Topic 1: Basics of Power Systems ECE 5332: Communications and Control for Smart Spring 2012 A.H. Mohsenian‐Rad (U of T) Networking and Distributed Systems 1 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. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 2 Power Systems • Power Production: Different Types: Traditional Renewable Capacity, Cost, Carbon Emission Step‐up Transformers Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 3 Power Systems • Power Transmission: High Voltage (HV) Transmission Lines Several Hundred Miles Switching Stations Transformers Circuit Breakers Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 4 Power Systems • The Power Transmission Grid in the United States: www.geni.org Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 5 Power Systems • Major Inter‐connections in the United States: www.geni.org Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 6 Power Systems • Power Distribution: Medium Voltage (MV) Transmission Lines (< 50 kV) Power Deliver to Load Locations Interface with Consumers / Metering Distribution Sub‐stations Step‐Down Transformers Distribution Transformers Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 7 Power Systems • Power Consumption: Industrial Commercial Residential Demand Response Controllable Load Non‐Controllable Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 8 Power Systems Generation Dr. Hamed Mohsenian-Rad Transmission Distribution Communications and Control in Smart Grid Load Texas Tech University 9 Power Systems • Power System Control: Data Collection: Sensors, PMUs, etc. Decision Making: Controllers Actuators: Circuit Breakers, etc. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 10 Power Grid Graph Representation Nodes: Buses Links: Transmission Lines Generator Load Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 11 Power Grid Graph Representation Nodes: Buses Links: Transmission Lines Buses (Voltage) Generator Load Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 12 Power Grid Graph Representation Nodes: Buses Links: Transmission Lines Generator Load Transmission Lines (Power Flow, Loss) Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 13 Power Grid Graph Representation Nodes: Buses Links: Transmission Lines Load Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Consumers Generator Texas Tech University 14 Power Grid Graph Representation Nodes: Buses Links: Transmission Lines Generator 10 MW 3 MW Load 7 MW Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 15 Transmission Line Admittance • Admittance y is defined as the inverse of impedance z: z=r+jx (r: Resistance, x: Reactance) y=g+jb (g: Conductance, b: Susceptance) y=1/z Parameter g is usually positive Parameter b: Positive: Capacitor Negative: Inductor Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 16 Transmission Line Admittance • For the transmission line connecting bus i to bus k: Addmitance: yik Example: yik = 1 – j 4 (per unit) Note that, yii is denoted by yi and indicates: Susceptance for any shunt element (capacitor) to ground at bus i. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 17 Y-Bus Matrix • We define: Ybus = [ Yij ] where Diagonal Elements: Off‐diagonal Elements: Yii yi N y k 1, k i ik Yij yij Note that Ybas matrix depends on the power grid topology and the admittance of all transmission lines. N is the number of busses in the grid. