## Description

Karlsruhe Institute of Technology

Institute for Automation and Applied Informatics

TUTORIAL II: NETWORK THEORY AND POWER FLOW

PROBLEM II.1 (PROGRAMMING) – NETWORK THEORY BASICS

Consider the simple network shown ins Figure 1. Calculate in Python or by hand:

(a) Compile the nodes list and the edge list.

Remark: While graph-theoretically both lists are unordered sets, let’s agree on an ordering now which can serve as basis for the matrices in the following exercises: we sort everything in ascending numerical order, i.e. node 1 before node 2 and edge (1,2) before (1,4) before (2,3).

(b) Determine the order and the size of the network.

(c) Compute the adjacency matrix A and check that it is symmetric.

(d) Find the degree kn of each node n and compute the average degree of the network.

(e) Determine the incidence matrix K by assuming the links are always directed from smaller-numbered node to largernumbered node, i.e. from node 2 to node 3, instead of from 3 to 2.

(f) Compute the Laplacian L of the network using kn and A. Remember that the Laplacian can also be computed as L = KKT and check that the two definitions agree.

(g) Find the diameter of the network by looking at Figure 1.

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Figure 1: Simple Network

PROBLEM II.2 (PROGRAMMING) – LINEAR POWER FLOW

2017 in hourly MW in the file imbalance.csv. They have been derived from physical flows as published by ENTSO-E.

The linear power flow is given by

pi =∑ L˜ i,jθj and fl = 1 ∑Ki,lθi, where L˜ i,j =∑Ki,l 1 Kj,l (1) j l i l xl

x

is the weighted Laplacian. For simplicity, we assume identity reactance on all links xl = 1.

(a) Compute the voltage angles θj and flows fl for the first hour in the dataset with the convention of θ0 = 0; i.e. the slack bus is at node 0.

Remark: Linear equation systems are solved efficiently using numpy.linalg.solve.

(b) Determine the average flow on each link for 01-2017 and draw it as a directed network.

Figure 2: Simple Network

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