) = h_Vm(bus) + sum(fOrderVm);
h(obj.numFIM.G+obj.numFIM.B+obj.idVm) = h_Vm(2:end);
% H_Vm(:, snap) = h_Vm;
% remove the source bus whose magnitude is not the state variable
% H_Vm(1, :) = [];
% h_VmLarge = reshape(H_Vm, [], 1);
% h(obj.numFIM.G+obj.numFIM.B+1:obj.numFIM.G+obj.numFIM.B+obj.numFIM.Vm) = h_VmLarge;
% Va
% H_Va = zeros(obj.numBus, obj.numSnap);
h_Va = obj.dataE.Vm(bus, snap) * obj.dataE.Vm(:, snap) .* (- obj.dataE.B(:, bus)) / obj.k.va; % (- obj.dataE.B(:, bus))
h_Va(bus) = ( - obj.dataE.Vm(bus, snap)^2 * obj.dataE.B(bus, bus)...
- obj.data.Q_noised(bus, snap)) / obj.k.va;
h(obj.numFIM.G+obj.numFIM.B+obj.idVa) = h_Va(2:end);
% H_Va(:, snap) = h_Va;
% remove the source bus whose magnitude is not the state variable
% H_Va(1, :) = [];
% h_VaLarge = reshape(H_Va, [], 1);
% h(obj.numFIM.G+obj.numFIM.B+obj.numFIM.Vm+1:end) = h_VaLarge;
% build FIMP
h = h / obj.sigma.P(bus);
[row,col,val] = find(h);
l = length(val);
obj.mRow(obj.spt:obj.spt+l-1) = row;
obj.mCol(obj.spt:obj.spt+l-1) = col*pt;
obj.mVal(obj.spt:obj.spt+l-1) = val;
obj.spt = obj.spt + l;
% obj.M(:, pt) = h;
% FIMPThis = h * h';
% obj.A_FIMP = obj.A_FIMP + FIMPThis;
end
function obj = approximateFIMQ(obj, bus, snap, pt)
% This method approximate the Q part of FIM. We ignore the sin
% part of the power flow equations.
h = zeros(obj.numFIM.Sum, 1);
theta_ij = obj.dataE.Va(bus, snap) - obj.dataE.Va(:, snap);
% Theta_ij = repmat(obj.dataE.Va(:, snap), 1, obj.numBus) - repmat(obj.dataE.Va(:, snap)', obj.numBus, 1);
% % G_ij\cos(\Theta_ij)+B_ij\sin(\Theta_ij)
% GBThetaP = obj.dataE.G .* cos(Theta_ij) + obj.dataE.B .* sin(Theta_ij);
% % G_ij\sin(\Theta_ij)-B_ij\cos(\Theta_ij)
% GBThetaQ = obj.dataE.G .* sin(Theta_ij) - obj.dataE.B .* cos(Theta_ij);
% G matrix
% H_G = zeros(obj.numBus, obj.numBus);
h_GG = obj.dataE.Vm(bus, snap) * obj.dataE.Vm(:, snap)' .* sin(theta_ij') / obj.k.G;
h(obj.idGB(bus, [1:bus-1 bus+1:end])) = h_GG([1:bus-1 bus+1:end]);
% H_G(bus, :) = h_GG;
% h_G = obj.matToColDE(H_G);
% h(1:obj.numFIM.G) = h_G;
% B matrix
% H_B = zeros(obj.numBus, obj.numBus);
h_BB = - obj.dataE.Vm(bus, snap) * obj.dataE.Vm(:, snap)' / obj.k.B;
h_BB = h_BB + obj.data.Vm(bus, snap)^2 / obj.k.B;
h(obj.numFIM.G+obj.idGB(bus, [1:bus-1 bus+1:end])) = h_BB([1:bus-1 bus+1:end]);
% H_B(bus, :) = h_BB;
% h_B = obj.matToColDE(H_B);
% h(obj.numFIM.G+1:obj.numFIM.G+obj.numFIM.B) = h_B;
% Vm
% the first order term of other Vm
% H_Vm = zeros(obj.numBus, obj.numSnap);
h_Vm = obj.dataE.Vm(bus, snap) * (-obj.dataE.B(:, bus)) / obj.k.vm; % (-obj.dataE.B(:, bus))
% the second order term of Vm(bus)
h_Vm(bus) = 2*obj.dataE.Vm(bus, snap) * (-obj.dataE.B(bus, bus)) / obj.k.vm; % (-obj.dataE.B(bus, bus))
% the first order term of Vm(bus)
fOrderVm = obj.dataE.Vm(:, snap) .* (-obj.dataE.B(:, bus)) / obj.k.vm; % (-obj.dataE.B(:, bus))
fOrderVm(bus) = 0;
h_Vm(bus) = h_Vm(bus) + sum(fOrderVm);
h(obj.numFIM.G+obj.numFIM.B+obj.idVm) = h_Vm(2:end);
% H_Vm(:, snap) = h_Vm;
% % remove the source bus whose magnitude is not the state variable
% H_Vm(1, :) = [];
% h_VmLarge = reshape(H_Vm, [], 1);
% h(obj.numFIM.G+obj.numFIM.B+1:obj.numFIM.G+obj.numFIM.B+obj.numFIM.Vm) = h_VmLarge;
% Va
% H_Va = zeros(obj.numBus, obj.numSnap);
h_Va = - obj.dataE.Vm(bus, snap) * obj.dataE.Vm(:, snap) .* obj.dataE.G(:, bus) / obj.k.va; % obj.dataE.G(:, bus)
h_Va(bus) = (- obj.dataE.Vm(bus, snap)^2 * obj.dataE.G(bus, bus) ...
