AI Power System Analysis and Design PDF

Chapter 2: Fundamentals This chapter reviews the fundamental concepts essential for understanding power system analysis. It covers phasors, power calculations (instantaneous, complex, real, reactive), network equations, three-phase circuits, and energy conversion. **Key Concepts and Equations:**...

Chapter 2: Fundamentals This chapter reviews the fundamental concepts essential for understanding power system analysis. It covers phasors, power calculations (instantaneous, complex, real, reactive), network equations, three-phase circuits, and energy conversion. **Key Concepts and Equations:** **2.1 Phasors:** - **Sinusoidal Quantities:** Representing sinusoidal voltages and currents as phasors simplifies calculations in AC circuits. - **RMS Phasor Representation:** - V = V∠δ = Vcos(δ) + jVsin(δ) (Rectangular) - V = Ve\^(jδ) (Exponential) - V = V∠δ (Polar)\ where: - V is the RMS value. - δ is the phase angle. - **Euler\'s Identity:** e\^(jφ) = cos(φ) + jsin(φ) **2.2 Instantaneous Power in Single-Phase AC Circuits:** - **Instantaneous Power:** p(t) = v(t)i(t) - **Resistive Load:** - p(t) = VI\[1 + cos(2(ωt + δ))\] - P = VI = V²/R = I²R (Average Power) - **Inductive Load:** - p(t) = VI sin(2(ωt + δ)) - P = 0 (Average Power) - **Capacitive Load:** - p(t) = -VI sin(2(ωt + δ)) - P = 0 (Average Power) - **RLC Load:** - p(t) = VIcos(δ - β){1 + cos\[2(ωt + δ)\]} + VIsin(δ - β)sin\[2(ωt + δ)\] - P = VIcos(δ - β) (Average Power) **2.3 Complex Power:** - **Complex Power:** S = VI\* = P + jQ\ where:\ \* S is complex power (VA).\ \* V is RMS voltage (V).\ \* I\* is the complex conjugate of RMS current (A).\ \* P is real power (W).\ \* Q is reactive power (var). - **Apparent Power:** \|S\| = VI (VA) - **Power Factor:** p.f. = cos(δ - β) = P/\|S\| - **Power Triangle:** A graphical representation of the relationship between S, P, and Q. - S = √(P² + Q²) - θ = δ - β = tan⁻¹(Q/P) (Power factor angle) - **Power Factor Correction:** Adding a capacitor in parallel with an inductive load to improve the power factor. **2.4 Network Equations:** - **Nodal Analysis:** A method for analyzing circuits by writing KCL equations at each node. - **Bus Admittance Matrix (Y):** A matrix that relates the bus voltages to the injected currents. - Diagonal elements (Ykk): Sum of admittances connected to bus k. - Off-diagonal elements (Ykn): Negative sum of admittances connected between buses k and n. - **Nodal Equations:** YV = I\ where:\ \* Y is the bus admittance matrix.\ \* V is the vector of bus voltages.\ \* I is the vector of injected currents. **2.5 Balanced Three-Phase Circuits:** - **Y-Connection:** Neutrals of each phase are connected. - **Line-to-Neutral Voltage (VLN):** Voltage between a line and the neutral. - **Line-to-Line Voltage (VLL):** Voltage between two lines. VLL = √3 VLN∠30° (for positive sequence) - **Line Current (IL):** Current flowing through a transmission line. - **Delta (Δ)-Connection:** Phases are connected in a closed loop. - **Δ-Y Conversion:** A balanced Δ-load can be converted to an equivalent Y-load by dividing the Δ-load impedance by 3: Zy = ZΔ/3 - **Equivalent Line-to-Neutral Diagram:** A simplified representation of a balanced three-phase circuit using only one phase. **2.6 Power in Balanced Three-Phase Circuits:** - **Instantaneous Power (Three-Phase):** p3φ(t) = P3φ = √3VLLIL cos(δ - β) = 3VLNIL cos(δ - β) (Constant under balanced conditions) - **Complex Power (Three-Phase):** S3φ = √3VLLIL∠(δ - β) = 3VLNIL∠(δ - β) = P3φ + jQ3φ\ where:\ \* S3φ is total three-phase complex power.\ \* P3φ is total three-phase real power. P3φ = √3VLLILcos(θ)\ \* Q3φ is total three-phase reactive power. Q3φ = √3VLLILsin(θ) **2.7 Advantages of Balanced Three-Phase versus Single-Phase Systems:** - **Reduced Conductor Material:** Three-phase systems require fewer conductors compared to equivalent single-phase systems. - **Constant Power:** Balanced three-phase systems deliver constant power, reducing vibrations and stress on equipment. - **Higher Power Transfer:** For the same amount of conductor material, three-phase systems can transmit more power. **2.8 Energy Conversion:** - **Non-Electric to Electric:** Power generation (fuel, wind, hydro, solar). - **Electric to Non-Electric:** Loads (mechanical motion, light, heat). - **Electric to Electric:** - Transformers (voltage change). - Rectifiers (AC to DC). - Inverters (DC to AC). - DC-DC Converters (DC voltage change). - AC-AC Variable Frequency Converters (frequency change). **Chapter 3: Power Transformers** **Introduction:** Power transformers are essential components of power systems, enabling efficient transmission by stepping voltage up for transmission and down for distribution and utilization. Their efficiency is remarkably high, approaching 100% in modern designs. This chapter covers transformer theory, equivalent circuits, and various transformer types. **3.1 The Ideal Transformer:** - **Assumptions:** Ideal transformers are simplified models with the following characteristics: - Zero winding resistance (no I²R losses). - Infinite core permeability (zero reluctance). - No leakage flux (all flux links both windings). - No core losses. - **Key Equations:** - N₁I₁ = N₂I₂ (Ampere-turns balance) - E₁/E₂ = N₁/N₂ (Voltage ratio) - S₁ = S₂ (Complex power conservation) - Z₁\' = a²Z₂ (Impedance reflection, where a = N₁/N₂) - **Dot Convention:** Dots are used to indicate winding polarity. Current entering the dotted terminal of one winding induces a voltage with positive polarity at the dotted terminal of the other winding. **3.2 Equivalent Circuits for Practical Transformers:** - **Deviations from Ideal:** Practical transformers exhibit: - Winding resistance (R₁, R₂). - Leakage reactance (X₁, X₂). - Finite core permeability (magnetizing current, Im). - Core losses (core loss current, Ic). - **Equivalent Circuit:** A practical transformer\'s equivalent circuit includes these elements, referred to either the primary or secondary side. Exciting current (Ie = Ic + Im) is represented by a shunt branch. - **Short-Circuit and Open-Circuit Tests:** These tests determine the equivalent circuit parameters. - Short-circuit test: Measures series impedance (Req, Xeq). - Open-circuit test: Measures shunt admittance (Gc, Bm). **3.3 The Per-Unit System:** - **Definition:** Expresses quantities as a ratio to base values, simplifying calculations and eliminating ideal transformer windings in per-unit circuits. - **Base Quantities:** Sbase (power) and Vbase (voltage) are chosen. Other base values are derived from these. - Ibase = Sbase/Vbase - Zbase = Vbase²/Sbase - Ybase = 1/Zbase - **Per-Unit Calculation:** Per-unit value = (Actual value)/(Base value). - **Transformer Relations in Per-Unit:** In per-unit, with appropriate base value selection, ideal transformer relations simplify to: - E₁p.u. = E₂p.u. - I₁p.u. = I₂p.u. **3.4 Three-Phase Transformer Connections and Phase Shift:** - **Connections:** Single-phase transformers can be connected in Y-Y, Y-Δ, Δ-Y, or Δ-Δ configurations. - **Phase Shift:** Y-Δ and Δ-Y connections introduce a 30° phase shift between primary and secondary line-to-neutral voltages. The American standard specifies that high-voltage quantities lead low-voltage quantities by 30°. **3.5 Per-Unit Equivalent Circuits of Balanced Three-Phase Transformers:** - **Simplification:** Per-unit circuits for three-phase transformers are similar to single-phase equivalents, with phase shift represented by an ideal phase-shifting transformer in Y-Δ and Δ-Y connections. **3.6 Three-Winding Transformers:** - **Equivalent Circuit:** Has three series impedances (Z₁, Z₂, Z₃) connected at a common node. These impedances are derived from short-circuit tests. **3.7 Autotransformers:** - **Construction:** A single winding tapped to provide both primary and secondary connections. - **Advantages:** Smaller size, lower weight, higher efficiency, lower cost (for moderate turns ratios) compared to two-winding transformers. - **Disadvantages:** Direct electrical connection between primary and secondary (potential for transient overvoltage transfer). **3.8 Transformers with Off-Nominal Turns Ratios:** - **Definition:** Occurs when the transformer voltage ratings are not proportional to the chosen base voltages. - **Modeling:** Can be represented using an ideal transformer in series with the leakage impedance or by an equivalent π-network. **Key Takeaways:** This chapter lays the groundwork for understanding and analyzing power transformers in power system studies. The per-unit system and equivalent circuits are fundamental tools for simplifying calculations and capturing the behavior of practical transformers. **Chapter 4: Transmission Line