Topic 1: Basics of Power Systems
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1 Topic : asics of Power Systems ECE 52: Communications and Control for Smart Spring 202 A.H. Mohsenian Rad (U of T) Networing and Distributed Systems
2 Power Systems The Four Main Elements in Power Systems: Power Production / Generation Power Transmission Power Distribution Power Consumption / Load Of course, we also need monitoring and control systems. Communications and Control in Smart Grid Texas Tech University 2
3 Power Systems Power Production: Different Types: Traditional Renewable Capacity, Cost, Carbon Emission Step up Transformers Communications and Control in Smart Grid Texas Tech University
4 Power Systems Power Transmission: High Voltage (HV) Transmission Lines Several Hundred Miles Switching Stations Transformers Circuit reaers Communications and Control in Smart Grid Texas Tech University 4
5 Power Systems The Power Transmission Grid in the United States: Communications and Control in Smart Grid Texas Tech University 5
6 Power Systems Major Inter connections in the United States: Communications and Control in Smart Grid Texas Tech University 6
7 Power Systems Power Distribution: Medium Voltage (MV) Transmission Lines (< 50 V) Power Deliver to Load Locations Interface with Consumers / Metering Distribution Sub stations Step Down Transformers Distribution Transformers Communications and Control in Smart Grid Texas Tech University 7
8 Power Systems Power Consumption: Industrial Commercial Residential Demand Response Controllable Load Non Controllable Communications and Control in Smart Grid Texas Tech University 8
9 Power Systems Generation Transmission Distribution Load Communications and Control in Smart Grid Texas Tech University 9
10 Power Systems Power System Control: Data Collection: Sensors, PMUs, etc. Decision Maing: Controllers Actuators: Circuit reaers, etc. Communications and Control in Smart Grid Texas Tech University 0
11 Power Grid Graph Representation Nodes: uses Lins: Transmission Lines Generator Load Communications and Control in Smart Grid Texas Tech University
12 Power Grid Graph Representation Nodes: uses Lins: Transmission Lines uses (Voltage) Generator Load Communications and Control in Smart Grid Texas Tech University 2
13 Power Grid Graph Representation Nodes: uses Lins: Transmission Lines Generator Load Transmission Lines (Power Flow, Loss) Communications and Control in Smart Grid Texas Tech University
14 Power Grid Graph Representation Nodes: uses Lins: Transmission Lines Generator Load Consumers Communications and Control in Smart Grid Texas Tech University 4
15 Power Grid Graph Representation Nodes: uses Lins: Transmission Lines Generator 0 MW MW Load 7 MW Communications and Control in Smart Grid Texas Tech University 5
16 Transmission Line Admittance Admittance y is defined as the inverse of impedance z: z = r + j x y = g + j b (r: Resistance, x: Reactance) (g: Conductance, b: Susceptance) y = / z Parameter g is usually positive Parameter b: Positive: Capacitor Negative: Inductor Communications and Control in Smart Grid Texas Tech University 6
17 Transmission Line Admittance For the transmission line connecting bus ito bus : Addmitance: y i Example: y i = j 4 (per unit) Note that, y ii is denoted by y i and indicates: Susceptance for any shunt element (capacitor) to ground at bus i. Communications and Control in Smart Grid Texas Tech University 7
18 Y-us Matrix We define: Y bus = [ Y ij ] where Diagonal Elements: Y ii y i N y, i i Off diagonal Elements: Y ij y ij Note that Y bas matrix depends on the power grid topology and the admittance of all transmission lines. N is the number of busses in the grid. Communications and Control in Smart Grid Texas Tech University 8
19 Y-us Matrix Example: For a grid with 4 buses, we have: Y bus y y 2 y y y y 2 4 y 4 y 2 y 2 y y y 2 y y 24 y y y y y 2 y 4 2 y 4 y 4 y y y y y 42 y 4 After separating the real and imaginary parts: Y bus G j Communications and Control in Smart Grid Texas Tech University 9
20 us Voltage Let V i denote the voltage at bus i: Note that, V i is a phasor, with magnitude and angle. V i V i i In most operating scenarios we have: V i V j i j Communications and Control in Smart Grid Texas Tech University 20
21 Power Flow Equations Let S i denote the power injection at bus i: S i = P i + j Q i Active Power Reactive Power Generation us: P i > 0 Load us: P i < 0 (negative power injection) Communications and Control in Smart Grid Texas Tech University 2