+ obj.data.P_noised(bus, snap)) / obj.k.va;
h(obj.numFIM.G+obj.numFIM.B+obj.idVa) = h_Va(2:end);
% H_Va(:, snap) = h_Va;
% % remove the source bus whose magnitude is not the state variable
% H_Va(1, :) = [];
% h_VaLarge = reshape(H_Va, [], 1);
% h(obj.numFIM.G+obj.numFIM.B+obj.numFIM.Vm+1:end) = h_VaLarge;
% build FIMQ
h = h / obj.sigma.Q(bus);
% obj.M(:, pt) = h;
[row,col,val] = find(h);
l = length(val);
obj.mRow(obj.spt:obj.spt+l-1) = row;
obj.mCol(obj.spt:obj.spt+l-1) = col*pt;
obj.mVal(obj.spt:obj.spt+l-1) = val;
obj.spt = obj.spt + l;
% FIMQThis = h * h';
% obj.A_FIMQ = obj.A_FIMQ + FIMQThis;
end
function obj = calABound(obj, varargin)
% this method calculate the bound from the A_FIM matrix;
if nargin == 3
obj.sparseOption = varargin{1};
obj.topoPrior = varargin{2};
elseif nargin == 2
obj.sparseOption = varargin{1};
obj.topoPrior = false(obj.numBus, obj.numBus);
elseif nargin == 1
obj.sparseOption = true;
obj.topoPrior = false(obj.numBus, obj.numBus);
end
% build the indexes we really care about
delCols = [obj.matToColDE(obj.topoPrior)>1e-4;obj.matToColDE(obj.topoPrior)>1e-4];
obj.numFIM.index = true(obj.numFIM.Sum, 1);
obj.numFIM.index(delCols) = false;
obj.numFIM.del = sum(delCols)/2;
% for [A B; B' C], we calculate A-B/C*B'
if obj.sparseOption
idCell = 2*(obj.numBus-1) * ones(1, obj.numSnap);
Cell = mat2cell(obj.A_FIM(obj.numFIM.index, obj.numFIM.index), ...
[obj.numFIM.G+obj.numFIM.B-2*obj.numFIM.del obj.numFIM.Vm+obj.numFIM.Va], ...
[obj.numFIM.G+obj.numFIM.B-2*obj.numFIM.del obj.numFIM.Vm+obj.numFIM.Va]);
Cell{1,2} = mat2cell(Cell{1,2},...
obj.numFIM.G+obj.numFIM.B-2*obj.numFIM.del, idCell);
Cell{2,2} = mat2cell(Cell{2,2}, ...