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 18 Y-Bus Matrix • Example: For a grid with 4‐buses, we have: Ybus y1 y12 y13 y14 y21 y31 y41 y12 y2 y21 y23 y24 y13 y23 y32 y42 y3 y31 y32 y34 y43 y14 y24 y34 y4 y41 y42 y43 • After separating the real and imaginary parts: Ybus G j B Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 19 Bus Voltage • Let Vi denote the voltage at bus i: • Note that, Vi is a phasor, with magnitude and angle. Vi Vi i • In most operating scenarios we have: Vi V j Dr. Hamed Mohsenian-Rad i j Communications and Control in Smart Grid Texas Tech University 20 Power Flow Equations • Let Si denote the power injection at bus i: Si = Pi + j Qi Active Power • Generation Bus: Pi > 0 • Load Bus: Pi < 0 Dr. Hamed Mohsenian-Rad Reactive Power (negative power injection) Communications and Control in Smart Grid Texas Tech University 21 Power Flow Equations • Using Kirchhoff laws, AC Power Flow Equations become: Pk V k V j G kj cos( k j ) B kj sin( k j ) N j 1 Q k V k V j G kj sin( k j ) B kj cos( k j ) N j 1 • Do we know all notations here? • If we know enough variables, we can obtain the rest of variables by solving a system of nonlinear equations. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 22 Power Flow Equations • The AC Power Flow Equations are complicated to solve. • Next, we try to simplify the equations in three steps. • Step 1: For most networks, G << B. Thus, we set G = 0: Pk Vk V j Bkj sin( k j ) N j 1 Qk Vk V j Bkj cos( k j ) N j 1 Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 23 Power Flow Equations • Step 2: For most neighboring buses: i j 10 to 15 . • As a result, we have: Sin ( k j ) k j Cos ( k j ) 1 Pk Vk V j Bkj ( k j ) N j 1 Qk Vk V j Bkj N j 1 Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 24 Power Flow Equations • Step 3: In per‐unit, |Vi| is very close to 1.0 (0.95 to 1.05). • As a result, we have: Vi V j 1 . N Pk Bkj ( k j ) j 1 Qk Bkj Bkk Bkj bk N N j 1 j 1, jk • Pk has a linear model and Qk is almost fixed. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 25 Power Flow Equations • Step 3: In per‐unit, |Vi| is very close to 1.0 (0.95 to 1.05). • As a result, we have: Vi V j 1 . DC Power Flow Equations N Pk Bkj ( k j ) j 1 Qk Bkj Bkk Bkj bk N N j 1 j 1, jk • Pk has a linear model and Qk is almost fixed. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 26 Power Flow Equations • Given the power injection values at all buses, we can use N Pk Bkj ( k j ) j 1 to obtain the voltage angles at all buses. • Let Pij denote the power flow from bus i to bus j, we have: Pij Bij ( i j ) Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 27 Power Flow Equations • Example: Obtain power flow values in the following grid: P2g 2 pu P1g 2 pu y12 j10 y14 j10 y13 j10 P2l 1 pu y23 j10 y34 j10 P4g 1 pu Dr. Hamed Mohsenian-Rad P3l 4 pu Communications and Control in Smart Grid Texas Tech University 28 Power Flow Equations • First, we obtain the Y‐bus matrix: Ybus b1 b12 b13 b14 b21 j b31 b41 j Dr. Hamed Mohsenian-Rad b12 b2 b21 b23 b24 b32 b42 jB b13 b23 b3 b31 b32 b34 b43 B11 B j 21 B31 B41 B12 B22 B32 B42 Communications and Control in Smart Grid B13 B23 B33 B43 b14 b24 b34 b4 b41 b42 b43 B14 B24 B34 B44 Texas Tech University 29 Power Flow Equations • Next, we write the (active) power flow equations: P1 B12 B13 B14 1 B12 2 B13 3 B14 4 P2 B 21 1 B 21 B 23 B 24 2 B 23 3 B 24 4 P3 B31 1 B32 2 B31 B32 B34 3 B34 4 P4 B 41 1 B 42 2 B 43 3 B 41 B 42 B 43 4 • This can be written as: P1 B12 B13 B14 P B21 2 P3 B31 B41 P4 Dr. Hamed Mohsenian-Rad B12 B21 B23 B24 B13 B23 B32 B31 B32 B34 B42 B43 Communications and Control in Smart Grid 1 2 3 B34 B41 B42 B43 4 B14 B24 Texas Tech University 30 Power Flow Equations • From the last two slides, we finally obtain: 1 2 3 4 • Therefore, the voltage angles are obtained as: 1 2 3 4 Dr. Hamed Mohsenian-Rad 1 Communications and Control in Smart Grid Texas Tech University 31 Power Flow Equations • However, the last matrix in the previous slide is singular! • Therefore, we cannot take 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 1) as reference bus: 1 0 . Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 32 Power Flow Equations • We should also remove the corresponding rows/columns: 2 30 10 10 10 1 1 10 20 10 0 2 4 10 10 30 10 3 1 10 0 10 20 4 1 20 10 0 2 4 10 30 10 3 1 0 10 20 4 • The angular differences (with respect to 1 ): 2 20 10 0 10 30 10 3 4 0 10 20 Dr. Hamed Mohsenian-Rad 1 1 0.025 4 0.15 1 0.025 Communications and Control in Smart Grid 2 1 0.025 3 1 0.15 4 1 0.025 Texas Tech University 33 Power Flow Equations • Finally, the power flow values are calculated as: P12 B12 (1 2 ) 10(0 0.025) 0.25 P13 B13 (1 3 ) 10(0 0.15) 1.5 P14 B14 (1 4 ) 10(0 0.025) 0.25 P23 B23 ( 2 3 ) 10(0.025 0.15) 1.25 P34 B34 ( 3 4 ) 10(0.15 0.025) 1.25 P2g 2 pu P1g 2 pu 0.25 0.25 P2l 1 pu 1.5 1.25 1.25 P4g 1 pu Dr. Hamed Mohsenian-Rad P3l 4 pu Communications and Control in Smart Grid Texas Tech University 34 Power Flow Equations • What if the generator connected to bus 1 is renewable? • What if the capacity of transmission link (1,3) is 1 pu? • What if we can apply demand response to load bus 3? • What if one of the transmission lines fails? Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 35 Economic Dispatch Problem • In the example we discussed earlier, we had: Power Supply = Power Load • In particular, we had: P1g P2g P4g P2l P3l • However, generation levels P1g , P2g , and P4g assumed given. • Q: What if the generators have different generation costs? Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 36 Economic Dispatch Problem • For thermal power plants, generation cost is quadratic: Generation Cost = C(P) = a1 + a2 x P + a3 x P2 • Example: a grid with three power plants: C1(P1) = 561 + 7.92 x P1 + 0.001562 x (P1)2 150 MW ≤ P1 ≤ 600 MW C2(P2) = 310 + 7.85 x P2 + 0.001940 x (P2)2 100 MW ≤ P2 ≤ 400 MW C3(P3) = 78 + 7.97 x P3 + 0.004820 x (P3)2 50 MW ≤ P3 ≤ 200 MW • Each power plant has some min and max generation levels. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 37 Economic Dispatch Problem • For thermal power plants, generation cost is quadratic: Generation Cost = C(P) = a1 + a2 x P + a3 x P2 • Example: a grid with three power plants: C1(P1) = 561 + 7.92 x P1 + 0.001562 x (P1)2 150 MW ≤ P1 ≤ 600 MW C2(P2) = 310 + 7.85 x P2 + 0.001940 x (P2)2 100 MW ≤ P2 ≤ 400 MW C3(P3) = 78 + 7.97 x P3 + 0.004820 x (P3)2 50 MW ≤ P3 ≤ 200 MW • Each power plant has some min and max generation levels. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 38 Economic Dispatch Problem • For thermal power plants, generation cost is quadratic: Generation Cost = C(P) = a1 + a2 x P + a3 x P2 • Example: a grid with three power plants: C1(P1) = 561 + 7.92 x P1 + 0.001562 x (P1)2 150 MW ≤ P1 ≤ 600 MW C2(P2) = 310 + 7.85 x P2 + 0.001940 x (P2)2 100 MW ≤ P2 ≤ 400 MW C3(P3) = 78 + 7.97 x P3 + 0.004820 x (P3)2 50 MW ≤ P3 ≤ 200 MW • Each power plant has some min and max generation levels. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 39 Economic Dispatch Problem • For thermal power plants, generation cost is quadratic: Generation Cost = C(P) = a1 + a2 x P + a3 x P2 • Example: a grid with three power plants: C1(P1) = 561 + 7.92 x P1 + 0.001562 x (P1)2 150 MW ≤ P1 ≤ 600 MW C2(P2) = 310 + 7.85 x P2 + 0.001940 x (P2)2 100 MW ≤ P2 ≤ 400 MW C3(P3) = 78 + 7.97 x P3 + 0.004820 x (P3)2 50 MW ≤ P3 ≤ 200 MW • Each power plant has some min and max generation levels. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 40 Economic Dispatch Problem • For thermal power plants, generation cost is quadratic: Generation Cost = C(P) = a1 + a2 x P + a3 x P2 • Example: a grid with three power plants: C1(P1) = 561 + 7.92 x P1 + 0.001562 x (P1)2 150 MW ≤ P1 ≤ 600 MW C2(P2) = 310 + 7.85 x P2 + 0.001940 x (P2)2 100 MW ≤ P2 ≤ 400 MW C3(P3) = 78 + 7.97 x P3 + 0.004820 x (P3)2 50 MW ≤ P3 ≤ 200 MW • Each power plant has some min and max generation levels. