Formula Sheet: Generation

Synchronous Generators

Basic Generator Relationships

Synchronous Speed: \[n_s = \frac{120f}{P}\]

• \(n_s\) = synchronous speed (rpm)
• \(f\) = frequency (Hz)
• \(P\) = number of poles

Angular Synchronous Speed: \[\omega_s = \frac{4\pi f}{P} = \frac{2\pi n_s}{60}\]

• \(\omega_s\) = angular synchronous speed (rad/s)
• \(f\) = frequency (Hz)
• \(P\) = number of poles
• \(n_s\) = synchronous speed (rpm)

Generated Voltage (Per Phase): \[E_A = K\phi\omega\]

• \(E_A\) = induced voltage per phase (V)
• \(K\) = machine constant
• \(\phi\) = magnetic flux per pole (Wb)
• \(\omega\) = angular velocity (rad/s)

Generator Equivalent Circuit

Terminal Voltage: \[V_\phi = E_A - I_A(R_A + jX_S)\]

• \(V_\phi\) = terminal voltage per phase (V)
• \(E_A\) = internal generated voltage per phase (V)
• \(I_A\) = armature current per phase (A)
• \(R_A\) = armature resistance per phase (Ω)
• \(X_S\) = synchronous reactance per phase (Ω)

Synchronous Impedance: \[Z_S = R_A + jX_S\]

• \(Z_S\) = synchronous impedance per phase (Ω)
• \(R_A\) = armature resistance per phase (Ω)
• \(X_S\) = synchronous reactance per phase (Ω)

Synchronous Reactance: \[X_S = X_L + X_A\]

• \(X_S\) = synchronous reactance (Ω)
• \(X_L\) = leakage reactance (Ω)
• \(X_A\) = armature reaction reactance (Ω)

Power Relationships

Real Power per Phase: \[P_\phi = V_\phi I_A \cos\theta\]

• \(P_\phi\) = real power per phase (W)
• \(V_\phi\) = terminal voltage per phase (V)
• \(I_A\) = armature current (A)
• \(\theta\) = power factor angle

Total Three-Phase Real Power: \[P = 3V_\phi I_A \cos\theta = \sqrt{3}V_{LL}I_L\cos\theta\]

• \(P\) = total three-phase real power (W)
• \(V_\phi\) = phase voltage (V)
• \(V_{LL}\) = line-to-line voltage (V)
• \(I_A\) = armature current (A)
• \(I_L\) = line current (A)
• \(\theta\) = power factor angle

Reactive Power per Phase: \[Q_\phi = V_\phi I_A \sin\theta\]

• \(Q_\phi\) = reactive power per phase (VAR)
• \(V_\phi\) = terminal voltage per phase (V)
• \(I_A\) = armature current (A)
• \(\theta\) = power factor angle

Total Three-Phase Reactive Power: \[Q = 3V_\phi I_A \sin\theta = \sqrt{3}V_{LL}I_L\sin\theta\]

• \(Q\) = total three-phase reactive power (VAR)
• \(V_\phi\) = phase voltage (V)
• \(V_{LL}\) = line-to-line voltage (V)
• \(I_A\) = armature current (A)
• \(I_L\) = line current (A)
• \(\theta\) = power factor angle

Apparent Power per Phase: \[S_\phi = V_\phi I_A\]

• \(S_\phi\) = apparent power per phase (VA)
• \(V_\phi\) = terminal voltage per phase (V)
• \(I_A\) = armature current (A)

Total Three-Phase Apparent Power: \[S = 3V_\phi I_A = \sqrt{3}V_{LL}I_L\] \[S = \sqrt{P^2 + Q^2}\]

• \(S\) = total three-phase apparent power (VA)
• \(V_\phi\) = phase voltage (V)
• \(V_{LL}\) = line-to-line voltage (V)
• \(I_A\) = armature current (A)
• \(I_L\) = line current (A)

Power Angle Relationship

Power Output (Neglecting \(R_A\)): \[P = \frac{3E_AV_\phi}{X_S}\sin\delta\]

• \(P\) = total three-phase real power (W)
• \(E_A\) = internal generated voltage per phase (V)
• \(V_\phi\) = terminal voltage per phase (V)
• \(X_S\) = synchronous reactance per phase (Ω)
• \(\delta\) = power angle (torque angle) in radians
• Valid for cylindrical rotor machines

