UNIT 4: POWER SYSTEM ANALYSIS, STABILITY, CONTROL & ECONOMIC OPERATION

1.0 POWER SYSTEM STRUCTURE & INTERCONNECTION

1.1 Necessity and Benefits of Interconnected Power Systems

• Necessity: To ensure reliable, continuous, and economical power supply by sharing generation reserves, load diversity, and accessing remote energy resources.

• Key Benefits:

  • Reliability: Mutual aid during contingencies (N-1 criterion).

  • Economy: Utilization of most efficient plants (economic dispatch over a larger area), reduced spinning reserve.

  • Improved Voltage & Frequency: Larger system inertia helps damp frequency fluctuations; reactive power support is easier.

  • Access to Remote Resources: Hydro, wind, solar farms far from load centers can be connected.

1.2 Problems Associated with Interconnected Systems

• Complexity: Protection coordination, stability analysis, and control become significantly more difficult.

• Stability Issues: Interarea oscillations, risk of cascading failures.

• Regulatory & Commercial: Need for open access, wheeling charges, and settlement mechanisms in a deregulated environment.

• Synchronism Loss: A major disturbance in one area can cause instability and loss of synchronism across the interconnection.

1.3 Modern Power System Concepts

• 1.3.1 Deregulation & Restructuring: Separation of generation, transmission, and distribution (unbundling). Introduction of competition in generation and retail supply. Open Access allows any generator/consumer to use the transmission network for a fee.

• 1.3.2 Distributed Generation (DG): Small-scale generation (solar PV, wind, microturbines) located close to the load center. Impacts power flow, voltage profile, and protection schemes.

• 1.3.3 Transmission Open Access (TOA): Non-discriminatory use of the transmission network by all market participants, governed by a central system operator.

[!TIP] Exam Focus: Be prepared to explain why interconnection is needed (benefits) and what new problems it introduces (stability, coordination, deregulation challenges).

2.0 LOAD FLOW ANALYSIS (POWER FLOW STUDIES)

2.1 Significance and Objectives

• Determines voltage magnitudes and angles at all buses, and real & reactive power flows in all lines for a given generation and load condition.

• Objectives: Planning (new lines, capacity), Operation (optimal dispatch, contingency analysis), Stability studies (initial operating point).

2.2 Classification of Buses

• Justification: Only one slack bus is needed as the reference. Generator buses with AVR are PV buses. All other buses are PQ buses.

2.3 Formation of Admittance Matrix (Y_BUS)

• Element-by-Element Addition Method:

  1. Start with a zero matrix.

  2. For each line k between buses i and j with series impedance z_series = R + jX:

     • Y_ii += 1/z_series

     • Y_jj += 1/z_series

     • Y_ij = Y_ji = -1/z_series

  3. For a shunt element (line charging jB/2 at each end):

     • Y_ii += jB/2

     • Y_jj += jB/2

  • Result: Symmetric, sparse matrix.

• Direct Formation: From network data using the above rules systematically.

Example Problem (Y_BUS Formation): Given 4-bus system with line impedances (p.u.), neglect shunt capacitance.

DiagramCANVAS: Draw a simple 4-bus diagram with lines 1-2, 1-3, 1-4, 2-3, 3-4. Label impedances as per data.

Solution Steps:

1. Y_12 = 1/(0.25+j0.1) = 3.882 - j1.553 p.u.

2. Y_13 = 1/(0.20+j0.8) = 1.176 - j4.706 p.u.

3. Y_14 = 1/(0.30+j1.2) = 0.784 - j3.137 p.u.

4. Y_23 = 1/(0.20+j0.8) = 1.176 - j4.706 p.u.

5. Y_34 = 1/(0.15+j0.6) = 1.569 - j6.275 p.u.

6. Assemble Ybus:

*   `Y_11 = Y_12 + Y_13 + Y_14`

*   `Y_22 = Y_12 + Y_23`

*   `Y_33 = Y_13 + Y_23 + Y_34`

*   `Y_44 = Y_14 + Y_34`

*   Off-diagonals are negatives of mutual admittances.