Parameters** **Introduction:** This chapter delves into the electrical characteristics of transmission lines, which are crucial for power system analysis. These characteristics are modeled as distributed parameters: resistance, inductance, capacitance, and conductance. The chapter focuses primarily on overhead lines. **4.1 Transmission Line Design Considerations:** Overhead transmission lines consist of conductors, insulators, support structures, and shield wires. Design involves optimizing various factors: - **Conductors:** Aluminum (often ACSR) is commonly used due to cost and weight advantages. Bundled conductors are used in EHV lines to reduce corona. - **Insulators:** Typically suspension-type disc insulators made of porcelain or glass. The number of discs increases with voltage. - **Support Structures:** Lattice steel towers for EHV, wood frames for lower voltages. - **Shield Wires:** Protect against lightning strikes. - **Electrical Factors:** Conductor size and bundling for thermal capacity and corona control. Insulator selection and clearances for insulation. Shield wire placement for lightning protection. Conductor spacing affects impedance and admittance. - **Mechanical Factors:** Conductor strength for ice and wind loading. Insulator strength. Tower design for load and conductor breakage. Vibration damping. - **Environmental Factors:** Land usage, visual impact, and biological effects of electric and magnetic fields. - **Economic Factors:** Minimizing total cost (installation + losses). **4.2 Resistance:** - **DC Resistance:** R\_dc = ρl/A, where ρ is resistivity, l is length, and A is cross-sectional area. Units: Ω-m (SI), Ω-cmil/ft (English). Circular mils (cmil) are used for area in English units. - **AC Resistance (Rac):** Higher than R\_dc due to skin effect (current crowding at higher frequencies). Calculated from power loss and rms current: Rac = P\_loss / I². - **Temperature Effect:** Resistivity varies linearly with temperature: ρ\_T2 = ρ\_T1\[ (T₂ + T) / (T₁ + T) \], where T is a temperature constant. - **Spiraling:** Stranded conductors have slightly higher resistance due to longer path length. **4.3 Conductance:** Represents leakage current losses at insulators and corona losses. Usually negligible for overhead lines. **4.4 Inductance: Solid Cylindrical Conductor:** - **Internal Inductance (Lint):** Due to flux within the conductor itself. Lint = μ₀ / 8π H/m. - **External Inductance:** Due to flux outside the conductor. For flux between distances D₁ and D₂: L\_12 = (μ₀ / 2π) ln(D₂/D₁) H/m. - **Total Inductance (Lp):** Lp = (μ₀ / 2π) ln(D/r\') H/m, where r\' = 0.7788r (r is conductor radius, D is distance to external point). - **Flux Linkage (λ\_p):** λ\_p = 2 x 10⁻⁷ I ln(D/r\') Wb-t/m. - **Array of Conductors:** For M conductors with currents I\_m summing to zero, flux linkage for conductor k is: λ\_k = 2 x 10⁻⁷ Σ(I\_m) ln(1/D\_km) Wb-t/m. **4.5 Inductance: Single-Phase and Three-Phase Lines with Equal Spacing:** - **Single-Phase Two-Wire Line:** L = (μ₀ / π) ln(D/r\') H/m per conductor (D is conductor spacing, r\' is GMR). Total loop inductance is 2L. - **Three-Phase Three-Wire Line:** L\_a = 2 x 10⁻⁷ ln(D/r\') H/m per phase (D is phase spacing). **4.6 Inductance: Composite Conductors, Unequal Spacing, Bundled Conductors:** - **Composite Conductors (GMR and GMD):** - GMR (Geometric Mean Radius): Represents the effective radius of a composite conductor. - GMD (Geometric Mean Distance): Represents the effective distance between conductors. - **Unequal Spacing (Transposition):** Transposition balances inductance by rotating phase positions. Equivalent spacing (D\_eq) is used: D\_eq = (D₁₂D₂₃D₃₁)¹/₃. - **Bundled Conductors:** Reduces electric field strength and reactance. GMR of bundle (D\_SL) is calculated based on the number of subconductors and their spacing. **4.7 Series Impedances: Three-Phase Line with Neutral Conductors and Earth Return:** - **Earth Return:** Earth can be modeled as image conductors carrying the opposite current of overhead conductors. - **Series Impedance Matrix (Z\_p):** Accounts for resistance, self-inductance, and mutual inductance between all conductors (including earth return). - **Simplified Circuit:** Z\_p is reduced to a 3x3 matrix for balanced three-phase operation. **4.8 Electric Field and Voltage: Solid Cylindrical Conductor:** - **Electric Field (Ex):** Ex = q / (2πεx) V/m (q is charge per unit length, ε is permittivity, x is distance). - **Voltage (V\_12):** V\_12 = (q / 2πε) ln(D₂/D₁) volts (between points at distances D₁ and D₂). **4.9 Capacitance: Single-Phase and Three-Phase Lines with Equal Spacing:** - **Single-Phase Two-Wire Line:** C\_xy = πε / ln(D/r) F/m (line-to-line). C\_n = 2C\_xy F/m (line-to-neutral with grounded center tap). - **Three-Phase Three-Wire Line:** C\_an = 2πε / ln(D/r) F/m (line-to-neutral). **4.10 Capacitance: Stranded Conductors, Unequal Spacing, Bundled Conductors:** Similar to inductance calculations, using GMR and D\_eq for stranded conductors and transposed lines. Bundled conductors increase capacitance. **4.11 Shunt Admittances: Lines with Neutral Conductors and Earth Return:** - **Shunt Admittance Matrix (Y\_p):** Accounts for capacitance between all conductors (including earth return). Calculated using the method of images. - **Simplified Circuit:** Y\_p is reduced to a 3x3 matrix for balanced three-phase operation. **4.12 Electric Field Strength at Conductor Surfaces and at Ground Level:** - **Conductor Surface:** E = q / (2π εr) V/m (r is conductor radius). For bundled conductors, E\_ave and E\_max are calculated. - **Ground Level:** The electric field at ground level is calculated by summing the contributions from all overhead conductors and their images. **4.13 Parallel Circuit Three-Phase Lines:** - **Mutual Coupling:** Parallel lines have mutual inductance and capacitance. - **Equivalent Circuits:** Z\_peq and Y\_peq matrices are derived to represent the combined impedance and admittance of parallel lines. **Chapter 5: Transmission Lines: Steady-State Operation** This chapter delves into the performance analysis of transmission lines under normal, balanced, steady-state conditions. It covers single-phase and three-phase lines, considering the distributed nature of line parameters. **5.1 Medium and Short Line Approximations** - **Two-Port Network Representation:** Transmission lines are conveniently represented as two-port networks, characterized by ABCD parameters: - Vs = AVR + BIR - Vs, Is: Sending-end voltage and current - VR, IR: Receiving-end voltage and current - A, B, C, D: Complex parameters depending on line constants (R, L, C, G) - **Short Line Approximation (Lines \< 25 km):** Neglects shunt admittance. - A = D = 1 - B = Z (total series impedance) - **Medium Line Approximation (25 km \< Lines \< 250 km):** Uses a nominal π circuit, lumping half the shunt capacitance at each end. - A = D = 1 + (YZ/2) - B = Z - Z: Total series impedance - Y: Total shunt admittance - **Voltage Regulation:** Quantifies voltage change at the receiving end as load varies. - VRNL: No-load receiving-end voltage - VRFL: Full-load receiving-end voltage - **Line Loadability:** Considers thermal limits, voltage drop limits, and steady-state stability limits. Short lines are usually limited by thermal ratings. **Example 5.1 (Medium Line):** This example calculates ABCD parameters, sending-end quantities, voltage regulation, and thermal limits for a 345-kV, 200-km line. It illustrates how voltage drop can limit the line\'s capacity. **5.2 Transmission-Line Differential Equations** This section derives the exact transmission line equations, accounting for the distributed nature of line parameters. - **Differential Equations:** - dV(x)/dx = zI(x) - z: Series impedance per unit length - y: Shunt admittance per unit length - x: Distance from receiving end - **Solutions:** Solving these equations yields expressions for voltage and current at any point x along the line, involving hyperbolic functions: - V(x) = cosh(γx)VR + Zc sinh(γx)IR - γ: Propagation constant = sqrt(zy) - Zc: Characteristic impedance = sqrt(z/y) - **ABCD Parameters (Long Line):** - A = D = cosh(γl) - B = Zc sinh(γl) - l: Line length **Example 5.2 (Long Line):** This example calculates exact ABCD parameters for a 765-kV, 300-km line. **5.3 Equivalent π Circuit** - **Equivalent π Circuit:** Provides a circuit representation with modified parameters Z\' and Y\' that match the ABCD parameters of the distributed line. - Z\' = Zc sinh(γl) **Example 5.3:** Calculates and compares the equivalent and nominal π circuits for the long line in the previous example. **5.4 Lossless Lines** Simplifies analysis by neglecting line losses (R = G = 0). - **Surge