Maximum Power (Pull-Out Power): \[P_{max} = \frac{3E_AV_\phi}{X_S}\]

• \(P_{max}\) = maximum power output (W)
• Occurs when \(\delta = 90°\)
• Represents static stability limit

Salient Pole Power Output: \[P = \frac{3E_AV_\phi}{X_d}\sin\delta + \frac{3V_\phi^2}{2}\left(\frac{1}{X_q} - \frac{1}{X_d}\right)\sin(2\delta)\]

• \(P\) = total three-phase real power (W)
• \(E_A\) = internal generated voltage per phase (V)
• \(V_\phi\) = terminal voltage per phase (V)
• \(X_d\) = direct-axis synchronous reactance (Ω)
• \(X_q\) = quadrature-axis synchronous reactance (Ω)
• \(\delta\) = power angle (rad)
• First term: field excitation power
• Second term: reluctance power

Torque Relationships

Electromagnetic Torque: \[T_{em} = \frac{P}{\omega_s}\]

• \(T_{em}\) = electromagnetic torque (N·m)
• \(P\) = total three-phase power (W)
• \(\omega_s\) = synchronous angular speed (rad/s)

Torque in Terms of Power Angle: \[T_{em} = \frac{3E_AV_\phi}{\omega_s X_S}\sin\delta\]

• \(T_{em}\) = electromagnetic torque (N·m)
• \(E_A\) = internal generated voltage per phase (V)
• \(V_\phi\) = terminal voltage per phase (V)
• \(\omega_s\) = synchronous angular speed (rad/s)
• \(X_S\) = synchronous reactance (Ω)
• \(\delta\) = power angle (rad)

Voltage Regulation

Voltage Regulation: \[VR = \frac{E_{A,NL} - V_\phi,FL}{V_\phi,FL} \times 100\%\]

• \(VR\) = voltage regulation (%)
• \(E_{A,NL}\) = internal voltage at no-load (V)
• \(V_\phi,FL\) = terminal voltage per phase at full-load (V)
• At rated frequency and field current
• Positive VR indicates voltage drops under load

Excitation and Field Current

Field Current Relationship: \[E_A = K_f I_f\]

• \(E_A\) = internal generated voltage (V)
• \(K_f\) = field constant
• \(I_f\) = field current (A)
• Linear relationship in unsaturated region

Field Power: \[P_f = V_f I_f\]

• \(P_f\) = field power (W)
• \(V_f\) = field voltage (V)
• \(I_f\) = field current (A)

Efficiency and Losses

Generator Efficiency: \[\eta = \frac{P_{out}}{P_{in}} \times 100\% = \frac{P_{out}}{P_{out} + P_{losses}} \times 100\%\]

• \(\eta\) = efficiency (%)
• \(P_{out}\) = output power (W)
• \(P_{in}\) = input power (W)
• \(P_{losses}\) = total losses (W)

Total Losses: \[P_{losses} = P_{Cu} + P_{core} + P_{fw} + P_{stray}\]

• \(P_{Cu}\) = copper losses (I²R losses) (W)
• \(P_{core}\) = core losses (hysteresis and eddy current) (W)
• \(P_{fw}\) = friction and windage losses (W)
• \(P_{stray}\) = stray load losses (W)

Armature Copper Losses (Three-Phase): \[P_{Cu,armature} = 3I_A^2 R_A\]

• \(P_{Cu,armature}\) = armature copper losses (W)
• \(I_A\) = armature current per phase (A)
• \(R_A\) = armature resistance per phase (Ω)

Field Copper Losses: \[P_{Cu,field} = I_f^2 R_f\]

• \(P_{Cu,field}\) = field copper losses (W)
• \(I_f\) = field current (A)
• \(R_f\) = field resistance (Ω)

Short Circuit Ratio

Short Circuit Ratio (SCR): \[SCR = \frac{I_{f,OC}}{I_{f,SC}}\]

• \(SCR\) = short circuit ratio (dimensionless)
• \(I_{f,OC}\) = field current for rated voltage at open circuit (A)
• \(I_{f,SC}\) = field current for rated armature current at short circuit (A)
• Typical values: 0.5 to 1.0
• Higher SCR indicates better voltage regulation