idCell, idCell);
% get the inversion of Cell{2,2}, we separate it into a
% single function
Cell{2,2} = obj.cell2diag(Cell{2,2});
% [D E; E' F]
disp('calculating invC22');
invC22 = cellfun(@inv, Cell{2,2},'UniformOutput',false);
% calculate the inv(A-B/CB')
disp('calculating (A-B/CB)^-1');
BCB = obj.cellMulSum(Cell{1,2}, invC22, Cell{1,2});
ABC = pinv(Cell{1,1} - BCB); % inv(A-B/CB')
diagABC = diag(ABC);
% Calculate the diag of C
diagC = obj.cellGetDiag(invC22);
% Calculate the var
var = [diagABC; diagC];
else
cov = full(obj.A_FIM(obj.numFIM.index, obj.numFIM.index))\eye(sum(obj.numFIM.index));
var = diag(cov);
% % we construct a Hermitian matrix H and use Cholesky
% % decomposition to compute the inverse matrix
% FIM = obj.A_FIM(obj.numFIM.index, obj.numFIM.index);
% H = FIM * FIM';
% U = chol(H);
% Uinv = U \ eye(size(U));
% Cov = H' * (Uinv * Uinv');
end
if min(var) < 0
var = abs(var);
% cov = cov - diag(diag(cov)) + diag(var);
fprintf('We use the absolute value of the variance.\n');
end
obj.boundA.total = sqrt(var);
obj.boundA.total(obj.boundA.total>obj.prior.Gmax) = obj.prior.Gmax;
% obj.boundA.cov = cov;
boundG = zeros(obj.numFIM.G, 1);
boundG(obj.numFIM.index(1:obj.numFIM.G)) = obj.boundA.total(1:obj.numFIM.G-obj.numFIM.del) / obj.k.G;
obj.boundA.total(1:obj.numFIM.G-obj.numFIM.del) = obj.boundA.total(1:obj.numFIM.G-obj.numFIM.del) / obj.k.G;
obj.boundA.G = obj.colToMatDE(boundG, obj.numBus);
boundB = zeros(obj.numFIM.B, 1);
boundB(obj.numFIM.index(1:obj.numFIM.G)) = ...
obj.boundA.total(obj.numFIM.G+1-obj.numFIM.del:obj.numFIM.G+obj.numFIM.B-2*obj.numFIM.del) / obj.k.B;
obj.boundA.total(obj.numFIM.G+1-obj.numFIM.del:obj.numFIM.G+obj.numFIM.B-2*obj.numFIM.del) = ...
obj.boundA.total(obj.numFIM.G+1-obj.numFIM.del:obj.numFIM.G+obj.numFIM.B-2*obj.numFIM.del) / obj.k.B;
obj.boundA.B = obj.colToMatDE(boundB, obj.numBus);
obj.boundA.G_relative = abs(obj.boundA.G ./ repmat(diag(obj.data.G), 1, obj.numBus));
obj.boundA.B_relative = abs(obj.boundA.B ./ repmat(diag(obj.data.B), 1, obj.numBus));
obj.boundA.G_relative_col = reshape(obj.boundA.G_relative, [], 1);
obj.boundA.B_relative_col = reshape(obj.boundA.B_relative, [], 1);
obj.boundA.VmVa = reshape(obj.boundA.total(obj.numFIM.G+obj.numFIM.B+1-2*obj.numFIM.del:end), 2*(obj.numBus-1), obj.numSnap);
obj.boundA.Vm = reshape(obj.boundA.VmVa(1:obj.numBus-1, :), [], 1) / obj.k.vm;
obj.boundA.VmBus = mean(obj.boundA.VmVa(1:obj.numBus-1, :), 2);
obj.boundA.Va = reshape(obj.boundA.VmVa(obj.numBus:end, :), [], 1) / obj.k.vm;
obj.boundA.VaBus = mean(obj.boundA.VmVa(obj.numBus:end, :), 2);
% obj.boundA.Vm = ...
% obj.boundA.total(obj.numFIM.G+obj.numFIM.B+1-2*obj.numFIM.del...
% :obj.numFIM.G+obj.numFIM.B+obj.numFIM.Vm-2*obj.numFIM.del) / obj.k.vm;
% obj.boundA.VmBus = mean(reshape(obj.boundA.Vm, obj.numBus-1, obj.numSnap), 2);
% obj.boundA.total(obj.numFIM.G+obj.numFIM.B+1-2*obj.numFIM.del...
% :obj.numFIM.G+obj.numFIM.B+obj.numFIM.Vm-2*obj.numFIM.del)...
% = obj.boundA.Vm;
% obj.boundA.Va = ...
% obj.boundA.total(obj.numFIM.G+obj.numFIM.B+obj.numFIM.Vm+1-2*obj.numFIM.del...
% :obj.numFIM.Sum-2*obj.numFIM.del) / obj.k.va;
% obj.boundA.VaBus = mean(reshape(obj.boundA.Va, obj.numBus-1, obj.numSnap), 2);
% obj.boundA.total(obj.numFIM.G+obj.numFIM.B+obj.numFIM.Vm+1-2*obj.numFIM.del...
% :obj.numFIM.Sum-2*obj.numFIM.del)...