Impedance:** Zc = sqrt(L/C) (purely resistive) - **ABCD Parameters (Lossless):** - A = D = cos(βl) - B = jZc sin(βl) - β: Phase constant = ωsqrt(LC) - **Wavelength:** λ = 2π/β - **Surge Impedance Loading (SIL):** Power delivered to a load equal to the surge impedance. SIL = Vrated\^2 / Zc. At SIL, the voltage profile is flat. **Example 5.4:** Calculates the theoretical steady-state stability limit (maximum power transfer) for a lossless line. **5.5 Maximum Power Flow (Lossy Lines)** Derives equations for maximum power flow, considering losses. **Example 5.5:** Calculates the theoretical maximum power that a lossy line can deliver. **5.6 Line Loadability (Practical Considerations)** Discusses practical line loadability, considering voltage drop and stability limits. Practical loadability is typically lower than the theoretical maximum. **Example 5.6:** Calculates practical line loadability and voltage regulation, demonstrating the impact of voltage drop on a long line. Example 5.7 explores how increasing voltage levels reduces the number of transmission lines needed. Example 5.8 discusses the use of intermediate substations. **5.7 Reactive Compensation Techniques** - **Shunt Reactors/Capacitors:** Used to control voltage and reactive power flow. - **Series Capacitors:** Increase line loadability by reducing series impedance. - **Static Var Compensators (SVCs):** Provide dynamic reactive power control. **Examples 5.9 and 5.10:** Illustrate the effects of shunt and series compensation on voltage regulation and line loadability, respectively. **Case Study: Xcel Energy and SVCs** The case study demonstrates the practical application of SVCs to enhance grid stability and voltage control. **6.1-6.11 Power Flows and Solution Methods** These sections delve into solving the power flow problem, which involves finding voltage magnitudes and angles at all buses in a power system. Key concepts include: - **Gauss Elimination:** A direct method for solving linear algebraic equations. - **Iterative Methods (Jacobi, Gauss-Seidel):** Iterative approaches for solving linear equations. - **Newton-Raphson Method:** An iterative method for solving nonlinear equations, commonly used for power flow. - **Fast Decoupled Power Flow:** An approximation of the Newton-Raphson method for faster solutions. - **DC Power Flow:** A further simplification for approximate real power flow analysis. - **Sparsity Techniques:** Methods to exploit the sparsity of power system matrices for efficient computation. - **Power Flow Modeling of Wind and Solar Generation:** Discusses how different wind turbine and solar PV models are incorporated into the power flow. - **Realistic and Large-Scale Power Flow Models:** Addresses more advanced power flow complexities. **Chapter 6: Power Flows** This chapter introduces the power flow (or load flow) analysis, a fundamental tool in power system operation and planning. It determines the steady-state operating point of a balanced three-phase power system, calculating voltage magnitudes and angles at all buses, as well as real and reactive power flows. **Learning Objectives** (already summarized in the initial response) **Successful Power System Operation Requirements** (already discussed in the initial response) **6.1 Direct Solutions to Linear Algebraic Equations: Gauss Elimination** - **Matrix Representation:** Linear algebraic equations, crucial for power flow solutions, are represented in matrix form: - **Gauss Elimination:** A direct method to solve for x by transforming A into an upper triangular matrix and then using back substitution. This is commonly used within the Newton-Raphson power flow to solve the update equation. **Example 6.1:** Solves a 2x2 system of equations using Gauss elimination.\ **Example 6.2:** Triangularizes a 3x3 matrix using Gauss elimination. **6.2 Iterative Solutions to Linear Algebraic Equations: Jacobi and Gauss-Seidel** - **Iterative Approach:** An alternative to direct solutions, involving repeated calculations until convergence. - **Jacobi Method:** Uses \"old\" values of x in each iteration to compute \"new\" values. All values are updated simultaneously. - **Gauss-Seidel Method:** Updates x values sequentially within each iteration, using the most recently calculated values. This often converges faster than Jacobi. **Example 6.3:** Solves the same 2x2 system as Example 6.1 using the Jacobi method.\ **Example 6.4:** Solves the same system using Gauss-Seidel, showing faster convergence.\ **Example 6.5:** Demonstrates divergence of Gauss-Seidel for a specific matrix. **6.3 Iterative Solutions to Nonlinear Algebraic Equations: Newton-Raphson** - **Nonlinear Equations:** Power flow equations are nonlinear. - **Newton-Raphson Method:** An iterative method to solve nonlinear equations, involving the Jacobian matrix of partial derivatives. **Example 6.6:** Solves a simple nonlinear equation (x² = 9) using Newton-Raphson and compares it with extended Gauss-Seidel.\ **Example 6.7:** Solves a 2x2 system of nonlinear equations using Newton-Raphson.\ **Example 6.8:** Illustrates the four steps of the Newton-Raphson method. **6.4 The Power Flow Problem** - **Bus Variables:** Each bus k has four associated variables: voltage magnitude (Vk), phase angle (δk), net real power (Pk), and net reactive power (Qk). - **Bus Types:** - **Swing Bus (Slack Bus):** Reference bus with known V and δ (usually V = 1.0 p.u., δ = 0°). The power flow solves for P and Q at the swing bus. - **Load (PQ) Bus:** P and Q are known; V and δ are unknown. - **Voltage Controlled (PV) Bus:** P and V are known; Q and δ are unknown. Often generator buses. - **Ybus Matrix:** The bus admittance matrix, formed from line and transformer data, is crucial for power flow calculations. **Example 6.9:** Illustrates power flow input data and Ybus formation for a 5-bus system. **6.5 Power Flow Solution by Gauss-Seidel** Applies the Gauss-Seidel iterative method to solve the power flow equations. **Example 6.10:** Demonstrates one iteration of Gauss-Seidel for the 5-bus system. **6.6 Power Flow Solution by Newton-Raphson** Applies the Newton-Raphson method, which is the most widely used approach for power flow solutions due to its strong convergence characteristics. **Example 6.11:** Calculates Jacobian elements and power mismatches for the 5-bus system.\ **Example 6.12:** Uses a power flow program to demonstrate changing generation within limits.\ **Example 6.13:** Analyzes a larger, 37-bus system with different voltage levels.\ **Example 6.14:** Shows the effect of shunt capacitor banks on the power system.\ **Example 6.15:** Demonstrates how to remove overloads by redispatching generation.\ **Example 6.16:** Visualizes sparsity in Ybus and Jacobian matrices for a 37-bus system.\ **Example 6.17:** Solves a DC power flow example. **6.7 Control of Power Flow** Discusses methods to control power flow, including: - Prime mover and excitation control of generators - Switching of shunt capacitors/reactors and static var systems - Control of tap-changing transformers **6.8 Sparsity Techniques** Explains how the sparsity of the Ybus and Jacobian matrices can be exploited for computational efficiency. **6.9 Fast Decoupled Power Flow** Presents the fast decoupled power flow, an approximation of the Newton-Raphson method that provides faster solutions, particularly useful for contingency analysis. **6.10 The \"DC\" Power Flow** Introduces the DC power flow, a simplified linear approximation for analyzing real power flows. **6.11 Power Flow Modeling of Wind and Solar Generation** Discusses the incorporation of wind and solar generation models into power flow analysis. **6.12 Realistic and Large-Scale Power Flow Models** Addresses complexities encountered in practical power flow models, such as control actions, discrete control devices, and area interchange. **Example 6.18:** Shows a more realistic 37-bus system with area interchange, phase shifters, and other realistic modeling features. **Chapter 7: Power System Economics and Optimization** This chapter explores the economic aspects of power system operation and planning, focusing on minimizing generation costs while meeting demand and satisfying operational constraints. **Learning Objectives** (already summarized in the initial response) **7.1 Generator and Load Economics** - **Generator Costs:** - **Fixed Costs:** Capital costs, maintenance, etc. (not considered in economic dispatch). - **Variable Costs:** Primarily fuel costs, often represented as a function of power output (C(P)). - **Incremental Heat Rate:** Measures fuel input per unit of electrical output (BTU/kWh). Its reciprocal indicates fuel efficiency. - **Incremental Operating Cost (IC):** dC/dP, representing