Relationship to Synchronous Reactance: \[SCR \approx \frac{1}{X_S(pu)}\]

• \(X_S(pu)\) = synchronous reactance in per-unit
• Approximate inverse relationship

Parallel Operation of Generators

Conditions for Paralleling:

• Equal terminal voltages: \(V_1 = V_2\)
• Equal frequencies: \(f_1 = f_2\)
• Same phase sequence
• Voltages in phase: \(\theta_{12} = 0°\)

Real Power Sharing: \[P_1 = \frac{S_{base,1}}{S_{base,1} + S_{base,2}} \times P_{total}\]

• \(P_1\) = real power supplied by generator 1 (W)
• \(S_{base,1}\) = rated apparent power of generator 1 (VA)
• \(S_{base,2}\) = rated apparent power of generator 2 (VA)
• \(P_{total}\) = total real power demand (W)
• For equal droop characteristics

Reactive Power Sharing: \[Q_1 = \frac{S_{base,1}}{S_{base,1} + S_{base,2}} \times Q_{total}\]

• \(Q_1\) = reactive power supplied by generator 1 (VAR)
• \(S_{base,1}\) = rated apparent power of generator 1 (VA)
• \(S_{base,2}\) = rated apparent power of generator 2 (VA)
• \(Q_{total}\) = total reactive power demand (VAR)
• Controlled by excitation adjustment

Frequency-Power Droop: \[f = f_{no-load} - m \times P\]

• \(f\) = operating frequency (Hz)
• \(f_{no-load}\) = no-load frequency (Hz)
• \(m\) = droop coefficient (Hz/W)
• \(P\) = real power output (W)
• Typical droop: 3-5%

Induction Generators

Basic Relationships

Slip: \[s = \frac{n_s - n_r}{n_s}\]

• \(s\) = slip (dimensionless or %)
• \(n_s\) = synchronous speed (rpm)
• \(n_r\) = rotor speed (rpm)
• For generator operation: \(s < 0\)="" (rotor="" speed="" exceeds="" synchronous="">

Rotor Speed: \[n_r = n_s(1 - s)\]

• \(n_r\) = rotor speed (rpm)
• \(n_s\) = synchronous speed (rpm)
• \(s\) = slip

Rotor Frequency: \[f_r = sf_s\]

• \(f_r\) = rotor frequency (Hz)
• \(s\) = slip
• \(f_s\) = stator (line) frequency (Hz)

Equivalent Circuit Parameters

Rotor Impedance Referred to Stator: \[Z_r' = \frac{R_r'}{s} + jX_r'\]

• \(Z_r'\) = rotor impedance referred to stator (Ω)
• \(R_r'\) = rotor resistance referred to stator (Ω)
• \(X_r'\) = rotor reactance referred to stator (Ω)
• \(s\) = slip

Input Current: \[I_1 = \frac{V_1}{Z_{in}}\] where \[Z_{in} = R_1 + jX_1 + \frac{jX_m \left(\frac{R_r'}{s} + jX_r'\right)}{jX_m + \frac{R_r'}{s} + jX_r'}\]

• \(I_1\) = stator current (A)
• \(V_1\) = applied stator voltage per phase (V)
• \(R_1\) = stator resistance per phase (Ω)
• \(X_1\) = stator leakage reactance per phase (Ω)
• \(X_m\) = magnetizing reactance (Ω)
• \(R_r'\) = rotor resistance referred to stator (Ω)
• \(X_r'\) = rotor reactance referred to stator (Ω)

Power Relationships

Air Gap Power: \[P_{ag} = 3I_r'^2 \frac{R_r'}{s}\]

• \(P_{ag}\) = air gap power (W)
• \(I_r'\) = rotor current referred to stator (A)
• \(R_r'\) = rotor resistance referred to stator (Ω)
• \(s\) = slip
• Power transferred across air gap from stator to rotor

Rotor Copper Losses: \[P_{rcl} = 3I_r'^2 R_r' = s \times P_{ag}\]

• \(P_{rcl}\) = rotor copper losses (W)
• \(I_r'\) = rotor current referred to stator (A)
• \(R_r'\) = rotor resistance referred to stator (Ω)
• \(s\) = slip
• \(P_{ag}\) = air gap power (W)