2.4 Load Flow Solution Methods

Gauss-Seidel Method (PQ Buses only):

For each PQ bus i, update voltage:

$$ V_i^{(k+1)} = \frac{1}{Y_{ii}} \left( \frac{P_i - jQ_i}{V_i^{(k)*}} - \sum_{j=1, j\neq i}^{n} Y_{ij} V_j^{(k)} \right) $$

• Iterate until $$\displaystyle |V_i^{(k+1)} - V_i^{(k)}| < \epsilon $$ for all buses.

• PV Bus Handling: Calculate Q from latest voltage. If Q exceeds limits, convert to PQ bus for that iteration.

Newton-Raphson Method:

• Rectangular Coordinates: Solve for ΔP, ΔQ in terms of Δe, Δf.

$$ \begin{bmatrix} \Delta P \\ \Delta Q \end{bmatrix} = \begin{bmatrix} J_{11} & J_{12} \\ J_{21} & J_{22} \end{bmatrix} \begin{bmatrix} \Delta e \\ \Delta f \end{bmatrix} $$

where Jacobian sub-blocks involve partial derivatives.

• Polar Coordinates: Solve for ΔP, ΔQ in terms of Δ|V|, Δδ. More common.

$$ \begin{bmatrix} \Delta P \\ \Delta Q \end{bmatrix} = \begin{bmatrix} \frac{\partial P}{\partial \delta} & \frac{\partial P}{\partial |V|} \\ \frac{\partial Q}{\partial \delta} & \frac{\partial Q}{\partial |V|} \end{bmatrix} \begin{bmatrix} \Delta \delta \\ \Delta |V| \end{bmatrix} $$

• Flow: Form Ybus → Calculate P, Q, mismatches → Form Jacobian → Solve for corrections → Update voltages → Repeat.

[!TIP] Common Pitfall: In GS, convergence is slow for large systems. In NR, initial guess must be close; singularity in Jacobian if slack bus is not properly defined.

3.0 POWER SYSTEM STABILITY

3.1 Fundamental Concepts

• Stability: Ability of a power system to regain a state of operating equilibrium after a disturbance.

• Steady-State (Small Signal) Stability: Ability to maintain synchronism for small disturbances (e.g., small load changes). Analyzed by linearized swing equation.

• Transient Stability: Ability to maintain synchronism after a large, sudden disturbance (e.g., fault, line outage). Analyzed by non-linear swing equation and Equal Area Criterion (EAC).

• Rotor Angle Stability: Concerned with maintaining synchronism between all synchronous machines. Root cause of instability.

3.2 Swing Equation

• Derivation: From Newton's 2nd law for rotor rotation.

$$ M \frac{d^2 \delta}{dt^2} = P_m - P_e = P_a $$

where:

*   $$\displaystyle M = 2H / \omega_s $$ (Inertia constant, kg-m² or MJ/MVA)

*   $\delta$ = Rotor angle (rad) relative to synchronously rotating reference.

*   $$\displaystyle P_m $$ = Mechanical power input (p.u.).

*   $$\displaystyle P_e $$ = Electrical power output (p.u.).

*   $$\displaystyle P_a $$ = Accelerating power (p.u.).

• Physical Significance: $M$ represents the stored kinetic energy in the rotating mass. $$\displaystyle P_a $$ determines acceleration/deceleration of the rotor.

Linearized Swing Equation (for small δ):

$$ M \frac{d^2 (\Delta \delta)}{dt^2} + D \frac{d (\Delta \delta)}{dt} + \omega_s^2 \frac{\partial P_e}{\partial \delta} \Delta \delta = 0 $$

• $D$ = Damping coefficient.

• $$\displaystyle \frac{\partial P_e}{\partial \delta} $$ = Synchronizing power coefficient.

• Steady-State Stability Limit: $$\displaystyle \frac{\partial P_e}{\partial \delta} > 0 $$. Maximum power transfer occurs at $$\displaystyle \frac{\partial^2 P_e}{\partial \delta^2} = 0 $$.

