An energy management system for PV-STATCOMs in power distribution networks via a complex-domain SDP relaxation

Abstract This article proposes an efficient energy management system for optimally scheduling distributed energy resources based on photovoltaic static synchronous compensators (PV-STATCOMs) in medium-voltage distribution networks. The complete nonlinear programming (NLP) model is reformulated as a semi-definite programming (SDP) relaxation in the complex domain. The key feature of our approximation lies in utilizing a set of Hermitian semi-definite matrices $$\mathbb {X}_{n,n,h}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>h</mml:mi> </mml:mrow> </mml:msub> </mml:math> , which allows transforming the NLP model from $$\mathbb {C}^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> into a linear programming one in the space of complex square matrices defined in $$\mathbb {C}^{n\times n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> . The IEEE 33-bus grid was selected as a test feeder in order to evaluate the performance of the proposed relaxation through numerical experiments. Two objective functions are considered: (i) the minimization of the energy purchasing costs at the substation terminals and (ii) the minimization of the expected daily energy losses. The results demonstrate the effectiveness and robustness of our proposed SDP relaxation approach.