EEEE 524/624: Fall 2017 Advances in Power Systems Lecture 6: Economic Dispatch with Network Constraints Prof. Luis Herrera Electrical and Microelectronic Engineering Rochester Institute of Technology
Topics that Will be Covered o Nodal Analysis o Power Flow Equations o DC Power Flow - simplification o Economic Dispatch with Network Constraints DC OPF o LP Formulation o Locational Marginal Price 2
Economic Dispatch with Network Constraints DC OPF In the previous lectures, we have not taken into account the network constraints However, it is important to take know the power flows throughout the system in order to: o Not overload equipment/lines o Keep voltages within bounds 3
Review of Complex Numbers and Notation Imaginary number: Euler equation: Rectangular form vs. Polar form vs Exponential form Operations in rectangular form Addition: Multiplication: 4
Review of Complex Numbers and Notation Vector view of a complex number: In polar coordinates: Multiplication Division 5
Review of Complex Numbers and Notation Given a complex number: The complex conjugate is defined as: 6
Review of Per Unit System The per unit system is helpful to analyze power systems with multiple areas! Having defined the bases for each area, the circuit becomes: 7
Review of Per Unit System Actual vs Base vs Per unit relationship Per unit equations: 8
Review of Per Unit System Convert the following circuit to per unit 9
Nodal Analysis Fundamentals Nodal analysis is a technique that utilizes Kirchhoff Current Law (KCL) at every node in the network Given the voltage between two points, we can calculate the current flowing through the line as: Convention: Always assume currents flow away from bus/node being studied 10
Nodal Analysis Admittance When we study ac systems, we need to consider the resistance and reactance of the line 11
Nodal Analysis 3 Bus System Compute the nodal equations at each bus in the following network: 12
Nodal Analysis - Admittance Matrix Compute the nodal equations at each bus in the following network: Rewrite the equations in matrix form: 13
Nodal Analysis Example 2 Write the admittance matrix 14
Topics that Will be Covered o Nodal Analysis o Power Flow Equations o DC Power Flow - simplification o Economic Dispatch with Network Constraints DC OPF o LP Formulation o Locational Marginal Price 15
Power Balance at Every Node Nodal Analysis 16
Power Balance at Every Node At every bus in a power system, power has to be balanced 17
Power Balance at Every Node At every bus in a power system, power has to be balanced 18
Decomposition into Active and Reactive Power Real and imaginary parts 19
Power Flow Equations Let s go back to the different nodes: 20
Power Flow Equations General Formula For a general power network the power flow equations are: 21
Power Flow Equations Caution! Be careful with the notation Keep track of the right angles and admittances to use 22
Power Flow Comments 23
Power Flow Example 24
Power Flow Example 25
Power Flow Example 26
Power Flow Example 27
Power Flow Example 28
Power Flow Example 29
Solving Systems of Equations 30
Newton-Raphson Method 31
Power Flow Example Numerical Solution 32
Power Flow Example Numerical Solution Matlab Code 33
Power Flow Example Numerical Solution Matlab Code 34
Power Flow Example - Results 35
Power Flow Example - Results 36
Power Flow Example - Results 37
Power Flow Example - PowerWorld 38
Topics that Will be Covered o Nodal Analysis o Power Flow Equations o DC Power Flow - simplification o Economic Dispatch with Network Constraints DC OPF o LP Formulation o Locational Marginal Price 39
Power Flow Equations 40
Power Flow Equations Assumptions 41
Power Flow Equations Assumption 1 42
Power Flow Equations Assumption 2 43
Power Flow Equations Assumptions 3 44
DC Power Flow Equations 45
DC Power Flow Equations Final Form 46
DC Power Flow Example 47
DC Power Flow Example 48
DC Power Flow Example 49
DC Power Flow Example 50
DC Power Flow Example 51
DC Power Flow Example - Comparison 52
DC Power Flow Summary 53
DC Power Flow Example 2 54
DC Power Flow Example 2 55
Topics that Will be Covered o Nodal Analysis o Power Flow Equations o DC Power Flow - simplification o Economic Dispatch with Network Constraints DC OPF o LP Formulation o Locational Marginal Price 56
DC Power Flow Line Power Flows Transmission lines are physically constrained in the power that they can deliver proportional to its size and related to material Therefore, when dispatching generation sources, it is important to make sure the transmission lines are not overloaded! 57