3.3 Equal Area Criterion (EAC)

• Geometric Interpretation: For a single machine-infinite bus system, stability is determined by comparing the accelerating area ($$\displaystyle A_{acc} $$) and decelerating area ($$\displaystyle A_{dec} $$) under the $$\displaystyle P_e(\delta) $$ curve during and after a fault.

$$ A_{acc} = \int_{\delta_0}^{\delta_{cr}} (P_m - P_e) d\delta $$

$$ A_{dec} = \int_{\delta_{cr}}^{\delta_{max}} (P_e - P_m) d\delta $$

• Condition for Stability: $$\displaystyle A_{acc} = A_{dec} $$ (system regains equilibrium). If $$\displaystyle A_{acc} > A_{dec} $$, machine loses synchronism.

• Critical Clearing Angle ($$\displaystyle \delta_{cr} $$): Angle at fault clearing where $$\displaystyle A_{acc} = A_{dec} $$.

• Critical Clearing Time ($$\displaystyle t_{cr} $$): Maximum fault duration for which system remains stable. Found by integrating swing equation numerically up to $$\displaystyle \delta_{cr} $$.

Example Problem (EAC): Generator delivers 50% of $$\displaystyle P_{max} $$ to infinite bus. Fault increases reactance to 500% of pre-fault value. Post-fault $$\displaystyle P_{max} $$ is 75% of original.

1. Pre-fault: $$\displaystyle P_{e0} = 0.5 P_{max0} $$ at $$\displaystyle \delta_0 $$.

2. During fault: $$\displaystyle P_{e,fault} = P_{max0}/5 = 0.2 P_{max0} $$ (since reactance 5x, $$\displaystyle P_{max} \propto 1/X $$).

3. Post-fault: $$\displaystyle P_{e,post} = 0.75 P_{max0} $$.

4. Calculate $$\displaystyle \delta_0 $$, $$\displaystyle \delta_1 $$ (fault on), $$\displaystyle \delta_2 $$ (fault cleared), $$\displaystyle \delta_3 $$ (max).

5. Set $$\displaystyle A_{acc} = A_{dec} $$ to find $$\displaystyle \delta_{cr} $$ (which is $$\displaystyle \delta_2 $$).

3.4 Factors Affecting Stability

3.5 Methods of Improving Stability

• Steady-State: Use AVRs (increase $$\displaystyle P_{max} $$), reduce transfer reactance (parallel lines, series capacitors), improve power factor.

• Transient:

  • Fast Fault Clearing: High-speed circuit breakers.

  • Single-Pole Switching: For single-line-to-ground faults, only faulty pole opens.

  • Braking Resistors: Insert resistance at generator terminals during fault to reduce acceleration.

  • FACTS Devices: SVC, STATCOM for dynamic voltage support; TCSC for reducing effective X.

  • Fast Governor Response: Fast valving.

  • High Inertia: More rotating mass.

4.0 FREQUENCY CONTROL & GOVERNOR MODELING

4.1 Necessity for Strict Frequency Control

• Equipment: Frequency affects speed of synchronous motors, induction motors, turbines, clocks.

• System Operation: Frequency is a real-time indicator of generation-load balance. Deviation indicates imbalance.

• Loads: Many industrial processes (paper mills, petrochemicals) are frequency-sensitive.

• Grid Codes: Strict limits (e.g., ±0.5 Hz in India) must be maintained.

4.2 Load Frequency Control (LFC) in Interconnected Systems

• Primary (Governor) Control:

  • Action: Speed governor responds to frequency change Δf by adjusting turbine valve.

  • Characteristic: Droop (Speed Regulation) R = Δf / ΔP_m (%). Provides inherent, instantaneous but incomplete regulation. Load sharing between parallel generators is inversely proportional to R.

  • Limitation: Leaves a steady-state frequency error.

• Secondary (ALFC) Control & Tie-Line Bias:

  • Area Control Error (ACE): ACE = ΔP_{tie} + B Δf

    • ΔP_{tie} = Net tie-line power deviation from scheduled.