% = obj.boundA.Va;
end
function obj = identifyTopo(obj)
% This method identifies the topology by the voltage magnitudes
% measurements. The initial topology may not be accurate.
T = obj.data.G~=0;
Vm = movmean(obj.data.Vm_noised, floor(obj.numSnap/20)+1, 2);
C = corrcoef(Vm');
C(isnan(C)) = 0;
C(1,1) = 1;
CT = corrcoef(obj.data.Vm');
CT(isnan(CT)) = 0;
CT(1,1) = 1;
end
function obj = approximateY(obj)
% This method approximates the Y matrix by the measurements. We
% use the simple Ohom's law to provide an initial value of Y.
% We also assume the G/B ratio is a constant.
% rng(103);
% randG = 0.75 + 0.5 * randn(size(obj.data.G));
% rng(104);
% randB = 0.75 + 0.5 * randn(size(obj.data.B));
% obj.dataE.G = obj.data.G .* randG;
% obj.dataE.B = obj.data.B .* randB;
% % The approximation of the diagonal elements
% diagG = diag(obj.dataE.G);
% diagB = diag(obj.dataE.B);
% approximate the topology using Vm data only
% ranking the Vm
% Vm = obj.data.Vm_noised;
if 1>2
IP = obj.data.IP_noised;
IQ = obj.data.IQ_noised;
G = IP * obj.data.Vm_noised' / (obj.data.Vm_noised * obj.data.Vm_noised');
B = - IQ * obj.data.Vm_noised' / (obj.data.Vm_noised * obj.data.Vm_noised');
else
IP = obj.data.IP_noised;
IQ = obj.data.IQ_noised;
Vm = obj.data.Vm;
Topo = logical(eye(obj.numBus));
VmMean = mean(Vm, 2);
[~, VmOrder] = sort(VmMean,'descend');
assert (VmOrder(1) == 1); % the first bus is the source bus
Vm = movmean(Vm, floor(obj.numSnap/20)+1, 2);
corr = corrcoef(Vm');
corr(isnan(corr)) = 0; % one can also simulate some disturbance in the source bus voltage
corr(obj.topoPrior) = -2;
for i = 2:obj.numBus
% iterate each bus
[~, loc] = max(corr(VmOrder(i), VmOrder(1:i-1))); % the location of the connected bus
Topo(VmOrder(i), VmOrder(loc)) = true;
Topo(VmOrder(loc), VmOrder(i)) = true;
end
% if obj.caseName == 'case123_R'
% Topo = ~obj.topoPrior;
% end
T = obj.data.G~=0;
% approximate the parameter
G_ols = zeros(obj.numBus, obj.numBus);
B_ols = zeros(obj.numBus, obj.numBus);
for i = 1:obj.numBus
j = VmOrder(i);
filter = Topo(:, j);
filter(j) = false;
previous = VmOrder(1:i);
previous = intersect(previous, find(filter));
VmDelta = Vm(filter, :) - repmat(Vm(j, :), sum(filter), 1);
yG = IP(i, :);
yB = IQ(i, :);
try
yG = yG - G_ols(previous, j)' * VmDelta(filter(previous), :);
yB = yB + B_ols(previous, j)' * VmDelta(filter(previous), :);
catch
assert (i == 1);
end
rng(i);
filter(previous) = false;
VmDelta = Vm(filter, :) - repmat(Vm(j, :), sum(filter), 1);
% G_ols(j, filter) = obj.tls(VmDelta', yG');
G_ols(j, filter) = yG * VmDelta' / (VmDelta * VmDelta');
outlier = G_ols(j,:) > -obj.prior.Gmin;
G_ols(j, filter & outlier') = - obj.prior.Gmin * (1+0.1*rand());
outlier = G_ols(j,:) < -obj.prior.Gmax;
G_ols(j, filter & outlier') = - obj.prior.Gmax * (1+0.1*rand());
outlier = isnan(G_ols(j,:));
G_ols(j, filter & outlier') = - obj.prior.Gmin * (1+0.1*rand());
G_ols(filter, j) = G_ols(j, filter);
G_ols(j, j) = -sum(G_ols(j, :));
B_ols(j, filter) = - yB * VmDelta' / (VmDelta * VmDelta');
outlier = B_ols(j,:) < obj.prior.Bmin;
B_ols(j, filter & outlier') = obj.prior.Bmin * (1+0.1*rand());
outlier = B_ols(j,:) > obj.prior.Bmax;
B_ols(j, filter & outlier') = obj.prior.Bmax * (1+0.1*rand());
B_ols(filter, j) = B_ols(j, filter);
outlier = isnan(B_ols(j,:));
B_ols(j, filter & outlier') = obj.prior.Bmin * (1+0.1*rand());
B_ols(j, j) = -sum(B_ols(j, :));
end
obj.dataE.G = G_ols;
obj.dataE.B = B_ols;
end
% obj.dataE.G = (G+G')/2;
% obj.dataE.B = (B+B')/2;
% obj.dataE.G = obj.data.G;
% obj.dataE.B = obj.data.B;
end
function obj = iterateY(obj)
% This method iterate Y matrix considering the measurement
% error from both inputs and outputs.