22 Power Flow Equations Using Kirchhoff laws, AC Power Flow Equations become: P Q N j N j V V V V j j G G j j cos( sin( j ) ) j j j sin( cos( ) j ) j Do we now all notations here? If we now enough variables, we can obtain the rest of variables by solving a system of nonlinear equations. Communications and Control in Smart Grid Texas Tech University 22
23 Power Flow Equations The AC Power Flow Equations are complicated to solve. Next, we try to simplify the equations in three steps. Step : For most networs, G <<. Thus, we set G = 0: P Q N j N j V V V j V j j sin( ) j j cos( ) j Communications and Control in Smart Grid Texas Tech University 2
24 Power Flow Equations Texas Tech University Communications and Control in Smart Grid 24 Step 2: For most neighboring buses:. As a result, we have: to5 0 j i ) ( ) ( j j j Cos Sin N j j j N j j j j V V Q V V P ) (
25 Power Flow Equations Step : In per unit, V i is very close to.0 (0.95 to.05). As a result, we have: V i V j. P Q N j N N j j b j j ( ) j j, j P has a linear model and Q is almost fixed. Communications and Control in Smart Grid Texas Tech University 25
26 Power Flow Equations Step : In per unit, V i is very close to.0 (0.95 to.05). As a result, we have: V i V j. DC Power Flow Equations P Q N j N N j j b j j ( ) j j, j P has a linear model and Q is almost fixed. Communications and Control in Smart Grid Texas Tech University 26
27 Power Flow Equations Given the power injection values at all buses, we can use P j ( j ) j to obtain the voltage angles at all buses. N Let P ij denote the power flow from bus ito bus j, we have: P ij ij ( i j ) Communications and Control in Smart Grid Texas Tech University 27
28 Power Flow Equations Example: Obtain power flow values in the following grid: y4 j0 y j0 P g 2 pu P g 2 2 pu y2 j0 y4 j0 P l 2 pu P g 4 pu P l 4 pu y2 j0 Communications and Control in Smart Grid Texas Tech University 28
29 Power Flow Equations First, we obtain the Y bus matrix: Y bus b j b 2 b b b b 2 4 b 4 b 2 b b 2 b b b b 24 b b b b b b b 4 b 4 b b b b b 42 b 4 j j j Communications and Control in Smart Grid Texas Tech University 29
30 Power Flow Equations Next, we write the (active) power flow equations: P P P P This can be written as: P P2 P P Communications and Control in Smart Grid Texas Tech University 0
31 Power Flow Equations Texas Tech University Communications and Control in Smart Grid From the last two slides, we finally obtain: Therefore, the voltage angles are obtained as:
32 Power Flow Equations However, the last matrix in the previous slide is singular! Therefore, we cannot tae the inverse. The system of equations would have infinite solutions. The problem is that the four angles are not independent. What matters is the angular/phase difference. We choose one bus (e.g., bus ) as reference bus: 0. Communications and Control in Smart Grid Texas Tech University 2
33 Power Flow Equations Texas Tech University Communications and Control in Smart Grid We should also remove the corresponding rows/columns: The angular differences (with respect to ):
34 Power Flow Equations Finally, the power flow values are calculated as: P P P P P ( ) 0( ) 0.25 ( ) 0(0 0.5).5 ( ) 0( ) 0.25 ( ) 0( ).25 2 ( ) 0( ) P g 2 pu P g 2 2 pu P l 2 pu P g 4 pu P l 4 pu Communications and Control in Smart Grid Texas Tech University 4
35 Power Flow Equations What if the generator connected to bus is renewable? What if the capacity of transmission lin (,) is pu? What if we can apply demand response to load bus? What if one of the transmission lines fails? Communications and Control in Smart Grid Texas Tech University 5
36 Economic Dispatch Problem In the example we discussed earlier, we had: In particular, we had: Power Supply = Power Load g g g l l P P2 P4 P2 P g g g However, generation levels, P and P assumed given. P 2, 4 Q: What if the generators have different generation costs? Communications and Control in Smart Grid Texas Tech University 6
37 Economic Dispatch Problem For thermal power plants, generation cost is quadratic: Generation Cost = C(P) = a + a 2 x P + a x P 2 Example: a grid with three power plants: C (P ) = x P x (P ) 2 C 2 (P 2 ) = x P x (P 2 ) 2 C (P ) = x P x (P ) 2 50 MW P 600 MW 00 MW P MW 50 MW P 200 MW Each power plant has some min and max generation levels. Communications and Control in Smart Grid Texas Tech University 7
38 Economic Dispatch Problem For thermal power plants, generation cost is quadratic: Generation Cost = C(P) = a + a 2 x P + a x P 2 Example: a grid with three power plants: C (P ) = x P x (P ) 2 C 2 (P 2 ) = x P x (P 2 ) 2 C (P ) = x P x (P ) 2 50 MW P 600 MW 00 MW P MW 50 MW P 200 MW Each power plant has some min and max generation levels. Communications and Control in Smart Grid Texas Tech University 8
