--- title: "Effects of real and reactive power mismatch" subject: "Power System Analysis II" chapter: 1 key: "semester_5/power_system_analysis_ii:pf-qv-mismatch-effects" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/pf-qv-mismatch-effects site: BCT Engineering --- # Effects of real and reactive power mismatch Power System Analysis II · Chapter 1: Interconnected Power System - Asked in: 2076 Chaitra, 4 marks; 2075 Chaitra, 8 marks; 2073 Chaitra, 8 marks; 2071 Chaitra, 5 marks (asked 4 times) - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/pf-qv-mismatch-effects - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-1 ## Question Explain how the mismatch in active power affects the system frequency and mismatch in reactive power affects the voltage magnitude in an interconnected power system. ## Short answer Active power mismatch changes system frequency: a deficit slows rotors, lowering frequency; a surplus raises it. Reactive power mismatch alters local bus voltage magnitudes: a deficit causes voltage drop across line reactance, surplus causes rise. The two are decoupled due to high X/R ratio. ## Answer In an interconnected system, a **mismatch of active power** (generation ≠ load + losses) shows up as a change in **frequency**, while a **mismatch of reactive power** shows up as a change in **bus voltage magnitude**. The two effects are nearly independent ("decoupled") because of the high X/R ratio of transmission lines. #### Active power mismatch and frequency All synchronous machines in an interconnected system run at the same electrical speed. The rotor of each machine obeys the swing equation: $$ \frac{2H}{\omega_s}\frac{d^2\delta}{dt^2} = P_m - P_e $$ - If load increases suddenly ($P_e > P_m$), the extra energy is first taken from the **kinetic energy** stored in the rotating masses ($\tfrac{1}{2}J\omega^2$). The rotors slow down and system frequency **falls**. - If load drops ($P_e < P_m$), the surplus energy accelerates the rotors and frequency **rises**. - Since frequency is common to the whole network, the mismatch is shared by all units in proportion to their inertia and then their governor droop. Frequency is restored in steps: 1. **Primary control (governor):** each governor senses the speed change and opens/closes the steam or water valve. Steady-state change is $\Delta f = -\dfrac{\Delta P_L}{\sum (1/R_i) + D}$, so a small offset remains. 2. **Secondary control (AGC / LFC):** the speed changer is adjusted until $\Delta f = 0$ and tie-line flows return to scheduled values (area control error $ACE = \Delta P_{tie} + B\,\Delta f \to 0$). #### Reactive power mismatch and voltage For a line between buses with voltages $V_1$, $V_2$ and reactance $X$ (R neglected, small $\delta$): $$ Q_{12} \approx \frac{V_1(V_1 - V_2)}{X} \quad\Rightarrow\quad \Delta V \approx \frac{Q X}{V} $$ - Reactive power flows from the higher-voltage bus to the lower-voltage bus, and the voltage drop across a line depends mainly on the Q it carries. - If reactive demand exceeds local supply, Q must be imported over the reactance, so the **voltage at that bus falls**. Excess Q (e.g. a lightly loaded long line with its charging) makes the **voltage rise** (Ferranti effect). - Unlike frequency, voltage is a **local** quantity. Reactive power cannot be sent far efficiently, so the mismatch must be fixed near where it occurs. Control: generator excitation (AVR), shunt capacitors/reactors, synchronous condensers, SVC/STATCOM and on-load tap-changing transformers. #### Why P–f and Q–V are decoupled Since $X \gg R$: $$ P \approx \frac{V_1V_2}{X}\sin\delta,\qquad Q \approx \frac{V_1(V_1 - V_2\cos\delta)}{X} $$ P depends mainly on the angle $\delta$ (linked to rotor position and frequency); Q depends mainly on the voltage magnitudes. So $\partial P/\partial \delta$ and $\partial Q/\partial |V|$ are large, while $\partial P/\partial |V|$ and $\partial Q/\partial \delta$ are small. | Mismatch | Affects | Nature | Corrected by | |---|---|---|---| | Active power P | Frequency f | System-wide | Governor, AGC | | Reactive power Q | Voltage magnitude | Local | AVR, capacitors, tap changers | ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In an interconnected system, an active power mismatch (generation not equal to load + losses) mainly appears as a change in: A. system frequency B. bus voltage magnitude C. line reactance D. power factor of generators only Answer: A. Active power imbalance is taken from or added to rotor kinetic energy, so speed and frequency change. (Key point: P mismatch shows as frequency) 2. If load suddenly increases so that Pe > Pm, what happens first? A. Rotors accelerate and frequency rises B. Energy is drawn from rotor kinetic energy, rotors slow and frequency falls C. Voltage rises at all buses while frequency stays constant D. Governors instantly restore frequency to exactly 50 Hz Answer: B. The deficit is supplied from stored kinetic energy, so rotors decelerate and frequency falls. (Key point: Load increase effect) 3. Why does a reactive power deficit at a bus lower its voltage? A. Q flows from the lower-voltage bus to the higher one B. Reactive power changes the rotor speed C. Q must be imported over line reactance, and the drop is about QX/V D. The line resistance drops to zero Answer: C. The voltage drop across a line depends mainly on the Q carried over its reactance. (Key point: Q mismatch and voltage) 4. P-f and Q-V are nearly decoupled because: A. R >> X, so P depends on voltage magnitude B. Frequency and voltage are always equal in a grid C. Q is a system-wide quantity like frequency D. X >> R, so P depends mainly on angle delta and Q mainly on voltage magnitude Answer: D. With high X/R, dP/d|V| and dQ/d(delta) are small. (Key point: Decoupling) ## Related in Power System Analysis II - [Real power/frequency control in interconnected system](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Areal-power-frequency-control) — Power System Analysis II, chapter 1 - [Interconnected system load-frequency and Q-V control](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Ainterconnection-control-schematic) — Power System Analysis II, chapter 1 - [Interconnection advantages and real power-frequency balance](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Ainterconnection-advantages-frequency) — Power System Analysis II, chapter 1, asked 3× - [Reactive power and voltage balance](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Areactive-power-voltage-balance) — Power System Analysis II, chapter 1 - [Generator reactive power at constant terminal voltage](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Agenerator-reactive-power-infinite-bus) — Power System Analysis II, chapter 1 - [Advantages of interconnected power system](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Ainterconnection-advantages) — Power System Analysis II, chapter 1, asked 3× ## Related in other subjects - [Direction of active and reactive power flow](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Adirection-active-reactive-flow) — Power System Analysis I, chapter 6 - [Line as source and sink of reactive power, effects](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aline-source-sink-reactive-effects) — Power System Analysis I, chapter 6 - [Reactive power proportional to voltage drop](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Areactive-power-voltage-drop) — Power System Analysis I, chapter 6 - [Interconnected power system and advantages](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Ainterconnected-system-advantages) — Power System Analysis I, chapter 1