Mathematical Programming for Power Systems Operation. Alejandro Garcés Ruiz

State estimation and grid identification12.1 Measurement units12.2 State estimation12.3 Topology identification12.4 Y_bus estimation12.5 Load model estimation12.6 Further readings12.7 Exercises13 Demand-side management13.1 Shifting loads13.2 Phase balancing13.3 Energy storage management13.4 Further readings13.5 Exercises

      10  A The nodal admittance matrix

      11  B Complex linearization

      12 C Some Python examplesC.1 Basic PythonC.2 NumPyC.3 MatplotLibC.4 Pandas

      13  Bibliography

      14  Index

      15  End User License Agreement

      List of Illustrations

      1 IntroductionFigure 0.1 Stages of solving an optimization problem.

      2 Chapter 1Figure 1.1 Types of optimization models.Figure 1.2 Schematic representation of the variables associated to a...Figure 1.3 Economic dispatch by areas considering network constraints with...Figure 1.4 Vehicle-to-grid concept with an aggregator that centralizes control actions...Figure 1.5 Example of a microgrid with a centralized...Figure 1.6 Set of possible configurations in a three-phase node.Figure 1.7 A possible architecture for implementing an optimization...

      3 Chapter 2Figure 2.1 Representation of the sets related to a general optimization problem.Figure 2.2 Three ways to measure the vector...Figure 2.3 Comparison among unit balls defined by norm-2, norm-1, and norm-∞Figure 2.4 Example of local and global optima: a) function with two local minima...Figure 2.5 Example of a function with several optimal points.Figure 2.6 A small photovoltaic system with three solar panels.Figure 2.7 Convergence of the gradient method.Figure 2.8 Intersection of an affine space with a cylinder.

      4 Chapter 3Figure 3.1 Example of a convex set Ω_A and a non-convex...Figure 3.2 Example of three different sets with a similar definition....Figure 3.3 Union and intersection of two convex sets...Figure 3.4 Example of a polytope in the plane.Figure 3.5 Comparison between a convex and a non-convex function.Figure 3.6 List of common convex functions.Figure 3.7 Composition of a convex function and a convex non-decreasing function.Figure 3.8 Schematic representation of the neighborhood of...Figure 3.9 Relation among the concepts of strict and strong convexity....Figure 3.10 Relation between primal and dual problems.Figure 3.11 Example of a primal non-convex problem and its corresponding dual function.Figure 3.12 Comparison between the primal and the dual problems.

      5 Chapter 4Figure 4.1 Using Python for mathematical optimization.Figure 4.2 Set of feasible solutions for the linear programming problem...Figure 4.3 Oriented graph for a transportation problem with four sources (s)...Figure 4.4 Example of two quadratic functions, q₁ (left)...Figure 4.5 Indication function and logarithmic barrier.Figure 4.6 Tree generated by the branch and bound algorithm.

      6 Chapter 5Figure 5.1 Example of a convex cone...Figure 5.2 Representation of the second order cone...Figure 5.3 Area of an ellipsoid seen as a linear transformation of a unit ballFigure 5.4 Venn diagram that represents the relation among different connic problemsFigure 5.5 Graphical representation for the problem Equation...Figure 5.6 Plot of polynomial p(x) = x⁴ −...Figure 5.7 Example of a Monte Carlo integration. We add only the points inside...

      7 Chapter 6Figure 6.1 Probability density function for a variable with mean...Figure 6.2 A multivariate normal distribution in...Figure 6.3 Confidence regions for a multivariate normal distribution in...

      8 Chapter 7Figure 7.1 Three thermal units with their respective cost functions for...Figure 7.2 Incremental cost for three thermal units considering capability limits.Figure 7.3 Linear approximation of quadratic cost functions.Figure 7.4 Example of a Pareto frontier for two contradictory...Figure 7.5 Power grid with six nodes and six generators. All lines....

      9 Chapter 8Figure 8.1 Simplified model for the starting-up of a thermal power plant.Figure 8.2 Unit commitment for a system with three thermal units.Figure 8.3 Grid constraints for the unit comment problem.

      10 Chapter 9Figure 9.1 Power demand and inflows for a hydrothermal system.Figure 9.2 Power demand and generated power.Figure 9.3 Quadratic function for a hydropower unit.Figure 9.4 Example of a hydraulic chain where...Figure 9.5 Schematic representation of a pumped hydroelectric storage system.Figure 9.6 Results for the Example 9.4.Figure 9.7 Three node system for hydrothermal scheduling.

      11 Chapter 10Figure 10.1 Example of a power distribution grid with distributed resources.Figure 10.2 Approximation of the voltage restriction as a box constraint.Figure 10.3 Cumulative distribution function of the linearization...Figure 10.4 Comparison between the constraint....Figure 10.5 Example of a capability curve for a conventional synchronous machine.

      12 Chapter 11Figure 11.1 Three-feeder test system for power system reconfiguration [84].Figure 11.2 Radial distribution network zkm = 0.01+ 0.005j.Figure 11.3 Example of two configurations of distributed generation.Figure 11.4 Schematic representation of the voltages and currents....Figure 11.5 Three-phase currents for a non-linear load with 5th harmonic.

      13 Chapter 12Figure 12.1 Power measurements at a given node i for dc state estimation.Figure 12.2 Three-bus system. ▾ represents points with power metering systems.Figure 12.3 Four-node grid with PMUs in all nodes for...Figure 12.4 Estimation error vs number of scenarios.Figure 12.5 Synthetic data of voltage and power for load estimation.

      14 Chapter 13Figure 13.1 Example of a shifting load. In both cases the shape....Figure 13.2 Results for the shifting load problem. Original loads...Figure 13.3 Phase balancing as an assignment problem.Figure 13.4 Load curve in a typical day: (- -) normal load without...Figure 13.5 Example of a grid-connected microgrid.

      15 Appendix AFigure A.1 Example of an oriented graph.

      16 Appendix BFigure B.1 Possible directions for taking the limit that defines the...

      List of Tables

      1 Chapter 2Table 2.1. Bounds of some ordered sets.

      2 Chapter 3Table 3.1. Summary of the main properties of convex optimization problems

      3 Chapter 4Table 4.1. Details of the node generated by the branch and bound problem.Table 4.2. Parameters for a transportation problem with...

      4 Chapter 5Table 5.1. Constraints that can be transformed to SDP.

      5 Chapter 6Table 6.1. Dual norms for the most common cases

      6 Chapter 7Table 7.1 Cost functions and operative limits for a...

      7 Chapter 8Table 8.1. Logic table for operation...

      8 Chapter 10Table 10.1 34-bus test system taken from [83]Table 10.2 Parameters of a 21-nodes DC power distribution grid

      9 Chapter 11Table 11.1 Parameters of the three-feeder test system for...Table 11.2 IEEE 33 nodes test distribution network [102].

      10 Chapter 13Table 13.1. Three industrial process and price of the energy in each timeTable 13.2. Feasible permutations for the phase-balancing problemTable 13.3. Expected generation, demand and price for 24h operation of a microgrid

      11 Appendix CTable C.1. Comparison of the installed power and increase....

      Guide

      1  Cover

      2