the cost to generate an additional MWh. - **Load Characteristics:** - **Elasticity of Demand:** Measures how much load changes in response to price changes. - **Energy Storage:** Allows shifting of energy consumption from peak to off-peak periods, improving system economics and facilitating integration of renewables. - **Area Transactions:** Considers power imports and exports between interconnected areas. - **Transmission System Considerations:** - **Transmission Losses:** Losses increase with generation and depend on the location and dispatch of generators. - **Transmission Constraints:** Line and transformer limits, voltage limits, and reactive power balance. **7.2 Economic Dispatch** - **Objective:** Minimize total variable cost (CT) while meeting load demand (PT). - **Unconstrained Economic Dispatch:** Assumes no generator limits and neglects losses. The optimal solution is when all generators operate at equal incremental cost (λ). - **Effect of Inequality Constraints (Generator Limits):** When limits are reached, units are held at their limits, and the remaining units operate at equal incremental cost. - **Effect of Transmission Losses:** Losses (PL) are included in the power balance equation: - **Loss Coefficients (B Coefficients):** Used to approximate transmission losses as a quadratic function of generator outputs. **Example 7.1:** Solves the economic dispatch problem neglecting generator limits and line losses.\ **Example 7.2:** Includes generator limits in the economic dispatch.\ **Example 7.3:** Uses PowerWorld Simulator to illustrate economic dispatch with generator limits.\ **Example 7.4:** Includes transmission losses in the economic dispatch calculation.\ **Example 7.5:** PowerWorld Simulator case with losses. **7.3 Optimal Power Flow (OPF)** - **Objective:** Minimize generation costs while satisfying various constraints (including transmission constraints). - **DC OPF (DCOPF):** Uses the linearized DC power flow, neglecting reactive power and assuming constant voltage magnitudes. Often formulated as a linear program. - **AC OPF (ACOPF):** Uses the full AC power flow equations, including voltage magnitudes and angles. More accurate but computationally more complex. - **Security-Constrained OPF (SCOPF):** Considers contingencies (e.g., N-1 security) to ensure system security under various operating conditions. **Example 7.6:** Formulates a DCOPF problem as a linear program.\ **Example 7.7:** Illustrates ACOPF using PowerWorld Simulator, showing the impact of line constraints on LMPs. **7.4 Unit Commitment and Longer Term Optimization** - **Unit Commitment:** Determines the on/off status of generators over a longer time horizon (e.g., a day). - **Solution Methods:** Dynamic programming, Lagrangian relaxation, mixed-integer programming (MIP). - **Longer-Term Optimization:** Considers factors like generator maintenance, fuel supply limitations, and seasonal variations in renewable resources. **7.5 Markets** Discusses the organization and operation of electricity markets, including: - **Wholesale Electricity Markets:** Competitive markets where generators bid to sell energy. - **Day-Ahead Market (DAM):** A forward market for scheduling generation a day in advance. - **Real-Time Market (RTM):** Balances real-time deviations from the day-ahead schedule. - **Locational Marginal Prices (LMPs):** Reflect the cost of delivering energy to a specific location. - **Ancillary Services:** Additional services provided by generators for system reliability (e.g., reserves, regulation). - **Financial Transmission Rights (FTRs):** Financial instruments to hedge against transmission congestion. - **Forward Capacity Market (FCM):** Ensures long-term resource adequacy. **Chapter 8: Symmetrical Faults** This chapter deals with the analysis of symmetrical (three-phase) faults in power systems. These faults, though less common than unsymmetrical faults, result in the largest fault currents and are therefore crucial for designing and setting protective devices. **Learning Objectives** (already summarized in the initial response) **8.1 Series R-L Circuit Transients** This section reviews the transient behavior of an R-L series circuit subjected to a sinusoidal voltage source, which serves as a simplified model for fault analysis. - **Circuit Equation:** - **Fault Current Components:** - **Symmetrical (AC) Component (iac):** A steady-state sinusoidal current. - **DC Offset Component (idc):** A decaying exponential current due to initial conditions. - **Total (Asymmetrical) Current (i):** The sum of the AC and DC components. - **RMS Fault Current:** - **RMS Symmetrical Fault Current (Iac):** Iac = V/Z - **RMS Asymmetrical Fault Current (Irms):** A time-varying RMS value due to the decaying DC component. Often approximated using an asymmetry factor K(τ): **Example 8.1:** Calculates fault currents (AC, momentary, and interrupting) for an R-L circuit. **8.2 Three-Phase Short Circuit---Unloaded Synchronous Machine** This section analyzes a three-phase fault at the terminals of an unloaded synchronous machine. - **Reactances:** The fault current behavior is determined by various machine reactances: - Subtransient reactance (X\"): The initial, very short duration reactance. - Transient reactance (X\'): The reactance during the first few cycles following the initial subtransient period. - Synchronous reactance (Xd): The steady-state reactance. - **Fault Current Components (AC):** The AC fault current has decaying components related to the different reactances. - **DC Offset Current:** Similar to the R-L circuit, a DC offset current exists, decaying with time. - **RMS Fault Current:** Similar to the R-L circuit case, except that the AC component also varies with time. **Example 8.2:** Calculates subtransient fault current, DC offset, and RMS asymmetrical fault current for an unloaded synchronous generator. **8.3 Power System Three-Phase Short Circuits** This section extends the analysis to three-phase faults in a power system. - **Assumptions:** Several simplifying assumptions are made (neglecting resistances, shunt admittances, load impedances, etc.) for easier calculation. These are not always valid, particularly at lower voltages. - **Superposition:** The superposition theorem is used to calculate fault currents. The fault is represented by two opposing voltage sources, and the system is analyzed in two steps: 1. All voltage sources are shorted except for the prefault voltage at the fault location. 2. The prefault voltage sources are reinserted, and the prefault load current is considered. **Example 8.3:** Calculates fault currents in a simple power system with a generator and a motor. **8.4 Bus Impedance Matrix** - **Zbus:** The bus impedance matrix is a powerful tool for calculating fault currents in multi-bus power systems. It relates bus voltages to injected currents. - **Fault Current Calculation:** The fault current at bus n can be calculated using: - **Voltage During Fault:** The voltage at any bus k during the fault can be calculated using: - **Rake Equivalent:** A circuit representation using Zbus elements to visualize fault current distribution. **Example 8.4:** Calculates fault currents and bus voltages using the Zbus matrix. Example 8.5 expands this using PowerWorld software. Example 8.6 continues this to explore changes with a new transmission line. **8.5 Circuit Breaker and Fuse Selection** - **Circuit Breakers:** Mechanical switches capable of interrupting fault currents. They have ratings for: - Voltage - Continuous current - Short-circuit current - Interrupting time - **E/X Simplified Method:** A simplified method for selecting circuit breakers. - **Fuses:** Simpler overcurrent devices that melt and clear faults. Fuses are inexpensive but must be replaced after operation. Current-limiting fuses can interrupt large fault currents before they reach their peak. **Example 8.7:** Illustrates circuit breaker selection using known X/R ratios.\ **Example 8.8:** Circuit breaker selection with unknown X/R ratios. **Chapter 9: Symmetrical Components** This chapter introduces symmetrical components, a powerful technique developed by C. L. Fortescue for analyzing unbalanced three-phase systems. The core idea is to transform phase components into a new set of components called *symmetrical components*. This transformation simplifies analysis because, in balanced systems, the sequence networks (equivalent circuits for symmetrical components) become uncoupled, and in unbalanced systems, they\'re only connected at points of unbalance. **9.1 Definition of Symmetrical Components** Any set of three-phase voltages (Va, Vb, Vc) can be decomposed into three sets of sequence components: 1. **Zero-Sequence Components (V0):** Three phasors with equal magnitudes and zero phase displacement. 