## Contributed Session Fri.1.MA 549

#### Friday, 10:30 - 12:00 h, Room: MA 549

**Cluster 18: Optimization in energy systems** [...]

### Power flow modelling and mechanism design

**Chair: Deepak Kumar Bagchi**

**Friday, 10:30 - 10:55 h, Room: MA 549, Talk 1**

**Stephan Lemkens**

Structural properties of power grid design

**Coauthor: Arie M.c.a. Koster**

**Abstract:**

The problem of designing a cost minimal power grid is often formulated as a mixed integer linear program using the well known DC power flow linearization. We consider its projection on the integral space, as every feasible integral point can be considered as a possible power grid design. We define the DC power grid design polytope as the convex hull of these integral points.

At first, we will consider the case in which the power flow on each line is not restricted by any means. We will show, that in this setting the convex hull is described by the connected subgraph polytope of the topology graph.

In addition, we will discuss the structural properties under the influence of bounded power flows, as every real world scenario requires bounded flows. Further, we will study the effects on the convex hull under the assumption of a metric topology.

Finally, we will discuss the impact on the stated results in the case where we use AC linear power flows instead of the DC power flow linearization.

**Friday, 11:00 - 11:25 h, Room: MA 549, Talk 2**

**Waqquas Bukhsh**

Local solutions of optimal power flow problem

**Coauthors: Andreas Grothey, Ken Mckinnon, Paul Trodden**

**Abstract:**

Optimal power flow (OPF) is a well studied optimization problem in electricity industry. Over the last two decades it has become a standard tool for planning, real time operations and market auctions.

OPF is nonlinear optimization problem and the existence of locally optimal solutions has been a question of interest for decades. Often it is conjectured that OPF feasible region is convex. In this talk, we present examples of local solutions of OPF on a range of power systems networks. We also show that a recent reformulation of OPF as SDP problem

sometimes fails to recover feasible solutions of OPF.

**Friday, 11:30 - 11:55 h, Room: MA 549, Talk 3**

**Deepak Kumar Bagchi**

Optimal combinatorial auction for smart grids with renewable energy resources

**Coauthor: Shantanu Biswas**

**Abstract:**

We present an optimal combinatorial auction mechanism for the virtual power plant (VPP) formation problem in a smart grid with renewable energy sources. The VPP planner can source electricity from various suppliers generating electricity from

renewable energy sources. The planner has to solve the VPP formation problem to determines which VPP to form at any given point of time. We take into consideration the uncertainty in availability of renewable energy sources due to changing weather patterns.

To the best of our knowledge this is the first attempt at developing an optimal mechanism for the VPP formation problem. We have incorporated the uncertainty in availability of energy from renewable resources in the auction formulation to minimize the associated risks.We have stated and proved the necessary and sufficient conditions for Myerson optimal auction for VPP formation problem in the presence of single

minded suppliers.