    • B = Frequency Bias Factor (MW/Hz), chosen as area load-frequency characteristic (-D - 1/R).

  • Action: ACE is integrated (Integral Controller) to generate a setpoint change for the governor, bringing ACE to zero → eliminates frequency error and corrects tie-line power.

• Modeling of Tie-Line Power Flow:

  For two areas, tie-line power from Area 1 to 2:

$$ P_{tie,12} = \frac{|V_1||V_2|}{X_{tie}} \sin(\delta_1 - \delta_2) $$

For small angle changes: `ΔP_{tie} = T_{12} Δδ`, where `T_{12}` is tie-line synchronizing coefficient.

4.3 Turbine Speed Governing System Modeling

• Components: Speed Governor → Amplifier/Pilot Valve → Servomotor → Turbine.

• Transfer Function Block Diagram:

  
  Δf → [1/(1+sT_g)] → ΔP_{m0} (Governor)
  
  ΔP_{m0} → [1/(1+sT_t)] → ΔP_m (Turbine)
  
  

  • T_g = Governor time constant (0.1-0.5 sec).

  • T_t = Turbine time constant (0.2-1.0 sec).

• Governor Time Constant & Speed Regulation: R is the steady-state droop. T_g represents the delay in valve movement.

4.4 Inertia Constant (H) & System Inertia

• Definition: H = (Stored Kinetic Energy at synchronous speed) / (Generator MVA base)

$$ H = \frac{0.5 J \omega_s^2}{S_{base}} \text{ (seconds or MJ/MVA)} $$

where `J` = moment of inertia (kg-m²), `ω_s` = synchronous speed (rad/s).

• Physical Meaning: Time (in seconds) for which a generator can supply full load from its stored kinetic energy if all input is lost.

• Equivalent Inertia Constant for Parallel Machines:

$$ H_{eq} = \frac{\sum (H_i \cdot S_{i,base})}{\sum S_{i,base}} $$

(Weighted average based on MVA ratings).

4.5 Numerical Problems

• Frequency Drop with Governor Delay (H given):

  Problem: 100 MVA, 50 Hz alternator, H=4.5 kW-sec/kVA. Load drop of 25 MW at t=0. Governor delay T_g=0.6 sec. Find frequency at t=0.6 sec.

  Solution:

  1. Convert H: H = 4.5 kW-sec/kVA = 4.5 sec (since 1 kW-sec/kVA = 1 sec).

  2. Stored energy: KE = H * S_base = 4.5 * 100 = 450 MJ.

  3. During delay (0 to 0.6 sec), P_m constant = P_e0 (assume initial load = 0? Or need initial load?). Typically, assume initial P_m = P_e0. Sudden load drop ΔP = 25 MW.

  4. Accelerating power P_a = P_m - P_e = 0 - (-25) = +25 MW? Clarify: If load drops, P_e decreases. For constant P_m, P_a = P_m - P_e becomes positive (accelerating).

  5. Swing equation: M d²δ/dt² = P_a. M = 2H/ω_s.

  6. Integrate: Δω = (P_a * t) / (2H) (for constant P_a, small Δf).

  7. Δf = (f_s / (2H)) * ∫ P_a dt (in p.u. or consistent units).

  8. For 0 ≤ t ≤ T_g, P_a = ΔP_load (since P_m hasn't changed).

  9. Δf(0.6) = (ΔP * t) / (2H * S_base) in p.u. → convert to Hz.

  Key Formula:

$$ \Delta f (Hz) = \frac{f_s \cdot \Delta P (MW) \cdot t (sec)}{2 \cdot H (MJ/MVA) \cdot S_{base} (MVA)} $$

Plug: `Δf = (50 * 25 * 0.6) / (2 * 4.5 * 100) = 750 / 900 = 0.833 Hz drop`. New freq = 49.167 Hz.

• Two-Area Load Sharing: Generation shares inversely proportional to their speed regulations R. If R_A and R_B, then P_{GA} / P_{GB} = R_B / R_A for a given total load and tie-line schedule.

