Electrical Engineering Fundamentals

A PID controller (Proportional–Integral–Derivative controller) is a closed-loop feedback controller that continuously computes an error signal — the difference between a measured process variable and a desired setpoint — and produces a control output by combining three separate actions: proportional, integral, and derivative. It is the most widely used controller in industrial and power system automation because it is simple, robust, and effective for a wide range of processes.

Mathematical form

The controller output is:

$u(t) = K_p\, e(t) + K_i \int_0^t e(\tau)\,d\tau + K_d\, \frac{de(t)}{dt}$

where $e(t) = SP - PV$ (setpoint minus process variable), and $K_p$, $K_i$, $K_d$ are the proportional, integral and derivative gains. In Laplace form:

$G_c(s) = K_p + \frac{K_i}{s} + K_d s = K_p\left(1 + \frac{1}{T_i s} + T_d s\right)$

where $T_i = K_p/K_i$ is the integral time and $T_d = K_d/K_p$ is the derivative time.

Working principle of the three actions

1. Proportional (P) action — $K_p e(t)$ Produces an output proportional to the present error. A large error gives a large corrective output. Its limitation is steady-state offset: a pure P controller settles at a point where some residual error is needed to hold the output. Increasing $K_p$ reduces offset but causes oscillation and eventual instability.

2. Integral (I) action — $K_i \int e\,dt$ Integrates the error over time. As long as any error persists, the integral term keeps accumulating and drives the output further, thereby eliminating steady-state error. Its drawback is integral windup (the term saturates during large disturbances) and reduced phase margin, which slows the response and can cause overshoot. Windup is handled by anti-windup clamping or back-calculation.

3. Derivative (D) action — $K_d\, de/dt$ Responds to the rate of change of error, providing a "lead" or anticipatory action. It damps overshoot and improves transient response and stability margin. Because it differentiates, it amplifies measurement noise, so in practice a first-order filter is added to the derivative term.

The three actions together give fast rise time (P), zero steady-state error (I), and reduced overshoot with better damping (D). Gains are tuned by Ziegler–Nichols, Cohen–Coon, or trial-and-error methods.

Significance in power system control

Power systems require continuous balance between generation and demand at nominal frequency and voltage. PID controllers are significant because:

• Frequency control (AGC/LFC): In automatic generation control, the area control error (ACE) — a combination of frequency deviation $\Delta f$ and tie-line power deviation $\Delta P_{tie}$ — is fed to a PID controller that adjusts the governor setpoint of generating units. The integral action removes the steady-state frequency error, while the derivative action damps the oscillatory response of interconnected areas.
• Voltage and reactive power control: PID controllers regulate excitation (AVR) and tap changers to hold bus voltage at setpoint.
• Governor/turbine control: Speed governors in hydro and thermal plants use PID (or PI) control to maintain speed and load sharing.
• Stability improvement: Properly tuned PID controllers increase damping of low-frequency oscillations and improve transient stability.
• Economic and reliable operation: By keeping frequency and voltage within limits, they protect equipment, reduce losses, and prevent cascading outages.

Applications

• Automatic generation control (AGC) and load frequency control in power systems
• Automatic voltage regulators (AVR) for synchronous generators
• Turbine speed governors in hydro, thermal and gas plants
• Boiler control (drum level, steam pressure, temperature) in thermal stations
• HVDC and FACTS device control (converter current, STATCOM voltage)
• Motor drives, process industries (flow, level, temperature, pressure), robotics and mechatronics

Limitations

PID controllers are tuned for a nominal operating point; performance degrades under wide parameter variation or strong nonlinearity. For such cases, advanced controllers (fuzzy, adaptive, model predictive) are preferred. Nevertheless, the PID remains the backbone of power system control due to its simplicity, low cost, and proven reliability.