% We first assume a flat diagonal element setting
W = ones(obj.numBus*2, 1);
obj = optimizeY(obj, W);
end
function [obj, Gopt, Bopt] = optimizeY(obj, W)
% This method use some convex optimization method and provide
% the G and B matrix
% control variables
G = sdpvar(obj.numBus, obj.numBus);
B = sdpvar(obj.numBus, obj.numBus);
% anxillary variables
Pres = sdpvar(obj.numBus, obj.numSnap);
Qres = sdpvar(obj.numBus, obj.numSnap);
% constraints
constP = Pres == G * obj.data.Vm_noised - obj.data.IP_noised;
constQ = Qres == - B * obj.data.Vm_noised - obj.data.IQ_noised;
constG = sum(G) == zeros(1, obj.numBus);
constB = sum(B) == zeros(1, obj.numBus);
constraints = [constP; constQ; constG; constB];
for i = 1:obj.numBus
for j = i+1:obj.numBus
constraints = [constraints; G(i,j)<=0];
constraints = [constraints; B(i,j)>=0];
end
end
% objective function
objective = sum(W(1:obj.numBus)' * (Pres .* Pres)...
+ W(1+obj.numBus:end)' * (Qres .* Qres));
options = sdpsettings('solver','gurobi');
sol = optimize(constraints,objective,options);
Gopt = value(G);
Bopt = value(B);
end
function obj = initValue(obj)
% This method provides the initial value (voltage angles?)
end
function obj = identifyOptNLP(obj)
% This method simply use the nonlinar programming techique to
% solve the maximum identification problem
% This version we simply assume we have all the measurements
% We should bound all the control variables and all the
% anxillary variables
% control variables
G = sdpvar(obj.numBus, obj.numBus);
B = sdpvar(obj.numBus, obj.numBus);
Pest = sdpvar(obj.numBus, obj.numSnap);
Qest = sdpvar(obj.numBus, obj.numSnap);
Vm = sdpvar(obj.numBus, obj.numSnap);
Va = sdpvar(obj.numBus, obj.numSnap);
% anxillary variables
e_P = sdpvar(obj.numBus, obj.numSnap);
e_Q = sdpvar(obj.numBus, obj.numSnap);
e_Vm = sdpvar(obj.numBus, obj.numSnap);
e_Va = sdpvar(obj.numBus, obj.numSnap);
Theta_ij = sdpvar(obj.numBus, obj.numBus, obj.numSnap);
GBThetaP = sdpvar(obj.numBus, obj.numBus, obj.numSnap);
GBThetaQ = sdpvar(obj.numBus, obj.numBus, obj.numSnap);
% some constaints
maxGB = 1000;
maxNoise = 10;
% constraints
Constraints = [];
% the power flow equation, P and Q injections
for snap = 1:obj.numSnap
Theta_ij(:,:,snap) = repmat(Va(:, snap), 1, obj.numBus) - repmat(Va(:, snap)', obj.numBus, 1);
% G_ij\cos(\Theta_ij)+B_ij\sin(\Theta_ij)
GBThetaP(:,:,snap) = G .* cos(Theta_ij(:,:,snap)) + B .* sin(Theta_ij(:,:,snap));
% G_ij\sin(\Theta_ij)-B_ij\cos(\Theta_ij)
GBThetaQ(:,:,snap) = G .* sin(Theta_ij(:,:,snap)) - B .* cos(Theta_ij(:,:,snap));
Constraints = [Constraints; Pest(:, snap) == (GBThetaP(:,:,snap) * Vm(:, snap)) .* Vm(:, snap)];
Constraints = [Constraints; Qest(:, snap) == (GBThetaQ(:,:,snap) * Vm(:, snap)) .* Vm(:, snap)];
end
% the anxillary variable constraints
Constraints = [Constraints; Pest + e_P == obj.data.P_noised];
Constraints = [Constraints; Qest + e_Q == obj.data.Q_noised];
Constraints = [Constraints; Vm + e_Vm == obj.data.Vm_noised];
Constraints = [Constraints; Va + e_Va == obj.data.Va_noised];
% zero noise for reference bus
Constraints = [Constraints; e_Va(1,:) == zeros(1, obj.numSnap)];
Constraints = [Constraints; e_Vm(1,:) == zeros(1, obj.numSnap)];
% the sum of G and B
Constraints = [Constraints; sum(G) == zeros(1, obj.numBus)];
