--- title: "Steady state and transient behaviour with governor" subject: "Power Plant Equipment" chapter: 2 key: "semester_7/power_plant_equipment:governor-steady-transient" source: https://www.bctengineer.com/q/semester_7/power_plant_equipment/governor-steady-transient site: BCT Engineering --- # Steady state and transient behaviour with governor Power Plant Equipment · Chapter 2: Power/Frequency Control in Hydro Generator System - Asked in: 2073 Shrawan, 6 marks; 2070 Asar, 8 marks (asked 2 times) - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/power_plant_equipment/governor-steady-transient - Whole chapter: https://www.bctengineer.com/md/semester_7/power_plant_equipment/chapter-2 ## Question Describe the steady state and transient behavior of a turbine generator coupled system with governor. ## Short answer Steady-state: frequency deviation Δf = -ΔP_L/(D+1/R), load shared per droop. Transient: after load increase, electrical power jumps instantly, mechanical power lags, frequency falls, governor then corrects; hydro power initially dips due to water inertia. ## Answer A turbine-generator with a speed governor is a closed loop: the governor senses speed and adjusts the gate so that mechanical input matches electrical load. #### Model and block diagram Rotor and load: $\Delta\Omega(s) = \dfrac{\Delta P_m - \Delta P_L}{2Hs + D}$ Governor with droop: $\Delta P_v(s) = \dfrac{1}{1+sT_g}\left[\Delta P_{ref} - \dfrac{\Delta\Omega}{R}\right]$ Turbine: $\Delta P_m = G_t(s)\,\Delta P_v$ with $G_t(s) = \dfrac{1 - T_w s}{1+0.5T_w s}$ for hydro. ```text dPref + +-------+ dPv +-------+ dPm ---->(+)---->| 1 |---->|turbine|---+ ^ - |1+sTg | | Gt(s) | | | +-------+ +-------+ v +-----+ -dPL --->(+) | 1/R | | +-----+ +-----------+ | ^ dw | 1 | | +---------| 2Hs + D |<---------+ +-----------+ ``` #### Steady state behaviour With $\Delta P_{ref} = 0$ and a step load $\Delta P_L$, the closed loop gives $$ \frac{\Delta\Omega(s)}{\Delta P_L(s)} = \frac{-1}{2Hs + D + \dfrac{G_t(s)}{R(1+sT_g)}} $$ Using the final value theorem ($G_t(0) = 1$): $$ \Delta\omega_{ss} = \frac{-\Delta P_L}{D + \frac{1}{R}} = \frac{-\Delta P_L}{\beta} $$ where $\beta = D + 1/R$ is the **area frequency response characteristic**. Since $1/R \gg D$ (e.g. $R = 0.05$ gives $1/R = 20$), the governor reduces the frequency error about 20 times compared with no governor. - **Speed-power relation:** $\Delta P_m = \Delta P_{ref} - \Delta\omega/R$. This is the droop line: as load rises, speed falls in proportion. - Units in parallel share a load change in proportion to $1/R$ (i.e. to rating, if all have the same per-unit droop). - Changing $\Delta P_{ref}$ shifts the droop line up or down; this restores frequency to nominal (secondary control). #### Transient behaviour When a load $\Delta P_L$ is suddenly added: 1. **Instant $t = 0^+$:** the electrical output $P_e$ rises at once by $\Delta P_L$, because the generator must supply the connected load immediately. The mechanical input $P_m$ cannot change instantly (gate and water need time). 2. **Deceleration:** the deficit $P_e - P_m$ is supplied from the kinetic energy of the rotor, so speed starts to fall with slope $-\Delta P_L/2H$. 3. **Load damping:** as speed falls, frequency-sensitive load drops by $D\Delta\omega$, so $P_e$ comes down slightly from its initial jump. 4. **Governor action:** the speed fall is sensed; through the droop $1/R$ the governor opens the gate. For a hydro turbine, $P_m$ first **dips** (water inertia, non-minimum phase) and then rises. 5. **Recovery:** when $P_m$ exceeds $P_e$, the rotor accelerates and the fall in speed is arrested. After some oscillation, $P_m = P_e$ and speed settles at a new, lower value. Steady state, from the block diagram ($s \to 0$): $$ \Delta\omega_{ss} = \frac{-\Delta P_L}{D + \frac{1}{R}}, \qquad \Delta P_{m,ss} = \Delta P_{e,ss} = \Delta P_L + D\,\Delta\omega_{ss} $$ ```text power | Pe P0+ | ._____ dPL | | '--.________ Pm = Pe (final) | | _.-' | | _.' Pm P0 |---'. .' | '.' (hydro: Pm dips first) +---------------------------> t speed |----. | \ .-------- | '._ _.' dw_ss = -dPL/(D+1/R) | '' +---------------------------> t ``` - **Electrical power $P_e$:** jumps by $\Delta P_L$ at once, then reduces slightly as frequency falls (load damping) and settles at $\Delta P_L + D\Delta\omega_{ss}$. - **Mechanical power $P_m$:** starts at the old value (dips first in a hydro unit), rises with the governor and turbine time constants, overshoots slightly, and settles equal to $P_e$. - **Speed:** falls, reaches a minimum (frequency nadir), recovers partly and settles at a lower value $\Delta\omega_{ss}$ fixed by the droop. With an isochronous governor ($R \to 0$) speed would return to nominal. Supplementary control (load reference change) is then used to bring the frequency back to 50 Hz. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the steady-state speed deviation for a step load dPL with governor droop R? A. -dPL/(D + 1/R) B. -dPL/D C. -dPL R D. -dPL/(D R) Answer: A. By the final value theorem, dw = -dPL/beta with beta = D + 1/R. (Key point: Steady-state error) 2. Why does the governor reduce frequency error about 20 times for R = 0.05? A. R reduces H B. 1/R = 20 is much larger than D C. D becomes zero D. Gt(0) is 20 Answer: B. beta = D + 1/R and 1/R (20) is much greater than D (1 to 2). (Key point: Area frequency response) 3. What happens at t = 0+ when load is suddenly added? A. Pm jumps by dPL B. Speed rises C. Pe rises at once while Pm stays, so rotor kinetic energy supplies the deficit D. The governor closes the gate Answer: C. Pe rises immediately, Pm cannot change instantly, and speed falls at -dPL/2H initially. (Key point: Instant after load added) 4. What is special about Pm in a hydro unit after a speed fall? A. Pm rises instantly B. Pm stays constant C. Pm oscillates only D. Pm first dips due to water inertia, then rises Answer: D. The non-minimum phase turbine makes Pm initially move opposite to the gate. (Key point: Hydro transient) ## Related in Power Plant Equipment - [Transient response of turbine-generator with governor](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Agovernor-transient-response) — Power Plant Equipment, chapter 2, asked 3× - [Isochronous generator transient response](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aisochronous-generator-transient) — Power Plant Equipment, chapter 2 ## Related in other subjects - [Steady state vs transient stability; enhancement](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Asteady-transient-stability-enhancement) — Power System Analysis II, chapter 6, asked 2× - [Purpose of governor and governing of Francis turbine](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Agoverning-francis-turbine) — Hydro Power, chapter 7 - [Steady state work vs steady flow; turbine, nozzle](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Asteady-state-work-vs-flow) — Fundamental of Thermodynamics & Heat Transfer, chapter 4 - [Requirement elicitation and system behavior](https://www.bctengineer.com/question.md?key=semester_6%2Fobject_oriented_analysis_and_design%3Arequirement-elicitation-in-ooa) — Object Oriented Analysis and Design, chapter 2