• Energy Stored in Rotor: KE = H * S_base (in MJ if H in MJ/MVA, S_base in MVA).

5.0 VOLTAGE CONTROL & REACTIVE POWER MANAGEMENT

5.1 Relationship between Reactive Power Flow and Voltage Profile

• Phasor Diagram & Equation: For a line sending end voltage V_s, receiving end V_r, sending angle δ_s, receiving δ_r.

$$ V_r^2 = (V_s \cos \delta + I_r X)^2 + (V_s \sin \delta + I_r R)^2 $$

Neglecting R (high voltage lines):

$$ V_r \approx V_s - \frac{Q_r X}{V_s} $$

(for small δ, inductive load Q_r > 0)

**Key Insight:** Voltage drop `ΔV ≈ (Q_r X) / V_s`. **Reactive power flow (Q) is the dominant factor** in voltage variation along a transmission line.

• Reactive Power Balance: Q_gen - Q_loss - Q_load = ΔQ_shunt (capacitors/ reactors). Voltage rises if Q_gen > Q_load + losses.

5.2 Generation and Absorption of Reactive Power

5.3 Methods of Voltage Control

1. Generator Voltage Control (Excitation Systems): Primary control. AVR adjusts field current to maintain terminal voltage.

2. Tap Changing Transformers:

   • OLTC (On-Load Tap Changer): Continuous voltage control at substations.

   • Off-Load Tap Changer: For coarse, infrequent adjustments.

3. Shunt Compensation:

   • Shunt Capacitors: Inject Q (raise voltage). Switched in blocks.

   • Shunt Reactors: Absorb Q (lower voltage). Used for light-load voltage control on long lines.

4. Series Compensation: Series capacitors reduce effective line reactance X, increasing power transfer capability and improving voltage profile.

5.4 Protection and Absorption of Reactive Power

• Need for Balance: Excess reactive power causes over-voltages (esp. light load); deficiency causes voltage collapse.

• Methods for Absorption (Q_consumer):

  • Shunt Reactors: Fixed or switched.

  • SVC (Static Var Compensator): Thyristor-controlled reactor (TCR) + Thyristor-switched capacitor (TSC). Fast, continuous.

  • STATCOM: Voltage-source converter based. Better performance at low voltages.

  • Synchronous Condensers: Over-excited synchronous motors running without load. Good inertia support.

• Methods for Support/Generation (Q_producer):

  • Shunt Capacitors: Switched.

  • SVC/STATCOM: Can both generate and absorb.

  • Synchronous Condensers: Over-excited.

  • Generator Over-excitation: Increase field current (within limits).

[!TIP] Exam Focus: Be able to draw the V-I characteristic of SVC and STATCOM. Know that voltage control is essentially reactive power control.

6.0 EXCITATION SYSTEMS

6.1 Need and Objectives

• Voltage Control: Maintain generator terminal voltage or system voltage at a bus.

• Stability Enhancement: Increase steady-state stability limit ($$\displaystyle P_{max} \propto E V / X $$). Improve transient stability by maintaining voltage during faults.

• Reactive Power Sharing: Control reactive power output in parallel operation.

6.2 Types of Excitation Systems

1. DC Excitation System: DC generator (exciter) driven by motor or turbine. Amplidyne is a high-gain DC amplifier used as voltage regulator. Largely obsolete.

2. AC Static (Brushless) Excitation System:

   • Functional Block Diagram:

     
     AVR → Rotating Diode Rectifier (on shaft) → Generator Field
     
     

   • Description: AC exciter (on same shaft) produces AC, rectified by rotating diodes, supplies DC to main generator field. No brushes/slip rings. Reliable, low maintenance.

3. Modern Static Excitation System:

   • Block Diagram:

     
     AVR → Thyristor Rectifier (static) → Generator Field
     
     

   • Description: Uses transformer from generator terminals, thyristor bridge provides controlled DC. Fast response, precise control. Often includes Power System Stabilizer (PSS).

6.3 Functional Block Diagram & Elements

• Exciter: Source of DC power (DC gen, AC gen+rectifier, static transformer+thyristor).