Mechanical Power Developed: \[P_{mech} = P_{ag} - P_{rcl} = P_{ag}(1 - s) = 3I_r'^2 \frac{R_r'(1-s)}{s}\]

• \(P_{mech}\) = mechanical power developed (W)
• \(P_{ag}\) = air gap power (W)
• \(P_{rcl}\) = rotor copper losses (W)
• \(s\) = slip
• For generator: \(P_{mech}\) is input, \(P_{ag}\) is negative (power flows to stator)

Output Power (Generator Mode): \[P_{out} = P_{mech} - P_{rot} - P_{scl} - P_{core}\]

• \(P_{out}\) = electrical output power (W)
• \(P_{mech}\) = mechanical input power (W)
• \(P_{rot}\) = rotational losses (friction and windage) (W)
• \(P_{scl}\) = stator copper losses (W)
• \(P_{core}\) = core losses (W)

Stator Copper Losses: \[P_{scl} = 3I_1^2 R_1\]

• \(P_{scl}\) = stator copper losses (W)
• \(I_1\) = stator current per phase (A)
• \(R_1\) = stator resistance per phase (Ω)

Torque Relationships

Electromagnetic Torque: \[T_{em} = \frac{P_{ag}}{\omega_s} = \frac{3I_r'^2 R_r'}{s\omega_s}\]

• \(T_{em}\) = electromagnetic torque (N·m)
• \(P_{ag}\) = air gap power (W)
• \(\omega_s\) = synchronous angular speed (rad/s)
• \(I_r'\) = rotor current referred to stator (A)
• \(R_r'\) = rotor resistance referred to stator (Ω)
• \(s\) = slip

Shaft Torque: \[T_{shaft} = \frac{P_{mech}}{\omega_r} = \frac{P_{ag}(1-s)}{\omega_s(1-s)} = \frac{P_{ag}}{\omega_s}\]

• \(T_{shaft}\) = shaft torque (N·m)
• \(P_{mech}\) = mechanical power (W)
• \(\omega_r\) = rotor angular speed (rad/s)
• Note: \(T_{shaft} = T_{em}\) neglecting rotational losses

Maximum Torque (Breakdown Torque): \[T_{max} = \frac{3V_1^2}{2\omega_s\left[R_1 + \sqrt{R_1^2 + (X_1 + X_r')^2}\right]}\]

• \(T_{max}\) = maximum torque (N·m)
• \(V_1\) = applied voltage per phase (V)
• \(\omega_s\) = synchronous angular speed (rad/s)
• \(R_1\) = stator resistance (Ω)
• \(X_1\) = stator leakage reactance (Ω)
• \(X_r'\) = rotor reactance referred to stator (Ω)

Slip at Maximum Torque: \[s_{max} = \pm\frac{R_r'}{\sqrt{R_1^2 + (X_1 + X_r')^2}}\]

• \(s_{max}\) = slip at maximum torque
• \(R_r'\) = rotor resistance referred to stator (Ω)
• \(R_1\) = stator resistance (Ω)
• \(X_1\) = stator leakage reactance (Ω)
• \(X_r'\) = rotor reactance referred to stator (Ω)
• Negative sign for generator mode

Efficiency

Generator Efficiency: \[\eta = \frac{P_{out}}{P_{in}} \times 100\% = \frac{P_{out}}{P_{mech}} \times 100\%\]

• \(\eta\) = efficiency (%)
• \(P_{out}\) = electrical output power (W)
• \(P_{mech}\) = mechanical input power (W)

Reactive Power Requirements

Reactive Power Consumed: \[Q = 3V_1I_1\sin\theta\]

• \(Q\) = reactive power consumed (VAR)
• \(V_1\) = terminal voltage per phase (V)
• \(I_1\) = stator current (A)
• \(\theta\) = phase angle between voltage and current
• Induction generators always consume reactive power
• Requires capacitor bank or grid for excitation

Capacitor Requirements for Self-Excitation: \[Q_C \geq Q_{magnetization} + Q_{load}\]

• \(Q_C\) = reactive power supplied by capacitors (VAR)
• \(Q_{magnetization}\) = reactive power for magnetization (VAR)
• \(Q_{load}\) = reactive power required by load (VAR)