Why a tertiary winding is introduced in a star-star (Y-Y) connected power transformer

A star-star connected transformer is the most economical for stepping up or stepping down between two high-voltage systems, but it suffers from three inherent limitations that a third (tertiary) winding is added to overcome:

1. Provision of a low-impedance path for third-harmonic currents. In a Y-Y transformer the magnetising currents must supply the third-harmonic component of the excitation, but the neutral is usually earthed (or the core is not provided with a path for triple-frequency flux). The third-harmonic mmf cannot be balanced, so the flux waveform becomes peaked and a third-harmonic voltage appears in the phase voltage. A delta-connected tertiary provides a closed path in which the third-harmonic currents circulate, flattening the flux and restoring an approximately sinusoidal output voltage.

2. Stabilisation of the neutral point. The delta tertiary carries the zero-sequence (third-harmonic and unbalanced) currents and thereby keeps the neutral of the star windings at a fixed potential, preventing neutral displacement under unbalanced load or earth-fault conditions.

3. Reduction of zero-sequence and unbalanced-load impedance. The delta tertiary offers a low-impedance path for zero-sequence currents, so the transformer presents a low zero-sequence impedance to earth faults on either side. This limits the rise of potential at the healthy phases and improves the effectiveness of earth-fault protection.

4. Supply of station auxiliaries and reactive compensation. The tertiary winding is used to feed the station's own auxiliary load (at a convenient voltage such as 11 kV or 6.6 kV) and to connect shunt capacitors, reactors or synchronous condensers for voltage and reactive-power control.

5. Reduction of voltage distortion and telephone interference. By absorbing the third-harmonic currents the tertiary reduces the harmonic content of the output voltage and hence the inductive interference with neighbouring communication circuits.

6. Improved parallel operation and load sharing. The low-impedance tertiary helps in the satisfactory parallel operation of Y-Y transformers by damping circulating harmonic currents between units.

Conditions under which the tertiary winding can be avoided

A tertiary winding is not always necessary. It may be omitted when:

1. The neutral of at least one star winding is solidly earthed and the core provides a low-reluctance path for third-harmonic flux. If the transformer is a three-phase, three-limb core type with a solidly earthed neutral, the third-harmonic currents can flow through the neutral and the earthed system, so the flux remains nearly sinusoidal and no tertiary is needed. (In a five-limb or shell-type core the third-harmonic flux has a return path through the core itself, so again the tertiary may be dispensed with.)

2. The transformer is connected to a system where the neutral is earthed through a low-impedance path and the third-harmonic currents can be supplied from the system itself.

3. The transformer is a delta-star or star-delta unit. In these connections the delta winding already provides the closed path for third-harmonic currents, so a separate tertiary is unnecessary.

4. The transformer is small and the harmonic content of the voltage is not critical (e.g. small distribution transformers, or where the load is not sensitive to waveform distortion).

5. The transformer is used only for interconnecting two systems of the same voltage and phase sequence where no auxiliary supply or reactive compensation is required and the neutral is earthed, so the zero-sequence and harmonic requirements are already met.

6. The transformer is a three-winding transformer in which the third winding is already provided for another purpose (e.g. a delta-connected stabilising winding or an auxiliary supply winding), so no additional tertiary is required.

7. Where the system is effectively earthed and the transformer is protected by other means (e.g. zig-zag earthing transformers or separate earthing transformers are installed at the substation), the zero-sequence path is provided externally and the tertiary can be avoided.

Conclusion

The tertiary winding in a Y-Y transformer is essentially a delta-connected stabilising winding that provides a low-impedance path for third-harmonic and zero-sequence currents, stabilises the neutral, limits earth-fault potential rise, and can supply station auxiliaries or reactive compensation. It can be avoided when the core and earthing arrangement already provide a path for third-harmonic flux (three-limb core with solidly earthed neutral, or five-limb/shell-type core), when the transformer connection already includes a delta winding (Δ-Y or Y-Δ), when the transformer is small and harmonic distortion is tolerable, or when the zero-sequence path is provided externally by earthing transformers.