About Me¶
BS and MS in Mathematics and Computing Sciences (July'12-June'17) at Indian Institute of Technology Kharagpur
Google Summer of Code 2016 and 2017 student under the Julia Language
GitHub: Ayush-iitkgp
Website: https://ayush-iitkgp.github.io
Outline¶
- Power Flow Optimization
- Fidelity in Quantum Information
Optimal Power Flow - Introduction¶
- Power flow study is an analysis of a connected electrical power system’s capability to adequately supply the connected load.
- Unknowns: - Voltages angle and magnitude information for each bus
- Knowns: - Load( such as appliances and lights.) and generator real power and voltage condition.
Optimal Power Flow - Mathematical Formulation¶
Constraints¶
- $p_{ij}$: Active power entering the line from node i
- $q_{ij}$: Reactive power entering the line from node i
- Let $x_{i}$ denote the complex voltage for node i of the network.
We have the following power balance equations which are non-linear in unknown $x_{i}$ and $x_{j}$.
Objective¶
Depends upon the business needs such as:
- Minimize power losses in an electrical network
- Minimize cost of generation
Optimal Power Flow - SDP Relaxation¶
- The original optimal power flow problem is non-convex in nature.
- Thanks to the lifting technique which converts the above optimization problem to a SemiDefinite Programming Problem.
- The relaxed SDP problem finds the near global solution of the original non-convex problem.
Example¶
- The data is taken from the IEEE 14 Bus test case which represents a portion of the American Electric Power System (in the Midwestern US) as of February, 1962.
using Convex # Read the input data
using FactCheck
using MAT #Pkg.add("MAT")
TOL = 1e-2;
input = matopen("Data.mat")
varnames = names(input)
Data = read(input, "inj","Y");
n=size(Data[2],1); # Create some intermediate variables
Y=Data[2];
inj=Data[1];
W = ComplexVariable(n,n); # W is the matrix of pairwise products of the voltages
objective = real(sum(diag(W))); # The objective is to minimize cost of generation
c1 = Constraint[]; # The constraints are power balance equations
for i=2:n
push!(c1,sum(W[i,:].*(Y[i,:]'))==inj[i]);
end
c2 = W in :SDP;
c3 = real(W[1,1])==1.06^2;
push!(c1, c2);
push!(c1, c3);
p = maximize(objective,c1); # Create the problem
solve!(p); # Solve the problem
p.optval #15.125857662600703
evaluate(objective) #15.1258578588357
output = matopen("Res.mat"); # Verify the results
names(output);
outputData = read(output, "Wres");
Wres = outputData;
real_diff = real(W.value) - real(Wres);
imag_diff = imag(W.value) - imag(Wres);
@fact real_diff => roughly(zeros(n,n), TOL);
@fact imag_diff => roughly(zeros(n,n), TOL)
Fidelity in Quantum Information Theory - Introduction¶
This example is inspired from a lecture of John Watrous in the course on Theory of Quantum Information.
Fidelity is a measure of the closeness of two quantum states.
The ability to distinguish between the quantum states is equivalent to the ability to distinguish between the classical probability distributions.
If fidelity between two states is 1, they are the same quantum state.
- Application
- The Fidelity between two Hermitian semidefinite matrices P and Q is defined as:
where the trace norm $||.||_{tr}$ is the sum of the singular values, and the maximization goes over the set of all unitary matrices U.
Fidelity in Quantum Information Theory - Mathematical Formulation¶
Fidelity can be expressed as the optimal value of the following complex-valued SDP:
$$ \textbf{maximize} \frac{1}{2} trace(Z+Z^*)$$$$\text{subject to } \left[\begin{array}{cc}P&Z\\{Z}^{*}&Q\end{array}\right] \succeq 0$$$$\text{where } Z \in \mathbf {C}^{n \times n}$$Example¶
n = 20 # Create the data
P = randn(n,n) + im*randn(n,n);
P = P*P';
Q = randn(n,n) + im*randn(n,n);
Q = Q*Q';
Z = ComplexVariable(n,n); # Declare convex variable
objective = 0.5*real(trace(Z+Z')); # Specify the objective
constraint = [P Z;Z' Q] ⪰ 0;
problem = maximize(objective,constraint);
solve!(problem) # Solve the problem
# Verify that computer fidelity is equal to actual fidelity
computed_fidelity = evaluate(objective)
P1,P2 = eig(P);
sqP = P2 * diagm([p1^0.5 for p1 in P1]) * P2'
Q1,Q2 = eig(Q)
sqQ = Q2 * diagm([q1^0.5 for q1 in Q1]) * Q2'
actual_fidelity = sum(svd(sqP * sqQ)[2])
diff = computed_fidelity - actual_fidelity
@fact diff => roughly(0, TOL)