DC Power Flow Line Power Flows 58
Economic Dispatch Revisited The variables in the economic dispatch optimization problem are the generator active powers The economic dispatch without network constraints is as follows: 59
Economic Dispatch with Network Constraints DC OPF If we take into account the simplified DC power flow equations, we can impose limits on the transmission lines! Economic Dispatch + DC Power flow = DC Optimal Power Flow 60
Example DC Optimal Power Flow For the network as shown, assume the generator parameters are as follows Assume line 13 is limited to 0.6 pu power flow Compute the DC OPF 61
Example DC Optimal Power Flow For the network as shown, assume the generator parameters are as follows Assume line 13 is limited to 0.6 pu power flow Compute the DC OPF 62
Example DC Optimal Power Flow Matlab Code For the network as shown, assume the generator parameters are as follows Assume line 13 is limited to 0.6 pu power flow Compute the DC OPF 63
Example 2 DC Optimal Power Flow Consider a three bus system with local loads The generator costs are as follows With limits as (pu): 64
Example 2 DC Optimal Power Flow Find the optimal operation of the generators to minimize the cost of operation Take into account dc power flow Assume line 1-2 is limited to 150 MW What kind of optimization problem is this? (LP or QP) 65
Example 2 DC Optimal Power Flow Find the optimal operation of the generators to minimize the cost of operation Take into account dc power flow Assume line 1-2 is limited to 150 MW 66
Economic Dispatch with Network Constraints DC OPF If we take into account the simplified DC power flow equations, we can impose limits on the transmission lines! Economic Dispatch + DC Power flow = DC Optimal Power Flow 67
Topics that Will be Covered o Nodal Analysis o Power Flow Equations o DC Power Flow - simplification o Economic Dispatch with Network Constraints DC OPF o LP Formulation o Locational Marginal Price 68
Concept of Locational Marginal Price Locational Marginal Pricing: A method to reflect the value of electric energy at different locations (buses) accounting for: o o o Generation Losses Transmission line limits 69
Concept of Locational Marginal Price Locational Marginal Pricing: A method to reflect the value of electric energy at different locations (buses) accounting for generation, losses, and transmission line limits 70
Locational Marginal Pricing Two Bus System Suppose that we have a simple two bus system without line constraints 71
Locational Marginal Pricing Two Bus System Suppose that we have a simple two bus system Assume line 1-2 is limited to 200 MW What is the LMP at bus 2? At bus 1? 72
Locational Marginal Pricing Derivation DC OPF The main idea with Locational Marginal Pricing (LMP) is to answer: What is the cost of increasing the power demand at any bus? What is this total cost and how can we find it? We need to consider: generation, losses, and transmission line limits 73
Locational Marginal Pricing Derivation DC OPF The main idea with Locational Marginal Pricing (LMP) is to answer: What is the cost of increasing the power demand at any bus? Remember the DC OPF: 74
Locational Marginal Pricing Derivation DC OPF The main idea with Locational Marginal Pricing (LMP) is to answer: What is the cost of increasing the power demand at any bus? Write the Lagrangian for the problem: The Lagrangian can be seen as the total cost that we are looking for! 75
Locational Marginal Pricing Derivation DC OPF The main idea with Locational Marginal Pricing (LMP) is to answer: What is the cost of increasing the power demand at any bus? Write the Lagrangian for the problem: What is the LMP at bus i? 76
Locational Marginal Pricing Derivation DC OPF The main idea with Locational Marginal Pricing (LMP) is to answer: What is the cost of increasing the power demand at any bus? Write the Lagrangian for the problem: Therefore, the Locational Marginal Price at bus i is: 77
LMP Example DC OPF Example: Solve using DC OPF and find LMP at every bus 78
LMP Example DC OPF Example: Solve using DC OPF and find LMP at every bus 79
Summary of Topics Covered Nodal Analysis Power Flow Equations DC Power Flow - simplification Economic Dispatch with Network Constraints DC OPF Formulation (LP or QP) Locational Marginal Price 80