Constraints = [Constraints; sum(B) == zeros(1, obj.numBus)];
% bound all the variables
% for i = 1:obj.numBus
% for j = i+1:obj.numBus
% Constraints = [Constraints; -maxGB <= G(i,j) <= 0];
% Constraints = [Constraints; 0 <= B(i,j) <= maxGB];
% end
% end
Constraints = [Constraints; -obj.sigma.P*ones(1, obj.numSnap)*maxNoise <= e_P <= obj.sigma.P*ones(1, obj.numSnap)*maxNoise];
Constraints = [Constraints; -obj.sigma.Q*ones(1, obj.numSnap)*maxNoise <= e_Q <= obj.sigma.Q*ones(1, obj.numSnap)*maxNoise];
Constraints = [Constraints; -obj.sigma.Vm*ones(1, obj.numSnap)*maxNoise <= e_Vm <= obj.sigma.Vm*ones(1, obj.numSnap)*maxNoise];
Constraints = [Constraints; -obj.sigma.Va*ones(1, obj.numSnap)*maxNoise <= e_Va <= obj.sigma.Va*ones(1, obj.numSnap)*maxNoise];
% assign the initial value
assign(G, obj.dataE.G);
assign(B, obj.dataE.B);
assign(Vm, obj.data.Vm_noised);
assign(Va, obj.data.Va_noised);
% objective function
objective = sum((obj.sigma.P.^-2)' * (e_P.*e_P) ...
+ (obj.sigma.Q.^-2)' * (e_Q.*e_Q)...
+ (obj.sigma.Vm(2:end).^-2)' * (e_Vm(2:end,:).*e_Vm(2:end,:))...
+ (obj.sigma.Va(2:end).^-2)' * (e_Va(2:end,:).*e_Va(2:end,:)));
options = sdpsettings('solver','ipopt','ipopt.max_iter',3000);
sol = optimize(Constraints,objective,options);
Gopt = value(G);
Bopt = value(B);
Pestopt = value(Pest);
Qestopt = value(Qest);
Vmopt = value(Vm);
Vaopt = value(Va);
e_Popt = value(e_P);
e_Qopt = value(e_Q);
e_Vmopt = value(e_Vm);
e_Vaopt = value(e_Va);
end
function obj = identifyOptGradient(obj)
% This method uses gradient-based method to solve the nonconvex
% optimization problem.
% Hopefully we could implement some power system domain
% knowledge into the process because we know the ground truth
% value.
obj.maxIter = 4000;
obj.step = 1e-4;
obj.stepMax = 1e-3;
obj.stepMin = 1e-3;
obj.momentRatio = 0.9;
obj.updateStepFreq = 20;
obj.vmvaWeight = 1;
obj.momentRatioMax = 0.95;
obj.momentRatioMin = 0.9;
obj.kZero = 0.0005;
obj.tuneGrad = false;
% we first initialize data
obj.dataO.G = obj.dataE.G;
obj.dataO.B = obj.dataE.B;
% note that we should replace the Vm ro Va data to some
% initialized data if we do not have the measurement devices
obj.dataO.Vm = obj.data.Vm;
% obj.dataO.Va = obj.data.Va;
obj.dataO.Vm(2:end, :) = bsxfun(@times, obj.data.Vm_noised(2:end, :), obj.isMeasure.Vm(2:end));
obj.dataO.Vm(obj.dataO.Vm == 0) = 1;
obj.dataO.Va = bsxfun(@times, obj.data.Va_noised, obj.isMeasure.Va);
% begin the iteration loop
% initialize the gradient numbers
obj.numGrad.G = (obj.numBus - 1) * obj.numBus / 2; % exclude the diagonal elements
obj.numGrad.B = (obj.numBus - 1) * obj.numBus / 2;
obj.numGrad.Vm = obj.numSnap * (obj.numBus - 1); % exclude the source bus
obj.numGrad.Va = obj.numSnap * (obj.numBus - 1);
obj.numGrad.Sum = obj.numGrad.G + obj.numGrad.B + obj.numGrad.Vm + obj.numGrad.Va;
obj.iter = 1;
obj.gradChain = zeros(obj.numGrad.Sum, obj.maxIter);
obj.lossChain = zeros(5, obj.maxIter);
obj.parChain = zeros(obj.numGrad.Sum, obj.maxIter);
obj.stepChain = zeros(1, obj.maxIter);
obj.isConverge = false;
while (obj.iter <= obj.maxIter && ~obj.isConverge)
disp(obj.iter);
% profile on;
% collect the paramter vector
obj = collectPar(obj);
% build the gradient
obj = buildGradient(obj);
% implement the re-weight techique.