2. **Positive-Sequence Components (V1):** Three phasors with equal magnitudes, 120° phase displacement, and positive sequence (a-b-c). 3. **Negative-Sequence Components (V2):** Three phasors with equal magnitudes, 120° phase displacement, and negative sequence (a-c-b). **Transformation Equations:** The transformation is defined using the operator \'a\' (a = 1∠120°): - **Phase to Sequence:** - \[V0\] \[ 1 1 1 \] \[Va\] - \[V1\] = \[ 1 a² a \] \[Vb\] - **Sequence to Phase:** - \[Va\] \[ 1 1 1 \] \[V0\] - \[Vb\] = \[ 1 a a²\] \[V1\] These transformations can also be applied to currents (Ia, Ib, Ic and I0, I1, I2). A key relationship is that the neutral current (In) is three times the zero-sequence current: In = 3I0. **Examples 9.1 - 9.3:** These examples demonstrate the calculation of sequence components for balanced and unbalanced voltage and current sets. **9.2 Sequence Networks of Impedance Loads** This section develops the sequence networks for impedance loads. For a balanced Y-load with impedance Zy and neutral impedance Zn: - **Zero-Sequence Impedance (Z0):** Z0 = Zy + 3Zn - **Positive-Sequence Impedance (Z1):** Z1 = Zy - **Negative-Sequence Impedance (Z2):** Z2 = Zy The sequence networks are uncoupled for balanced loads. **Example 9.4:** This example shows how to draw and analyze sequence networks for balanced Y and Δ loads. **9.3 Sequence Networks of Series Impedances** This section describes sequence networks for series impedances (like transmission lines or transformers): - **Zero-Sequence Impedance (Z0):** Z0 = Zaa + 2Zab (where Zaa is self-impedance, Zab is mutual impedance) - **Positive/Negative-Sequence Impedance (Z1=Z2):** Z1 = Z2 = Zaa - Zab Again, the sequence networks are uncoupled for symmetrical series impedances. **9.4 Sequence Networks of Three-Phase Lines** The series impedance and shunt admittance matrices of transmission lines can be transformed to the sequence domain. For transposed lines, the sequence impedance matrix is diagonal, leading to uncoupled sequence networks. **9.5 Sequence Networks of Rotating Machines** The sequence impedances of rotating machines (generators and motors) are typically *not* equal: - **Z0:** Zero-sequence impedance (due to leakage and other effects). - **Z1:** Positive-sequence impedance (synchronous, transient, or subtransient). - **Z2:** Negative-sequence impedance. **Example 9.5:** This example illustrates fault current calculations using sequence networks for a simple system. **9.6 Per-Unit Sequence Models of Three-Phase, Two-Winding Transformers** This section presents per-unit sequence networks for various transformer connections (Y-Y, Y-Δ, Δ-Δ). Phase shifts are included for Y-Δ transformers. **Example 9.7:** This example demonstrates how to solve unbalanced network problems using per-unit sequence components, including a transformer. **9.7 Per-Unit Sequence Models of Three-Phase, Three-Winding Transformers** Similar to two-winding transformers, this section presents sequence networks for three-winding transformers. **9.8 Power in Sequence Networks** The total three-phase complex power (Sp) is related to the sequence powers (Ss) by: Sp = 3Ss = 3(V0I0\* + V1I1\* + V2I2\*) **Chapter 10: Unsymmetrical Faults** This chapter extends the analysis of faults to *unsymmetrical faults*, which are more common than balanced three-phase faults. The chapter uses the method of symmetrical components and sequence networks (developed in Chapter 9) to calculate fault currents and voltages. **10.1 System Representation** The following assumptions simplify fault analysis (but aren\'t always applicable in real-world scenarios): 1. **Prefault balanced system:** The system is in balanced steady-state operation before the fault. 2. **Negligible load current:** Prefault load current is ignored, simplifying calculations. 3. **Negligible transformer/line resistance and shunt admittance:** These are often small compared to reactances and are neglected. 4. **Simplified synchronous machine model:** Armature resistance, saliency, and saturation are ignored. 5. **Negligible nonrotating impedance loads:** These loads don\'t contribute significantly to fault currents. 6. **Simplified induction motor model:** Small motors are neglected, while larger ones are treated similarly to synchronous machines. **10.2 Single Line-to-Ground Fault** This is the most common fault type. Consider a fault on phase \'a\': - **Phase domain conditions:** Ib = Ic = 0 and Vag = Zf \* Ia (Zf is fault impedance) - **Sequence domain conditions:** I0 = I1 = I2 and V0 + V1 + V2 = 3Zf \* I1 The sequence networks are connected in *series* through the impedance 3Zf. The fault current is: I1 = Vf / (Z0 + Z1 + Z2 + 3Zf) Ia = 3I1 (since I0 = I1 = I2) **Example 10.3:** Calculates fault current and voltages for a single line-to-ground fault. Shows how to use PowerWorld Simulator for fault analysis. **10.3 Line-to-Line Fault** Consider a fault between phases \'b\' and \'c\': - **Phase domain conditions:** Ia = 0, Ib = -Ic, and Vbg - Vcg = Zf \* Ib - **Sequence domain conditions:** I0 = 0, I2 = -I1, and V1 - V2 = Zf \* I1 The positive and negative sequence networks are connected in *parallel* through Zf. The fault current is: I1 = -I2 = Vf / (Z1 + Z2 + Zf) Ib = (a² - a)I1 = -j√3 \* I1 **Example 10.4:** Calculates fault current for a line-to-line fault and again demonstrates PowerWorld Simulator. **10.4 Double Line-to-Ground Fault** Consider a fault between phases \'b\', \'c\', and ground: - **Phase domain conditions:** Ia = 0, Vbg = Vcg = Zf \* (Ib + Ic) - **Sequence domain conditions:** I0 + I1 + I2 = 0, V1 = V2, and V0 - V1 = 3Zf \* I0 The sequence networks are connected in *parallel*, with 3Zf in the zero-sequence network. The fault current is more complex to calculate due to the parallel combination: I1 = Vf / (Z1 + \[Z2 // (Z0 + 3Zf)\]) I2 = -I1 \* (Z0 + 3Zf) / (Z0 + 3Zf + Z2) I0 = -I1 \* Z2 / (Z0 + 3Zf + Z2) **Example 10.5:** Calculates fault currents and contributions from different parts of the network for a double line-to-ground fault. **Example 10.6:** Explores the effect of Δ-Y transformer phase shifts on fault currents. The key point is that phase shifts *do* affect fault contributions on the Y-side of the transformer but not on the Δ-side. **10.5 Sequence Bus Impedance Matrices** This section extends the concept of bus impedance matrices (Zbus) to sequence networks. A Zbus matrix can be formed for each sequence (Zbus0, Zbus1, Zbus2). These matrices simplify fault calculations, especially in larger systems. The equations for fault currents and voltages are derived in terms of the Zbus elements. **Example 10.7:** Demonstrates fault analysis using the Zbus method for single line-to-ground faults. It confirms the same results as previous examples, showing the equivalence of the methods. **Example 10.8:** Shows an example of symmetrical components and sequence networks calculation. **Key Takeaways:** - Unsymmetrical faults are analyzed using symmetrical components. - Sequence networks provide a simplified way to represent the system during faults. - Fault calculations become more complex for double line-to-ground faults due to the parallel network connections. - Sequence bus impedance matrices provide a powerful tool for fault analysis, especially in larger networks. - PowerWorld Simulator can be used to automate fault calculations, including contributions from various network components. **Chapter 11: System Protection** **Introduction** Short circuits (faults) are unavoidable in power systems. They arise from insulation failures caused by overvoltages (lightning, switching surges), contamination, or mechanical issues. While careful design and maintenance can minimize occurrences, protection systems are essential to mitigate their impact. Short-circuit currents can be vastly larger than normal operating currents, leading to damage, fires, and explosions if not quickly removed. **Protection System Components** Protection systems comprise three primary components: 1. **Instrument Transformers:** These scale down system voltages and currents to safer, measurable levels for relays. They also provide electrical isolation. *Voltage transformers (VTs)* reduce voltage, while *current transformers (CTs)* reduce current. - VT: V\' = (1/n)V (where V\' is the secondary voltage, n is the turns ratio, and V is the primary voltage) - CT: I\' ≈ (n)I (where I\' is the secondary current and I is the primary current---this is an approximation due to the excitation current) 2. **Relays:** These devices sense abnormal conditions and initiate control actions (e.g., tripping a circuit breaker). Relays can be electromechanical, solid-state, or numeric. Types include overcurrent, directional, distance, differential, and pilot relays. 