39 Economic Dispatch Problem For thermal power plants, generation cost is quadratic: Generation Cost = C(P) = a + a 2 x P + a x P 2 Example: a grid with three power plants: C (P ) = x P x (P ) 2 C 2 (P 2 ) = x P x (P 2 ) 2 C (P ) = x P x (P ) 2 50 MW P 600 MW 00 MW P MW 50 MW P 200 MW Each power plant has some min and max generation levels. Communications and Control in Smart Grid Texas Tech University 9
40 Economic Dispatch Problem For thermal power plants, generation cost is quadratic: Generation Cost = C(P) = a + a 2 x P + a x P 2 Example: a grid with three power plants: C (P ) = x P x (P ) 2 C 2 (P 2 ) = x P x (P 2 ) 2 C (P ) = x P x (P ) 2 50 MW P 600 MW 00 MW P MW 50 MW P 200 MW Each power plant has some min and max generation levels. Communications and Control in Smart Grid Texas Tech University 40
41 Economic Dispatch Problem For thermal power plants, generation cost is quadratic: Generation Cost = C(P) = a + a 2 x P + a x P 2 Example: a grid with three power plants: C (P ) = x P x (P ) 2 C 2 (P 2 ) = x P x (P 2 ) 2 C (P ) = x P x (P ) 2 50 MW P 600 MW 00 MW P MW 50 MW P 200 MW Each power plant has some min and max generation levels. Communications and Control in Smart Grid Texas Tech University 4
42 Economic Dispatch Problem We should select P, P 2, and P to: Meet total load P load = 850 MW Minimize the total cost of generation Economic Dispatch Problem: minimize P, P2, P subject to C P CP CP 50 P 00 P2 50 P P P P Communications and Control in Smart Grid Texas Tech University 42
43 Economic Dispatch Problem Is the formulated problem a convex program? Why? Convex programs can be solved efficiently. An useful software is CVX for Matlab ( The optimal economic dispatch solution: P = 9.2 MW P 2 = 4.6 MW Q: Do they satisfy all constraints? P = 22.2 MW Communications and Control in Smart Grid Texas Tech University 4
44 Economic Dispatch Problem Is the formulated problem a convex program? Why? Convex programs can be solved efficiently. An useful software is CVX for Matlab ( The optimal economic dispatch solution: P = 9.2 MW P 2 = 4.6 MW P = 22.2 MW Minimum Cost = = 894. Communications and Control in Smart Grid Texas Tech University 44
45 Economic Dispatch Problem What if we have to satisfy topology constraints? P P 2 P MW P 400 MW P P P ( ) Communications and Control in Smart Grid Texas Tech University 45
46 Economic Dispatch Problem The same optimal solutions are still valid: P 9.2 MW P MW MW 4.4 P MW 400 MW P 200 Communications and Control in Smart Grid Texas Tech University 46
47 Economic Dispatch Problem The same optimal solutions are still valid: P 9.2 MW P MW MW 4.4 What if P 70 P MW 400 MW P 200 Communications and Control in Smart Grid Texas Tech University 47
48 Economic Dispatch Problem Then the economic dispatch problem becomes: minimize P, P2, P,, 2, subject to C P CP CP 50 P 00 P2 50 P P P2 P P P Communications and Control in Smart Grid Texas Tech University 48
49 Economic Dispatch Problem Then the economic dispatch problem becomes: minimize P, P2, P,, 2, subject to C P CP CP P P P 2 P P 2 P P P Still a Convex Program? Communications and Control in Smart Grid Texas Tech University 49
50 Economic Dispatch Problem The new optimal solutions are obtained as: P 280 MW P MW MW 60 P MW 400 MW The total generation cost becomes: $8,2.66 > $8,94. Here, we had to sacrifice cost for implementation. Communications and Control in Smart Grid Texas Tech University 50
51 Economic Dispatch Problem The new optimal solutions are obtained as: P 280 MW P MW MW 60 What if P 50 P MW 400 MW The total generation cost becomes: $8,2.66 > $8,94. Here, we had to sacrifice cost for implementation. Communications and Control in Smart Grid Texas Tech University 5
52 Unit Commitment Economic Dispatch is solved a few hours ahead of operation. On the other hand, we need to decide about the choice of power plants that we want to turn on for the next day. This is done by solving the Unit Commitment problem. We particularly decide on which slow starting power plants we should turn on during the next day given various constraints. The mathematical concepts are similar to the E D problem. Communications and Control in Smart Grid Texas Tech University 52
53 References W. J. Wood and. F. Wollenberg, Power Generation, Operation, and Control, John Wiley & Sons, 2 nd Ed., 996. J. McCalley and L. Tesfatsion, "Power Flow Equations", Lecture Notes, EE 458, Department of Electrical and Computer Engineering, Iowa State University, Spring 200. Communications and Control in Smart Grid Texas Tech University 5
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