obj = tuneGradient(obj);
% update the chains
try
obj.gradChain(:, obj.iter) = obj.grad * (1-obj.momentRatio) + obj.gradChain(:, obj.iter-1) * obj.momentRatio;
catch
obj.gradChain(:, obj.iter) = obj.grad;
end
obj.lossChain(:, obj.iter) = [obj.loss.total; obj.loss.P; obj.loss.Q; obj.loss.Vm; obj.loss.Va];
% update the parameters
obj = updatePar(obj);
% if converge
if mod(obj.iter, obj.updateStepFreq) == 0 %obj.iter > 10
if (mean(obj.lossChain(1, obj.iter-9:obj.iter-5)) < mean(obj.lossChain(1, obj.iter-4:obj.iter)))
obj.step = max(obj.step / 2, obj.stepMin);
obj.momentRatio = min(obj.momentRatio + 0.1, obj.momentRatioMax);
elseif (((obj.lossChain(1, obj.iter) - obj.lossChain(1, obj.iter-1)) < 0) ...
&& ((obj.lossChain(1, obj.iter-1) - obj.lossChain(1, obj.iter-2)) < 0)...
&& ((obj.lossChain(1, obj.iter-2) - obj.lossChain(1, obj.iter-3)) < 0))
% && ((obj.lossChain(1, obj.iter-3) - obj.lossChain(1, obj.iter-4)) < 0))
obj.step = min(obj.step * 1.2, obj.stepMax);
obj.momentRatio = max(obj.momentRatio - 0.1, obj.momentRatioMin);
elseif (((obj.lossChain(1, obj.iter) - obj.lossChain(1, obj.iter-1)) *...
(obj.lossChain(1, obj.iter-1) - obj.lossChain(1, obj.iter-2)) < 0)...
&& ((obj.lossChain(1, obj.iter-7) - obj.lossChain(1, obj.iter-8)) *...
(obj.lossChain(1, obj.iter-8) - obj.lossChain(1, obj.iter-9)) < 0))
obj.step = max(obj.step / 2, obj.stepMin);
obj.momentRatio = min(obj.momentRatio + 0.1, obj.momentRatioMax);
% disp('tune the weight');
% obj.tuneGrad = true;
% obj = buildMeasure(obj);
% obj.boundTuned = sqrt(abs(diag(full(obj.H)\eye(obj.numGrad.Sum))));
end
% if (mean(obj.lossChain(1, obj.iter-9:obj.iter-5)) < mean(obj.lossChain(1, obj.iter-4:obj.iter)))
% isConverge = true;
% end
end
obj.stepChain(obj.iter) = obj.step;
obj.iter = obj.iter + 1;
% profile off;
% profile viewer;
end
end
function obj = buildGradient(obj)
% This method build the gradient of the squared loss function
% Initialize the gradient matrix
obj.grad = zeros(obj.numGrad.Sum, 1);
obj.gradP = zeros(obj.numGrad.Sum, 1);
obj.gradQ = zeros(obj.numGrad.Sum, 1);
obj.gradVm = zeros(obj.numGrad.Sum, 1);
obj.gradVa = zeros(obj.numGrad.Sum, 1);
obj.loss.total = 0;
obj.loss.P = 0;
obj.loss.Q = 0;
obj.loss.Vm = 0;
obj.loss.Va = 0;
% Initialize the idGB
obj.idGB = zeros(obj.numBus, obj.numBus);
id = 1;
for i = 1:obj.numBus
obj.idGB(i, i+1:end) = id:id+obj.numBus-i-1;
obj.idGB(i+1:end, i) = id:id+obj.numBus-i-1;
id = id+obj.numBus-i;
end
for i = 1:obj.numSnap
% calculate some basic parameters at present state
Theta_ij = repmat(obj.dataO.Va(:, i), 1, obj.numBus) - repmat(obj.dataO.Va(:, i)', obj.numBus, 1);
% G_ij\cos(\Theta_ij)+B_ij\sin(\Theta_ij)