3. **Circuit Breakers:** These interrupt fault currents, isolating the faulted section. They are designed for reliability, selectivity, speed, economy, and simplicity. **Relay Types** - **Overcurrent Relays:** Operate when current exceeds a set *pickup current* (Ip). Can be instantaneous or time-delayed. Time delay is inversely related to current magnitude. - **Directional Relays:** Operate only for fault currents flowing in a specific direction. They utilize voltage and current as inputs. - **Distance (Impedance) Relays:** Operate based on the *impedance* (voltage-to-current ratio) seen by the relay. Used for transmission line protection. Have directional capabilities. - **Differential Relays:** Compare currents entering and leaving a protected zone (e.g., generator, transformer, bus). Operate when the difference exceeds a threshold. - **Pilot Relays:** A form of differential relaying using a communication channel (pilot wires, power-line carrier, microwave, fiber optic) to compare quantities at the terminals of a protected line. **System Protection Schemes** - **Radial System Protection:** Uses time-delay overcurrent relays coordinated to operate in sequence. Closest breaker to the fault trips first, minimizing load interruption. - **Two-Source System Protection:** Requires directional relays to ensure correct operation when a fault can be fed from both sources. - **Zones of Protection:** Dividing a system into overlapping zones, each with its own protection. Circuit breakers isolate faulted zones. - **Line Protection (Impedance Relays):** Uses impedance relays to provide fast and selective fault clearing on transmission lines. Multiple zones of protection with increasing reach and time delay are common. **Other Protection Devices** - **Reclosers:** Automatically interrupt and reclose a circuit. Used on distribution systems to clear temporary faults. - **Fuses:** Overcurrent devices that melt to interrupt current. Provide backup protection for reclosers. - **Sectionalizers:** Isolate faulted sections of a distribution circuit after an upstream device has operated. **Numeric Relaying** Numeric relays use microprocessors and software for fault detection. They offer numerous advantages, including: - Compact size, programmability, low burden, multi-functionality, flexibility, sensitivity, data storage, self-checking diagnostics, and adaptive relaying. **Improving Power System Stability** Design methods for enhancing stability include: - **Improved Steady-State Stability:** Higher voltages, additional lines, smaller reactances, series compensation, and FACTS devices. - **High-Speed Fault Clearing and Reclosing:** Minimize fault duration and improve post-fault power transfer. - **Single-Pole Switching:** Isolates only the faulted phase, maintaining partial service. - **Larger Machine Inertia:** Reduces angular swings. - **Fast Excitation and Valving:** Improve transient stability. - **Braking Resistors and RASs:** Provide additional damping and corrective actions. **Calculations** Throughout the chapter, several examples illustrate the calculations involved in: - CT and VT sizing and error analysis. - Relay operating time calculations. - Coordination of overcurrent relays. - Setting distance relay zones. - Determining critical clearing times using the equal-area criterion. - Numerically integrating the swing equation. **Chapter 12: Power System Stability** **Introduction** Power system stability refers to a system\'s ability to regain equilibrium after a disturbance, maintaining its integrity. This chapter focuses on three types of stability: *rotor angle stability*, *frequency stability*, and *voltage stability*. Classical stability studies analyze the first swing of machine rotor angles, while modern studies consider longer durations, incorporating generator, load, and renewable energy dynamics. **The Swing Equation** The swing equation governs the motion of a synchronous generator rotor: - *Jαm(t) = Tm(t) - Te(t) = Ta(t)* - J = rotor moment of inertia - αm = rotor angular acceleration - Tm = mechanical torque (prime mover minus losses) - Te = electrical torque (output power plus losses) - Ta = net accelerating torque This equation is usually expressed in per-unit and uses the power angle (δ), which is the angle of the internal machine voltage relative to the infinite bus: - *(2H/ωsyn)ωp.u.(t)(d²δ(t)/dt²) = Pmp.u.(t) - Pep.u.(t) - (D/ωsyn)(dδ(t)/dt) = Pap.u.(t)* - H = inertia constant - ωsyn = synchronous electrical radian frequency - ωp.u.(t) = per-unit rotor angular velocity (often approximated as 1.0) - Pmp.u. = per-unit mechanical power - Pep.u. = per-unit electrical power - D = damping coefficient - Pap.u. = per-unit accelerating power **Simplified Synchronous Machine Model and System Equivalents** For basic stability studies, the *classical model* represents a synchronous machine as a constant internal voltage (E\') behind its transient reactance (Xd\'). The system is often simplified as an *infinite bus* (constant voltage, phase, and frequency) behind a system reactance. - *Pe = (E\'Vbus/Xeq)sinδ* - Pe = real power delivered - Vbus = infinite bus voltage - Xeq = equivalent reactance between machine and infinite bus - δ = power angle **The Equal-Area Criterion** This graphical method determines transient stability for a single machine connected to an infinite bus. Stability is maintained if the *accelerating area* (A1) equals the *decelerating area* (A2) on the power-angle curve. This allows for determining the critical clearing angle (δcr) and time (tcr). **Numerical Integration of the Swing Equation** For multi-machine systems, numerical methods like Euler\'s method or modified Euler\'s method are used to solve the swing equation iteratively. These methods approximate the solution step-by-step using a defined time step (Δt). **Multimachine Stability** Multi-machine stability analysis requires solving the swing equation for each machine while considering the network\'s power flow. The nodal admittance matrix (Ybus) is modified to include load admittances and inverted generator impedances, creating Y11, Y22, and Y12 matrices. These matrices are used to calculate machine electrical power outputs during the numerical integration. **A Two-Axis Synchronous Machine Model** More detailed stability studies employ a two-axis model, which accounts for field winding and damper winding dynamics. The model uses a rotating d-q reference frame aligned with the rotor. The electrical behavior is represented by algebraic and differential equations that account for transient and subtransient effects. **Wind Turbine and Solar PV Machine Models** - **Type 1 and 2 Wind Turbines:** Modeled using induction machine equations, considering slip (S). - **Type 3 Wind Turbines (DFAGs/DFIGs):** Often approximated as voltage-source converters (VSCs). - **Type 4 Wind Turbines and Solar PV:** Represented as VSCs, decoupled from machine dynamics. **Load Models** Load models can be *static* (constant impedance, ZIP, exponential), *dynamic* (induction motor models), or *composite* (combinations of static and dynamic models). **Design Methods for Improving Power System Stability** Strategies for enhancing stability include increased voltage levels, additional transmission lines, reduced reactances, series compensation, high-speed fault clearing and reclosing, single-pole switching, larger machine inertia, fast excitation, fast valving, braking resistors, and remedial action schemes (RASs). **Chapter 13: Power System Controls** This chapter discusses the automatic control systems employed in power systems under normal operation. It covers generator voltage control, turbine-governor control, load-frequency control (LFC), and power system stabilizers. **13.1 Generator-Voltage Control** - **Synchronous Generators:** The exciter provides DC power to the field winding on the rotor. Older generators used DC generators driven by the rotor, transferring power via slip rings and brushes. Newer generators use static or brushless exciters. Static exciters obtain AC power, rectify it using thyristors, and transfer it to the rotor via slip rings and brushes. Brushless exciters use an \"inverted\" synchronous generator on the rotor, eliminating the need for slip rings and brushes. - **Voltage Regulator:** The voltage regulator controls the generator terminal voltage (Vt). It compares Vt to a reference voltage (Vref) and adjusts the exciter output (Efd) to maintain the desired Vt. A voltage transformer and rectifier provide feedback to the regulator. A block diagram representing the IEEE Type 1 exciter (using a shaft-driven DC generator) is