• Automatic Voltage Regulator (AVR): Compares terminal voltage V_t with reference V_ref, amplifies error, controls exciter output. Main control loop.

• Power System Stabilizer (PSS): Auxiliary control. Takes input (Δω or ΔP) and produces a signal to AVR to damp power oscillations (rotor angle swings) by modulating field voltage.

6.4 Limiting Features in Excitation Systems

To protect generator from overheating and instability:

• Over-excitation Limit (OEL): Prevents field current/thermal overload (V/Hz or field current limit).

• Under-excitation Limit (UEL): Prevents instability from lagging power factor operation (stability limit).

• V/Hz Limit (Volts/Hz): Prevents core saturation (over-fluxing) during low-frequency/high-voltage conditions.

• Stator Current Limit: Prevents stator overheating.

7.0 ECONOMIC OPERATION OF POWER SYSTEMS

7.1 Economic Dispatch Problem

• Definition: Allocate total load demand among available generating units to minimize total fuel cost while satisfying constraints (generation limits, power balance).

• Input-Output Characteristics (Heat Rate Curve): F_i(P_i) = Fuel cost ($/hr) for generator i at output P_i (MW). Usually quadratic: F_i = a_i + b_i P_i + c_i P_i^2.

7.2 Economic Dispatch without Transmission Losses

• Lagrangian Multiplier Method:

  Minimize ∑ F_i(P_i) subject to ∑ P_i = P_D.

  Lagrangian: L = ∑ F_i(P_i) + λ (P_D - ∑ P_i)

  Coordination Equation: ∂L/∂P_i = 0 → F_i'(P_i) = λ

  • λ = Incremental Fuel Cost ($/MWh). At optimum, all units operate at same λ.

• Solution: For each unit, plot F_i'(P_i) vs P_i. Find intersection with common λ line satisfying ∑ P_i = P_D.

7.3 Economic Dispatch considering Transmission Losses

• Loss Coefficients (B-coefficients): Approximate transmission loss as:

$$ P_{loss} = \sum_{i} \sum_{j} B_{ij} P_i P_j + \sum_{i} B_{i0} P_i + B_{00} $$

`B_ij` are loss coefficients (symmetric, `B_ii > 0`, `B_ij < 0`), determined from system impedance matrix.

• Penalty Factor (λ_i):

$$ \lambda_i = \frac{\lambda}{1 - \frac{\partial P_{loss}}{\partial P_i}} = \frac{\lambda}{1 - (2 \sum_j B_{ij} P_j + B_{i0})} $$

*   At optimum: `F_i'(P_i) = λ_i` (not λ). Units with higher loss contribution (large ∂P_loss/∂P_i) have higher effective incremental cost.

• Iterative Solution Procedure:

  1. Assume initial P_i (e.g., equal sharing), calculate P_loss and ∂P_loss/∂P_i.

  2. Calculate λ_i for each unit.

  3. Find new P_i from F_i'(P_i) = λ_i.

  4. Check ∑ P_i - P_D - P_loss ≈ 0. If not, repeat from step 1.

Example Problem: Given two units with cost functions and B-coefficients, find optimal dispatch.

Steps:

1. Write F_1' = b_1 + 2c_1 P_1, F_2' = b_2 + 2c_2 P_2.

2. Calculate ∂P_loss/∂P_1 = 2B_11 P_1 + B_12 P_2 + B_10.

3. Set F_1' / (1 - ∂P_loss/∂P_1) = F_2' / (1 - ∂P_loss/∂P_2) = λ.

4. Solve simultaneous equations with P_1 + P_2 = P_D + P_loss(P_1,P_2).

5. Iterate until convergence.

8.0 ENERGY PRICING & TRANSMISSION SERVICE

8.1 Pricing of Energy (Generation Side)

• Marginal Cost: Cost of producing one additional MWh. Equals incremental fuel cost (λ) at optimal dispatch. Basis for spot pricing.

• Average Cost: Total cost / total MWh. Includes fixed (capital) and variable (fuel, O&M) costs.