Prime Movers and Turbines

Steam Turbines

Ideal Turbine Power: \[P = \dot{m}(h_1 - h_2)\]

• \(P\) = power output (W)
• \(\dot{m}\) = mass flow rate of steam (kg/s)
• \(h_1\) = inlet enthalpy (J/kg)
• \(h_2\) = outlet enthalpy (J/kg)

Turbine Efficiency: \[\eta_t = \frac{h_1 - h_2}{h_1 - h_{2s}}\]

• \(\eta_t\) = turbine isentropic efficiency
• \(h_1\) = inlet enthalpy (J/kg)
• \(h_2\) = actual outlet enthalpy (J/kg)
• \(h_{2s}\) = isentropic outlet enthalpy (J/kg)

Steam Rate: \[SR = \frac{\dot{m}}{P_{out}}\]

• \(SR\) = steam rate (kg/kWh or lb/kWh)
• \(\dot{m}\) = steam mass flow rate (kg/h or lb/h)
• \(P_{out}\) = turbine power output (kW)

Heat Rate: \[HR = \frac{Q_{in}}{P_{out}}\]

• \(HR\) = heat rate (kJ/kWh or Btu/kWh)
• \(Q_{in}\) = heat input (kJ/h or Btu/h)
• \(P_{out}\) = power output (kW)
• Lower heat rate indicates better performance

Hydraulic Turbines

Theoretical Power: \[P = \rho g Q H\]

• \(P\) = theoretical power (W)
• \(\rho\) = water density (kg/m³, typically 1000 kg/m³)
• \(g\) = acceleration due to gravity (9.81 m/s²)
• \(Q\) = flow rate (m³/s)
• \(H\) = head (m)

Actual Power Output: \[P_{out} = \eta_t \rho g Q H\]

• \(P_{out}\) = actual power output (W)
• \(\eta_t\) = turbine efficiency
• \(\rho\) = water density (kg/m³)
• \(g\) = acceleration due to gravity (m/s²)
• \(Q\) = flow rate (m³/s)
• \(H\) = head (m)

Specific Speed: \[n_s = \frac{n\sqrt{P}}{H^{5/4}}\]

• \(n_s\) = specific speed (dimensionless or customary units)
• \(n\) = rotational speed (rpm)
• \(P\) = power output (hp or kW)
• \(H\) = head (ft or m)
• Used to classify turbine type
• Pelton: 4-16; Francis: 40-400; Kaplan: 340-1000 (US customary units)

Turbine Efficiency: \[\eta_t = \frac{P_{shaft}}{\rho g Q H}\]

• \(\eta_t\) = turbine efficiency
• \(P_{shaft}\) = shaft power output (W)
• \(\rho\) = water density (kg/m³)
• \(g\) = acceleration due to gravity (m/s²)
• \(Q\) = flow rate (m³/s)
• \(H\) = head (m)

Gas Turbines

Brayton Cycle Efficiency: \[\eta_{Brayton} = 1 - \frac{1}{r_p^{(\gamma-1)/\gamma}}\]

• \(\eta_{Brayton}\) = ideal Brayton cycle efficiency
• \(r_p\) = pressure ratio (dimensionless)
• \(\gamma\) = specific heat ratio (typically 1.4 for air)

Pressure Ratio: \[r_p = \frac{P_2}{P_1}\]

• \(r_p\) = pressure ratio
• \(P_2\) = compressor outlet pressure (Pa)
• \(P_1\) = compressor inlet pressure (Pa)

Compressor Work: \[W_c = \dot{m}c_p(T_2 - T_1)\]

• \(W_c\) = compressor work (W)
• \(\dot{m}\) = air mass flow rate (kg/s)
• \(c_p\) = specific heat at constant pressure (J/kg·K)
• \(T_2\) = compressor outlet temperature (K)
• \(T_1\) = compressor inlet temperature (K)

Turbine Work: \[W_t = \dot{m}c_p(T_3 - T_4)\]

• \(W_t\) = turbine work (W)
• \(\dot{m}\) = gas mass flow rate (kg/s)
• \(c_p\) = specific heat at constant pressure (J/kg·K)
• \(T_3\) = turbine inlet temperature (K)
• \(T_4\) = turbine outlet temperature (K)