presented, showing the components and their interconnections. The diagram includes a feedback loop to enhance dynamic response and reduce overshoot. - **Wind Turbine-Generators:** Voltage control in wind turbines depends on the type. Type 1 (squirrel cage induction) has no direct voltage control. Type 2 (wound rotor induction) uses variable external resistance, which is modeled as a type of exciter to maintain constant power output. Type 3 and 4 wind turbine generators can perform voltage or reactive power control, including constant power factor, coordinated control across a wind farm, and constant reactive power control. **13.2 Turbine-Governor Control** - **Synchronous Generators:** Turbine-generator units store kinetic energy due to rotation. During load changes, this energy is released or absorbed, impacting rotor speed and frequency. Frequency deviation is used as a control signal for the turbine\'s mechanical power output. - **Steady-State Frequency-Power Relation:** The equation ΔPm = ΔPref - Δf/R describes the relationship between frequency change (Δf), turbine mechanical power change (ΔPm), and reference power setting change (ΔPref). R is the regulation constant. A graph illustrating this relationship shows a family of droop curves, with ΔPref as a parameter. The regulation constant R (Hz/MW or per unit) is the negative slope of these curves. - **Area Frequency Response:** For an interconnected system area, the characteristic β (MW/Hz or per unit) represents the area\'s frequency response. It\'s the sum of the individual unit responses (1/R) and accounts for losses and load frequency dependency. - **Turbine-Governor Block Diagram (TGOV1):** A block diagram for a steam turbine governor shows the various components and delays associated with the governor and turbine dynamics. The diagram includes parameters for reheat and non-reheat turbines and a damping coefficient. **13.3 Load-Frequency Control (LFC)** - **Objectives:** LFC aims to return frequency error (Δf) to zero after a load change and maintain scheduled tie-line power flow to neighboring areas, ensuring each area handles its own load changes. - **Area Control Error (ACE):** ACE is defined as ACE = (Ptie - Ptie,sched) + B\*Δf, where Ptie is the tie-line power flow, Ptie,sched is the scheduled tie-line flow, and B is the frequency bias constant. - **LFC Control Strategy:** The reference power setting change (ΔPref) for each turbine-governor is proportional to the integral of the ACE: ΔPref = -Ki ∫ ACE dt, where Ki is the integrator gain. The negative sign indicates that generation should increase when ACE is negative (low frequency or tie-line flow). **13.4 Power System Stabilizer (PSS) Control** - **Power System Oscillations:** Power systems experience oscillations (0.1-5 Hz), categorized as inter-area (0.1-1 Hz) and local/plant (1-5 Hz). Oscillations can be positively damped, negatively damped, or undamped. - **Power System Stabilizer (PSS):** A PSS modulates the exciter voltage to damp oscillations. It adds a control signal (Vs) in phase with speed variation. A block diagram of a simple PSS illustrates the components, including input signal delay, washout filter, output limits, gain (Ks), and time constants (T1 to T4) used for phase-lead compensation. **Chapter 14: Transmission Lines: Transient Operation** This chapter delves into the transient behavior of transmission lines, crucial for understanding and mitigating overvoltages caused by lightning or switching operations. It covers traveling waves, reflections, refractions, and insulation coordination. **14.1 Traveling Waves on Single-Phase Lossless Lines** - **Lossless Line Model:** A single-phase, two-wire lossless line is analyzed by considering a small section of length Δx, with inductance LΔx and capacitance CΔx. - **KVL and KCL Equations:** Kirchhoff\'s Voltage Law (KVL) and Kirchhoff\'s Current Law (KCL) equations are written for the line section. These equations relate the voltage and current at different points along the line and consider the time derivatives due to inductance and capacitance. - **Wave Equations:** By taking the limit as Δx approaches zero, the KVL and KCL equations transform into partial differential equations (wave equations) governing voltage and current along the line as functions of position (x) and time (t). These equations describe the propagation of waves along the line. - **Laplace Transform:** Applying the Laplace transform simplifies the wave equations, converting them into ordinary differential equations. - **General Solution:** The general solutions for voltage V(x,s) and current I(x,s) in the Laplace domain reveal forward and backward traveling waves, represented by exponential terms e\^(-sx/v) and e\^(sx/v), respectively. \'v\' represents the velocity of wave propagation (v = 1/√(LC)). - **Forward and Backward Waves:** The terms V+(s) and V-(s) are functions of \'s\' and represent the magnitudes of the forward and backward traveling voltage waves, respectively, and are determined by boundary conditions. Similarly, I+(s) and I-(s) represent the magnitudes of the forward and backward traveling current waves. The inverse Laplace transform reveals these waves as functions of time: v⁺(t − x/v), v¯(t + x/v), i⁺(t − x/v), and i¯(t + x/v). - **Characteristic Impedance:** The characteristic impedance of the lossless line (Zc = √(L/C)) determines the relationship between voltage and current waves traveling in the same direction. **14.2 Boundary Conditions for Single-Phase Lossless Lines** - **Terminations:** The chapter considers a line terminated by impedance ZR(s) at the receiving end (x = l) and a source with voltage Eg(s) and impedance Zg(s) at the sending end (x = 0). - **Receiving End Condition:** At the receiving end, the voltage and current are related by the load impedance: V(l, s) = ZR(s)I(l, s). This condition helps determine the relationship between forward and backward waves. - **Reflection Coefficient:** The receiving-end voltage reflection coefficient ΓR(s) is defined as (ZR(s) - Zc) / (ZR(s) + Zc). It quantifies the reflected wave\'s magnitude and phase relative to the incident wave. - **Sending End Condition:** At the sending end, the voltage and current are related by the source voltage and impedance: V(0, s) = Eg(s) -- Zg(s)I(0, s). This condition, along with the receiving end condition, fully defines the transient behavior. - **Sending End Reflection Coefficient:** The sending-end voltage reflection coefficient Γs(s) is defined as (Zg(s) - Zc) / (Zg(s) + Zc). It quantifies the reflection at the sending end. - **Complete Solution:** The complete solution for voltage and current anywhere on the line at any time is derived using the boundary conditions and reflection coefficients. **14.3 Bewley Lattice Diagram** - **Visualizing Reflections:** The Bewley lattice diagram provides a graphical way to organize and visualize the multiple reflections that occur at the line\'s terminations. - **Diagram Construction:** The diagram\'s vertical axis represents time (scaled in transit time τ), and the horizontal axis represents distance along the line. Diagonal lines represent forward and backward traveling waves. Each reflection is represented by multiplying the incident wave by the corresponding reflection coefficient. - **Determining Voltage and Current:** The voltage or current at any point on the line and at any time is found by summing the contributions of all waves that have reached that point by that time. **14.4 Discrete-Time Models of Single-Phase Lossless Lines and Lumped RLC Elements** - **Discrete-Time Analysis:** This section presents discrete-time models suitable for computer simulation of transmission line transients. - **Lossless Line Model:** The lossless line is modeled as a combination of resistors and current sources in the discrete-time domain. The current sources represent the \"past history\" of the line, capturing the effect of traveling waves. - **Lumped RLC Elements:** Discrete-time models are also developed for lumped resistance, inductance, and capacitance. These models are crucial for simulating circuits containing both transmission lines and lumped elements. - **Nodal Equations:** Nodal analysis is applied to the discrete-time equivalent circuit to obtain a set of linear algebraic equations that can be solved iteratively to find the voltage at each node at each time step. **14.5 Lossy Lines** - **Attenuation, Distortion, and Losses:** Real transmission lines have series resistance (R) and shunt conductance (G), leading to attenuation (reduction of wave magnitude), distortion (change of wave shape), and power losses. - **Distortionless Line:** A special case, the distortionless line (R/L = G/C), is