GBThetaP = obj.dataO.G .* cos(Theta_ij) + obj.dataO.B .* sin(Theta_ij);
% G_ij\sin(\Theta_ij)-B_ij\cos(\Theta_ij)
GBThetaQ = obj.dataO.G .* sin(Theta_ij) - obj.dataO.B .* cos(Theta_ij);
% P estimate
Pest = (GBThetaP * obj.dataO.Vm(:, i)) .* obj.dataO.Vm(:, i);
% Q estimate
Qest = (GBThetaQ * obj.dataO.Vm(:, i)) .* obj.dataO.Vm(:, i);
% the id of Vm and Va
obj.idVmVa = obj.numSnap * (0:obj.numBus-2) + i;
% calculate the sub-vector of P of all buses
for j = 1:obj.numBus
if obj.isMeasure.P(j)
obj = buildGradientP(obj, i, j, GBThetaP, GBThetaQ, Pest);
end
end
% calculate the sub-vector of Q of all buses
for j = 1:obj.numBus
if obj.isMeasure.Q(j)
obj = buildGradientQ(obj, i, j, GBThetaP, GBThetaQ, Qest);
end
end
% calculate the sub-vector of Vm of all buses
for j = 1:obj.numBus
if obj.isMeasure.Vm(j)
obj = buildGradientVm(obj, i, j);
end
end
% calculate the sub-vector of Va of all buses
for j = 1:obj.numBus
if obj.isMeasure.Va(j)
obj = buildGradientVa(obj, i, j);
end
end
end
% collect the gradients and the losses
obj.grad = obj.gradP + obj.gradQ + obj.gradVm + obj.gradVa;
obj.loss.total = obj.loss.P + obj.loss.Q + obj.loss.Vm + obj.loss.Va;
end
function obj = buildGradientP(obj , snap, bus, GBThetaP, GBThetaQ, Pest)
% This method builds the gradient from the measurement of P
theta_ij = obj.dataO.Va(bus, snap) - obj.dataO.Va(:, snap);
g = zeros(obj.numGrad.Sum, 1);
% G matrix
h_GG = obj.dataO.Vm(bus, snap) * obj.dataO.Vm(:, snap)' .* cos(theta_ij');
h_GG = h_GG - obj.dataO.Vm(bus, snap)^2;
g(obj.idGB(bus, [1:bus-1 bus+1:end])) = h_GG([1:bus-1 bus+1:end]);
% B matrix
h_BB = obj.dataO.Vm(bus, snap) * obj.dataO.Vm(:, snap)' .* sin(theta_ij');
g(obj.numGrad.G+obj.idGB(bus, [1:bus-1 bus+1:end])) = h_BB([1:bus-1 bus+1:end]);
% Vm
% the first order term of other Vm
h_Vm = obj.dataO.Vm(bus, snap) * GBThetaP(:, bus);
% the second order term of Vm(bus)
h_Vm(bus) = 2*obj.dataO.Vm(bus, snap) * GBThetaP(bus, bus);
% the first order term of Vm(bus)
fOrderVm = obj.dataO.Vm(:, snap) .* GBThetaP(:, bus);
fOrderVm(bus) = 0;
h_Vm(bus) = h_Vm(bus) + sum(fOrderVm);
g(obj.numGrad.G+obj.numGrad.B+obj.idVmVa) = h_Vm(2:end);
% Va
h_Va = obj.dataO.Vm(bus, snap) * obj.dataO.Vm(:, snap) .* GBThetaQ(:, bus);
h_Va(bus) = - obj.dataO.Vm(bus, snap)^2 * obj.dataO.B(bus, bus)...
- obj.data.Q_noised(bus, snap);
% h_Va(bus) = h_Va(bus)-sum(GBThetaQ(bus, :) * obj.dataO.Vm(:, snap) * obj.dataO.Vm(bus, snap));
g(obj.numGrad.G+obj.numGrad.B+obj.numGrad.Vm+obj.idVmVa) = h_Va(2:end);
% build GradientP and loss.P
lossThis = (Pest(bus) - obj.data.P_noised(bus, snap));
obj.loss.P = obj.loss.P + lossThis^2 * obj.sigma.P(bus).^(-2);
gradPThis = obj.sigma.P(bus).^(-2) * lossThis * g;
obj.gradP = obj.gradP + gradPThis;
end