Net Power Output: \[P_{net} = W_t - W_c\]

• \(P_{net}\) = net power output (W)
• \(W_t\) = turbine work (W)
• \(W_c\) = compressor work (W)

Wind Turbines

Power in Wind: \[P_{wind} = \frac{1}{2}\rho A v^3\]

• \(P_{wind}\) = power in wind (W)
• \(\rho\) = air density (kg/m³, typically 1.225 kg/m³)
• \(A\) = swept area of rotor (m²)
• \(v\) = wind velocity (m/s)

Swept Area: \[A = \pi R^2\]

• \(A\) = swept area (m²)
• \(R\) = rotor blade radius (m)

Power Extracted by Turbine: \[P = C_p \times \frac{1}{2}\rho A v^3\]

• \(P\) = power extracted (W)
• \(C_p\) = power coefficient (dimensionless)
• \(\rho\) = air density (kg/m³)
• \(A\) = swept area (m²)
• \(v\) = wind velocity (m/s)
• Maximum theoretical \(C_p = 0.593\) (Betz limit)
• Practical \(C_p\) typically 0.35-0.45

Tip Speed Ratio: \[\lambda = \frac{\omega R}{v}\]

• \(\lambda\) = tip speed ratio (dimensionless)
• \(\omega\) = angular velocity of rotor (rad/s)
• \(R\) = rotor blade radius (m)
• \(v\) = wind velocity (m/s)
• \(C_p\) is a function of \(\lambda\)

Torque: \[T = \frac{P}{\omega}\]

• \(T\) = torque (N·m)
• \(P\) = power (W)
• \(\omega\) = angular velocity (rad/s)

Excitation Systems

DC Excitation Systems

Exciter Voltage: \[V_f = K_e I_f + R_f I_f\]

• \(V_f\) = field voltage (V)
• \(K_e\) = exciter constant
• \(I_f\) = field current (A)
• \(R_f\) = field resistance (Ω)

Field Time Constant: \[\tau_f = \frac{L_f}{R_f}\]

• \(\tau_f\) = field time constant (s)
• \(L_f\) = field inductance (H)
• \(R_f\) = field resistance (Ω)

Static Excitation Systems

Exciter Response: \[V_f(t) = V_{f,final}\left(1 - e^{-t/\tau_e}\right)\]

• \(V_f(t)\) = field voltage at time t (V)
• \(V_{f,final}\) = final field voltage (V)
• \(t\) = time (s)
• \(\tau_e\) = exciter time constant (s)

Automatic Voltage Regulator (AVR)

Voltage Error: \[\Delta V = V_{ref} - V_t\]

• \(\Delta V\) = voltage error (V)
• \(V_{ref}\) = reference voltage (V)
• \(V_t\) = terminal voltage (V)

Proportional Control: \[V_f = K_A \Delta V\]

• \(V_f\) = field voltage (V)
• \(K_A\) = regulator gain
• \(\Delta V\) = voltage error (V)

Response Ratio (IEEE Definition): \[RR = \frac{\Delta V_f}{T_{\Delta V_f}} \times \frac{1}{V_{f,rated}}\]

• \(RR\) = response ratio (per second)
• \(\Delta V_f\) = change in exciter voltage (V)
• \(T_{\Delta V_f}\) = time to achieve voltage change (s)
• \(V_{f,rated}\) = rated field voltage (V)
• Typical value: 0.5 to 2.0 per second

Renewable Energy Generation

Solar Photovoltaic Systems

Solar Cell Output Power: \[P = \eta \times A \times G\]

• \(P\) = output power (W)
• \(\eta\) = conversion efficiency (decimal)
• \(A\) = cell area (m²)
• \(G\) = solar irradiance (W/m²)
• Standard test conditions: G = 1000 W/m², T = 25°C

Temperature Effect on Voltage: \[V(T) = V_{ref} + \beta(T - T_{ref})\]

• \(V(T)\) = voltage at temperature T (V)
• \(V_{ref}\) = voltage at reference temperature (V)
• \(\beta\) = temperature coefficient (V/°C, typically negative)
• \(T\) = operating temperature (°C)
• \(T_{ref}\) = reference temperature (°C, typically 25°C)