discussed, where waves are attenuated but maintain their shape. - **High-Frequency Behavior:** At high frequencies (above 1 MHz), practical transmission lines tend to behave like distortionless lines, simplifying the analysis. **14.6 Multiconductor Lines** - **Modal Analysis:** This section extends the analysis to three-phase lines, where waves travel in different modes, each with its own velocity and surge impedance. - **Transformation Matrices:** Transformation matrices are used to convert between phase quantities (voltages and currents) and modal quantities. This simplifies the analysis by decoupling the equations for each mode. **14.7 Power System Overvoltages** - **Lightning Surges:** Lightning strikes are a major cause of transmission line outages. The mechanism of lightning strikes, the characteristics of lightning surge currents, and the use of lightning detection networks are discussed. - **Switching Surges:** Switching operations in power systems can also generate transient overvoltages. Energizing an open-circuited line is presented as a key example, as well as the importance of pre-insertion resistors in circuit breakers to mitigate these overvoltages. - **Power Frequency Overvoltages:** Sustained overvoltages at power frequency can be caused by load rejection, ferroresonance, or faults. **14.8 Insulation Coordination** - **BIL and Protective Devices:** Basic Insulation Level (BIL) represents the equipment\'s ability to withstand transient overvoltages. Protective devices, like surge arresters, are used to limit overvoltages and protect equipment. - **Insulation Coordination:** The process of coordinating equipment BIL with protective device characteristics is discussed to ensure effective protection. - **Surge Arresters:** Different types of surge arresters, including rod gaps and surge arresters with nonlinear resistors (with and without air gaps), are presented. **Chapter 15: Power Distribution** This chapter explores the power distribution system, the part of the power network responsible for delivering electricity from substations to end consumers. It covers primary and secondary distribution systems, transformer applications, capacitor usage, distribution reliability, automation, and the evolution towards smart grids. **15.1 Introduction to Distribution** - **System Components:** A typical power system comprises generation, transmission, and distribution. The distribution system extends from distribution substations to customer service entrances. - **Transmission System:** The transmission system delivers energy from generators to substations, facilitating energy interchange and supplying the subtransmission and distribution systems. It uses high voltages (230 kV to 765 kV) for efficient long-distance power transfer. - **Subtransmission System:** The subtransmission system uses step-down transformers to reduce transmission voltages (typically 69 kV to 138 kV) and connect bulk power substations to distribution substations. - **Distribution Substations:** Distribution substations further lower the voltage (2.2 kV to 46 kV) for local distribution using distribution substation transformers. They also contain equipment for voltage regulation and protection. - **Primary and Secondary Distribution:** Primary distribution carries power from the substation to distribution transformers, while secondary distribution operates at utilization voltages (120 V to 480 V) from transformers to customer meters. **15.2 Primary Distribution** - **Voltage Levels:** Typical primary distribution voltages in the U.S. are in the 15-kV class (12.47 kV, 13.2 kV, 13.8 kV). Older systems may use 2.5 kV or 5 kV, while newer systems or those in rural areas might use 25 kV to 34.5 kV. - **System Types:** - **Radial Systems:** Simplest and most economical, with feeders emanating radially from the substation. Sectionalizing fuses and reclosers improve reliability by isolating faults. Normally open tie switches allow connection to adjacent feeders during emergencies. Figure 15.2 illustrates a typical radial system. - **Loop Systems:** Feeders loop through the load area and return to the substation, offering two-way feed and higher reliability. Reclosers and tie switches sectionalize the loop for fault isolation. Figure 15.8 shows an overhead loop system. - **Primary Network Systems:** A grid of interconnected feeders supplied by multiple substations, providing the highest reliability but at higher cost. Common in high-density urban areas. Network transformers and protectors are key components. Figure 15.10 illustrates a primary network. **15.3 Secondary Distribution** - **Voltage Levels and Applications:** Common secondary voltages include: - 120/240 V, single-phase for residential service. - 208Y/120 V, three-phase for residential/commercial service. - 480Y/277 V, three-phase for commercial/industrial/high-rise buildings. - **System Types:** - **Individual Transformer per Customer:** Used in rural areas or for large loads. Simple but with higher transformer costs and losses. Figure 15.12. - **Common Secondary Main:** Multiple customers are served from a common main, leveraging load diversity and reducing transformer capacity needs. Figure 15.13. - **Secondary Network (Grid):** High reliability for dense urban areas. Network transformers and protectors are used. Figure 15.14. - **Spot Network:** A secondary network dedicated to a single large load (e.g., a high-rise building). Offers high reliability. Figure 15.15. **15.4 Transformers in Distribution Systems** - **Distribution Substation Transformers:** Step down voltage from subtransmission to primary distribution levels. They have multiple MVA ratings (OA, FA, FOA) based on different cooling methods (Oil-Air, Forced-Air, Forced-Oil-Air). Impedance is typically given as a percentage of the OA rating. LTCs are common for voltage regulation. - **Distribution Transformers:** Connect primary distribution to secondary utilization voltages. Types include: - **Pole-Mount:** Liquid-filled, single- or three-phase, for overhead distribution. - **Padmount:** Liquid-filled or dry-type, single- or three-phase, for underground systems. - **Network Transformers:** Large, liquid-filled, three-phase for secondary networks and spot networks. **15.5 Shunt Capacitors in Distribution Systems** - **Benefits:** Shunt capacitors supply reactive power to loads, offsetting lagging currents from inductive loads. This improves power factor, reduces line losses, and improves voltage regulation. - **Placement:** Optimal placement is crucial for maximizing benefits. The \"two-thirds rule\" is a common heuristic for uniformly distributed loads. - **Fixed and Switched Banks:** Fixed banks compensate for reactive power at light loads, while switched banks are added during peak load periods. Switching methods include voltage, current, var, time, temperature, or SCADA control. **15.6 Distribution Software** This section lists various software functionalities used for distribution system planning, design, and operation, including power flow analysis, fault analysis, capacitor placement optimization, reliability evaluation, and more. **15.7 Distribution Reliability** - **Reliability Indices:** Key reliability indices include: - **SAIFI (System Average Interruption Frequency Index):** Average number of interruptions per customer. - **SAIDI (System Average Interruption Duration Index):** Average interruption duration per customer. - **CAIDI (Customer Average Interruption Duration Index):** Average duration of an interruption for customers who experienced an interruption. - **ASAI (Average Service Availability Index):** Percentage of time service is available. - **Outage Reporting:** Utilities track and report outage data to monitor reliability, identify weaknesses, and make improvements. **15.8 Distribution Automation** - **Benefits:** DA improves reliability, reduces outage durations, and enhances voltage control through remote control and automation of devices like sectionalizers, reclosers, and voltage regulators. - **Volt-Var Optimization:** DA enables more effective voltage control and loss reduction by coordinating voltage control devices across the feeder. **15.9 Smart Grids** - **Definition and Attributes:** Smart grids utilize digital technology to enhance reliability, flexibility, security, and efficiency of the electric system, incorporating distributed generation and storage, demand response, smart metering, and advanced communication. - **Impact on Distribution:** The smart grid transforms the distribution system from a passive conduit to an active, intelligent network, enabling bidirectional power and information flow, integration of DERs, and greater customer participation.