• Components of Energy Price:

  • Fuel Cost: Variable, major component.

  • O&M Cost: Fixed and variable.

  • Capital Recovery: Fixed charge (interest, depreciation).

  • Profit Margin: In deregulated markets.

  • Ancillary Services: Cost of regulation, spinning reserve.

8.2 Pricing of Transmission Service (Wheeling Charges)

• Need: In open access, transmission owner must recover fixed costs (investment, maintenance) for wheeling (transmitting) third-party power.

• Methods:

  • Postage Stamp: Flat fee per MW of wheeled power, regardless of distance. Simple, non-economic.

  • Contract Path: Charge based on agreed path (distance × rate). Ignores physical flows.

  • MW-Mile: Charge = Rate ($/MW-mile) × MW × Miles. Better reflects usage.

  • Marginal Cost Pricing: Charge based on marginal cost of using the network (congestion cost). Economically efficient but complex.

• Embedded Cost vs. Marginal Cost:

  • Embedded Cost: Allocates historical/ sunk costs (depreciation, fixed O&M). Used for "revenue requirement".

  • Marginal Cost: Cost of serving one more customer. Provides efficient price signals but may not cover fixed costs.

9.0 SYNTHESIS & APPLICATION OF CONCEPTS

9.1 Interrelationship: Frequency, Real Power, Governor Control

• Frequency (f) ∝ Net Rotating Speed.

• Real Power Imbalance (ΔP = P_m - P_e) causes Accelerating Power (P_a) → changes Rotor Angle (δ) and Frequency (f).

• Governor Control (Primary) adjusts P_m in response to Δf (droop characteristic). ALFC (Secondary) adjusts setpoints to bring f and tie-line power to scheduled values.

• Key Equation: Swing equation links P_a to d²δ/dt²; frequency deviation Δf ∝ dδ/dt.

9.2 Interrelationship: Voltage, Reactive Power, Excitation/Reactive Compensation

• Voltage (V) at a bus is determined by Reactive Power (Q) balance.

• Reactive Power Flow causes voltage drops (ΔV ∝ Q X / V).

• Excitation System controls generator E (internal voltage) → controls Q output → controls local V.

• Shunt Capacitors/Reactors, SVC, STATCOM directly inject/absorb Q → control V profile.

• Series Capacitors reduce X → improve V profile and power transfer.

9.3 Integrated View: Load Flow, Stability, Control Studies

• Load Flow: Provides steady-state operating point (V, δ, P, Q) for a dispatch. This is the initial condition for stability studies.

• Stability Studies (Transient): Start from load flow solution, apply a disturbance (fault), simulate using swing equation (for each machine) coupled with network algebraic equations (from Ybus). Requires dynamic models of governors (turbine-governor) and exciters (AVR).

• Control Systems: Governor and AVR models are integral parts of transient stability simulation. Their settings (R, time constants, limits) directly affect stability margins.

• Economic Dispatch: Determines P_i setpoints used in load flow. Must consider transmission losses (B-coefficients) which come from network model (Ybus).

9.4 Application of EAC in Multi-Machine Systems (Conceptual Extension)

• For a multi-machine system, EAC is applied between each machine and the system center of inertia (COI).

• COI Angle (δ_COI): Weighted average of all machine angles.

• Swing Equation relative to COI:

$$ M_{total} \frac{d^2 \delta_{COI}}{dt^2} = \sum (P_{mi} - P_{ei}) = 0 \text{ (if COI is inertial frame)} $$

• For machine i: M_i d²δ_i/dt² = P_{mi} - P_{ei} - (M_i/M_total) Σ(P_m - P_e).

• Multi-Machine EAC: Plot P_{ei} - P_{mi} vs δ_i - δ_COI for each machine. Stability requires all machines to have A_acc = A_dec relative to COI. A machine losing synchronism with COI means system instability.

[!TIP] Exam Synthesis: Be ready to explain how a load flow solution feeds into a transient stability study, and how governor/AVR settings from control studies affect the stability result.