Fill Factor: \[FF = \frac{V_{mp} \times I_{mp}}{V_{oc} \times I_{sc}}\]

• \(FF\) = fill factor (dimensionless, 0 to 1)
• \(V_{mp}\) = voltage at maximum power point (V)
• \(I_{mp}\) = current at maximum power point (A)
• \(V_{oc}\) = open circuit voltage (V)
• \(I_{sc}\) = short circuit current (A)
• Typical values: 0.7-0.85

PV Array Configuration:

• Series connection: \(V_{total} = N_s \times V_{module}\)
• Series connection: \(I_{total} = I_{module}\)
• Parallel connection: \(V_{total} = V_{module}\)
• Parallel connection: \(I_{total} = N_p \times I_{module}\)
• \(N_s\) = number of modules in series
• \(N_p\) = number of parallel strings

Energy Storage

Battery Energy Capacity: \[E = V \times Q\]

• \(E\) = energy capacity (Wh)
• \(V\) = nominal voltage (V)
• \(Q\) = charge capacity (Ah)

State of Charge (SOC): \[SOC = \frac{Q_{remaining}}{Q_{rated}} \times 100\%\]

• \(SOC\) = state of charge (%)
• \(Q_{remaining}\) = remaining charge capacity (Ah)
• \(Q_{rated}\) = rated charge capacity (Ah)

Depth of Discharge (DOD): \[DOD = 100\% - SOC\]

• \(DOD\) = depth of discharge (%)
• \(SOC\) = state of charge (%)

Round-Trip Efficiency: \[\eta_{rt} = \frac{E_{discharge}}{E_{charge}}\]

• \(\eta_{rt}\) = round-trip efficiency
• \(E_{discharge}\) = energy discharged (Wh)
• \(E_{charge}\) = energy charged (Wh)

Generator Protection and Control

Generator Capability Curves

Armature Current Limit (Circular): \[P^2 + Q^2 = (S_{rated})^2\]

• \(P\) = real power (W)
• \(Q\) = reactive power (VAR)
• \(S_{rated}\) = rated apparent power (VA)
• Represents stator current heating limit

Field Current Limit (Lagging): \[Q = P\tan(\cos^{-1}(PF_{lag}))\]

• \(Q\) = reactive power (VAR)
• \(P\) = real power (W)
• \(PF_{lag}\) = lagging power factor limit
• Represents rotor heating limit

Minimum Excitation Limit (Leading):

• Prevents loss of synchronism and stator end-region heating
• Typically 0.90-0.95 leading power factor at rated power

Over/Under Excitation

Reactive Power vs. Excitation:

• Overexcited: Generator delivers reactive power (lagging current)
• Underexcited: Generator absorbs reactive power (leading current)
• Unity power factor: \(Q = 0\)

V-Curve Relationship:

• Plot of armature current vs. field current at constant power
• Minimum armature current occurs at unity power factor
• Increased field current → lagging power factor
• Decreased field current → leading power factor

Governor Control

Speed Droop (Regulation): \[R = \frac{n_{NL} - n_{FL}}{n_{FL}} \times 100\%\]

• \(R\) = speed regulation or droop (%)
• \(n_{NL}\) = no-load speed (rpm)
• \(n_{FL}\) = full-load speed (rpm)
• Typical values: 3-5% for parallel operation

Governor Droop Characteristic: \[f = f_{set} - K_{droop} \times P\]

• \(f\) = operating frequency (Hz)
• \(f_{set}\) = set-point frequency (Hz)
• \(K_{droop}\) = droop constant (Hz/MW)
• \(P\) = power output (MW)

Droop Setting: \[Droop\% = \frac{\Delta f}{f_{rated}} \times \frac{P_{rated}}{\Delta P} \times 100\%\]

• \(Droop\%\) = droop percentage
• \(\Delta f\) = frequency change from no-load to full-load (Hz)
• \(f_{rated}\) = rated frequency (Hz)
• \(P_{rated}\) = rated power (W)
• \(\Delta P\) = change in power output (W)

Power Plant Performance

Thermal Plant Efficiency

Overall Plant Efficiency: \[\eta_{overall} = \frac{P_{electrical}}{Q_{fuel}}\]