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 41 Economic Dispatch Problem • We should select P1, P2, and P3 to: • Meet total load Pload = 850 MW • Minimize the total cost of generation • Economic Dispatch Problem: minimize CP1 CP2 CP3 P1 , P2 , P3 subject to 150 P1 600 100 P2 400 50 P3 200 P1 P2 P3 850 Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 42 Economic Dispatch Problem • Is the formulated problem a convex program? Why? • Convex programs can be solved efficiently. • An useful software is CVX for Matlab (http://cvxr.com/cvx). • The optimal economic dispatch solution: P1 = 393.2 MW P2 = 334.6 MW Q: Do they satisfy all constraints? P3 = 122.2 MW Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 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 (http://cvxr.com/cvx). • The optimal economic dispatch solution: P1 = 393.2 MW P2 = 334.6 MW Minimum Cost = 3916.6 + 3153.8 + 1123.9 = 8194.3 P3 = 122.2 MW Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 44 Economic Dispatch Problem • What if we have to satisfy topology constraints? P2 P1 P13 200 P3 2 20 10 0 10 30 10 3 4 0 10 20 Dr. Hamed Mohsenian-Rad 1 P2 450 400 P3 450 MW 400 MW P13 B13 (1 3 ) 10 3 20 3 20 Communications and Control in Smart Grid Texas Tech University 45 Economic Dispatch Problem • The same optimal solutions are still valid: P2 334.6 MW P1 393.2 MW 156.8 38.1 450 MW 41.4 198.3 160.3 P3 122.2 MW 2 15.685 19.830 3 4 3.805 Dr. Hamed Mohsenian-Rad 400 MW 3 20 Communications and Control in Smart Grid P13 200 Texas Tech University 46 Economic Dispatch Problem • The same optimal solutions are still valid: P2 334.6 MW P1 393.2 MW 156.8 38.1 What if P13 170 160.3 P3 122.2 MW 2 15.685 19.830 3 4 3.805 Dr. Hamed Mohsenian-Rad 450 MW 41.4 198.3 400 MW 3 20 Communications and Control in Smart Grid P13 200 Texas Tech University 47 Economic Dispatch Problem • Then the economic dispatch problem becomes: minimize CP1 CP2 CP3 P1 , P2 , P3 ,1 , 2 , 3 subject to 150 P1 600 100 P2 400 50 P3 200 P1 P2 P3 850 2 20 10 0 10 30 10 3 4 0 10 20 3 17 Dr. Hamed Mohsenian-Rad 1 Communications and Control in Smart Grid P2 450 400 P3 Texas Tech University 48 Economic Dispatch Problem • Then the economic dispatch problem becomes: minimize CP1 CP2 CP3 Still a Convex Program? P1 , P2 , P3 ,1 , 2 , 3 subject to 150 P1 600 100 P2 400 50 P3 200 P1 P2 P3 850 2 20 10 0 10 30 10 3 4 0 10 20 3 17 Dr. Hamed Mohsenian-Rad 1 Communications and Control in Smart Grid P2 450 400 P3 Texas Tech University 49 Economic Dispatch Problem • The new optimal solutions are obtained as: P2 400 MW P1 280 MW 110 0 450 MW 60 170 170 P3 170 MW 400 MW • The total generation cost becomes: $8,233.66 > $8,194.3 • Here, we had to sacrifice “cost” for “implementation”. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 50 Economic Dispatch Problem • The new optimal solutions are obtained as: P2 400 MW P1 280 MW 110 0 What if P13 150 450 MW 60 170 170 P3 170 MW 400 MW • The total generation cost becomes: $8,233.66 > $8,194.3 • Here, we had to sacrifice “cost” for “implementation”. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 51 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. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 52 References • W. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control, John Wiley & Sons, 2nd Ed., 1996. • J. McCalley and L. Tesfatsion, "Power Flow Equations", Lecture Notes, EE 458, Department of Electrical and Computer Engineering, Iowa State University, Spring 2010. Dr. Hamed Mohsenian-Rad Communications and Control in Smart Grid Texas Tech University 53