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I am working on the Alternating Current- Optimal Power Flow problem. It is actually a non convex optimization problem because of a few constraints which are non-convex in nature.

$min$ $x^THx + h^Tx + const\\ s.t.\\1.v^TMv-Pg+d_p\le0;-v^TMv+Pg-d_p\ge0\\2.v^TMv-Qg+d_q\le0;-v^TMv+Qg-d_q\ge0\\3.v^TF_pv-P_f\le0;-v^TF_pv+P_f\ge0\\4.v^TF_qv-Q_f\le0;-v^TF_qv+Q_f\ge0\\5.v^TT_pv-P_f\le0;-v^TT_pv+P_f\ge0\\6.v^TT_qv-Q_f\le0;-v^TT_qv+Q_f\ge0\\7.Pg_l\le Pg\le Pg_u\\8.Qg_l\le Qg\le Qg_u\\9.V_l\le V\le V_u\\ x=[v;Pf;Qf;Pt;Qt;Pg;Qg]$

Equations 1,2,3,4,5,6 are non convex in nature because matrices $M,F_p,F_q,T_p and T_q$ are not positive semi definite matrices. The problem is harder for 3,4,5 and 6 because $F_p, F_q, T_p and T_q$ aren't symmetric. Hence their eigen decomposition produces complex eigen values.

$M$ is a symmetric matrix with positive and negative eigen values. $F_p,F_q,T_p $ and $T_q$ are non symmetric matrices.

One way I thought was to compute $M = M_+ - M_-$ where $M_+ = \lambda_+(v_+)^T(v_+)$ and $M_- = -\lambda_-(v_-)^T(v_-)$ where $\lambda_+$ and $\lambda_-$ are the positive and negative eigen values of the matrix and $v_+$ and $v_-$ are the eigen vectors corresponding to the eigen values of $M$.

The eigen decomposition of the matrices produces an interesting structure where there are 4 non zero eigen values, 2 negative, 2 positive, the negative eigen values are equal to each other and the positive eigen values are each other.

That way $M_+$ is a positive semi definite matrix and the part corresponding to the negative eigen value will be linearized as $v^TM_-v = (v_k)^TM_-v_k + 2(v_k)^TM_-(v-v_k)$ where $v_k$ is the known value of v or initial value. But doing so does not satisfy the constraint as in it isn't $\le0$ when tested with an already known solution.

I was wondering if it possible to convexify 1,2,3,4,5 and 6 only to work this in a convex optimization problem. How do I go about this? I'm using MATLAB's fmincon function to solve this problem.

Viswanath Hariharan
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  • What do you mean by convexify? Are you hoping for an exact reformulation or an approximation? – LinAlg Apr 28 '18 at 19:07
  • Approximation. Ideally exact reformulation but I guess that isn't possible. – Viswanath Hariharan Apr 28 '18 at 20:34
  • I do not understand why the linear approximation doesn't work. – LinAlg Apr 28 '18 at 20:42
  • Me too. I was hoping there was some other approach I could take towards approximation. – Viswanath Hariharan Apr 28 '18 at 21:52
  • Most people don't know that ACOPF stands for "alternating current optimal power flow". – littleO Apr 28 '18 at 22:06
  • Made that change – Viswanath Hariharan Apr 28 '18 at 22:22
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    A quadratic form with a non-symmetric matrix can always be replaced by an equivalent quadratic form with a symmetric matrix (with real eigenvalues). – Rodrigo de Azevedo Apr 29 '18 at 10:25
  • @RodrigodeAzevedo Any suggestions on how I can do that? Also, 'replaced' or 'approximated'? – Viswanath Hariharan Apr 29 '18 at 15:31
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    @ViswanathHariharan Just replace the matrix by its symmetric part. Its skew-symmetric part contributes nothing to the quadratic form. Take a look at this answer. – Rodrigo de Azevedo Apr 29 '18 at 15:40
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    The change @RodrigodeAzevedo is suggesting is not an approximation. It is always the case that $v^TMv=v^T(M+M^T)v/2$ for real quantities. So you can always replace $F_p$, etc. with their symmetric part $(F_p+F_p^T)/2$, etc. It doesn't change convexity of course. – Michael Grant Apr 29 '18 at 20:43
  • Alright. I have made a couple of edits to the question about M, Fp and other matrices structures and how I used the eigen decomposition to work with them. I linearize a term after this as one of the terms will be convex and the other will be concave. But this approximation results in c(x) having a value greater than 0 when c(x) <= 0 is the constraint. Should I introduce an error correction term? – Viswanath Hariharan Apr 29 '18 at 20:59

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