• \(\eta_{overall}\) = overall plant efficiency
• \(P_{electrical}\) = electrical power output (W)
• \(Q_{fuel}\) = fuel energy input rate (W)

Component Efficiencies: \[\eta_{overall} = \eta_{boiler} \times \eta_{turbine} \times \eta_{generator}\]

• \(\eta_{boiler}\) = boiler efficiency
• \(\eta_{turbine}\) = turbine efficiency
• \(\eta_{generator}\) = generator efficiency

Carnot Efficiency: \[\eta_{Carnot} = 1 - \frac{T_C}{T_H}\]

• \(\eta_{Carnot}\) = Carnot efficiency (theoretical maximum)
• \(T_C\) = absolute temperature of cold reservoir (K)
• \(T_H\) = absolute temperature of hot reservoir (K)

Capacity Factor and Availability

Capacity Factor: \[CF = \frac{E_{actual}}{E_{rated}} \times 100\% = \frac{P_{avg}}{P_{rated}} \times 100\%\]

• \(CF\) = capacity factor (%)
• \(E_{actual}\) = actual energy produced over time period (kWh)
• \(E_{rated}\) = energy at rated capacity over same period (kWh)
• \(P_{avg}\) = average power output (kW)
• \(P_{rated}\) = rated power capacity (kW)

Availability Factor: \[AF = \frac{t_{available}}{t_{total}} \times 100\%\]

• \(AF\) = availability factor (%)
• \(t_{available}\) = time unit is available for operation (h)
• \(t_{total}\) = total time in period (h)

Forced Outage Rate: \[FOR = \frac{t_{forced\,outage}}{t_{forced\,outage} + t_{service}} \times 100\%\]

• \(FOR\) = forced outage rate (%)
• \(t_{forced outage}\) = hours of forced outage
• \(t_{service}\) = hours in service

Fuel Consumption

Fuel Energy Content: \[Q_{fuel} = \dot{m}_{fuel} \times HHV\]

• \(Q_{fuel}\) = fuel energy input rate (W)
• \(\dot{m}_{fuel}\) = fuel mass flow rate (kg/s)
• \(HHV\) = higher heating value (J/kg)
• Can also use LHV (lower heating value)

Specific Fuel Consumption: \[SFC = \frac{\dot{m}_{fuel}}{P_{output}}\]

• \(SFC\) = specific fuel consumption (kg/kWh)
• \(\dot{m}_{fuel}\) = fuel mass flow rate (kg/h)
• \(P_{output}\) = power output (kW)

Combined Heat and Power (Cogeneration)

CHP Performance Metrics

Electrical Efficiency: \[\eta_e = \frac{P_{electrical}}{Q_{fuel}}\]

• \(\eta_e\) = electrical efficiency
• \(P_{electrical}\) = electrical power output (W)
• \(Q_{fuel}\) = fuel energy input (W)

Thermal Efficiency: \[\eta_t = \frac{Q_{useful}}{Q_{fuel}}\]

• \(\eta_t\) = thermal efficiency
• \(Q_{useful}\) = useful thermal energy output (W)
• \(Q_{fuel}\) = fuel energy input (W)

Overall CHP Efficiency: \[\eta_{CHP} = \eta_e + \eta_t = \frac{P_{electrical} + Q_{useful}}{Q_{fuel}}\]

• \(\eta_{CHP}\) = overall CHP efficiency
• \(P_{electrical}\) = electrical power output (W)
• \(Q_{useful}\) = useful thermal energy output (W)
• \(Q_{fuel}\) = fuel energy input (W)
• Typical values: 70-90%

Power-to-Heat Ratio: \[PHR = \frac{P_{electrical}}{Q_{useful}}\]

• \(PHR\) = power-to-heat ratio (dimensionless)
• \(P_{electrical}\) = electrical power output (W)
• \(Q_{useful}\) = useful thermal energy output (W)

Fuel Utilization Effectiveness: \[FUE = \frac{P_{electrical} + Q_{useful}}{Q_{fuel}} \times 100\%\]

• \(FUE\) = fuel utilization effectiveness (%)
• \(P_{electrical}\) = electrical power output (W)
• \(Q_{useful}\) = useful thermal energy recovered (W)
• \(Q_{fuel}\) = fuel energy input (W)