Var/Voltage Control in Hydrogenerating Systems

The excitation system supplies DC field current to the synchronous generator and controls it to hold the terminal voltage and share reactive power. A conventional system uses a DC exciter driven by an amplifier, with a stabilizing transformer giving rate feedback to keep the loop stable.

          +---------+   +----------+   +---------+
 Vref --->|comparat.|-->|amplifier |-->| exciter |
          +---------+   +----------+   +---------+
            ^    ^                         |  Efd
            |    |  +-----------------+    |
            |    +--| stabilizing     |<---+
            |       | transformer     |    v
            |       +-----------------+ +----------+
            |                           |generator |
            |   +-------------------+   | field    |
            +---| PT + rectifier    |<--+----------+
                +-------------------+   terminals Vt

Working: the terminal voltage is measured, rectified and compared with the reference. The error is amplified and drives the exciter field; the exciter output $E_{fd}$ feeds the main field. If $V_t$ falls, $E_{fd}$ rises and restores it. The stabilizing transformer feeds back a signal proportional to the rate of change of $E_{fd}$, which damps overshoot and hunting.

Components and their transfer functions

1. Voltage sensor (PT + rectifier + filter): $\dfrac{\Delta V_s}{\Delta V_t} = \dfrac{K_R}{1 + sT_R}$ ($T_R$ is very small, 0.01–0.06 s).
2. Comparator: $\Delta V_e = \Delta V_{ref} - \Delta V_s - \Delta V_F$.
3. Amplifier (magnetic, rotating or thyristor): $\dfrac{\Delta V_R}{\Delta V_e} = \dfrac{K_A}{1 + sT_A}$ ($T_A$ = 0.02–0.1 s).
4. Exciter (separately excited DC exciter): the field winding has resistance $R_{ef}$ and inductance $L_{ef}$; output voltage is proportional to field current. So $\dfrac{\Delta E_{fd}}{\Delta V_R} = \dfrac{K_E}{1 + sT_E}$, with $T_E = L_{ef}/R_{ef}$.
5. Generator field (on no load): $\dfrac{\Delta V_t}{\Delta E_{fd}} = \dfrac{K_G}{1 + sT_G}$, with $T_G = T'_{do}$ (open-circuit field time constant, 1–10 s).

Stabilizing transformer

The primary of the stabilizing transformer is connected across the exciter output ($E_{fd}$) and the secondary is connected in series with the amplifier input. The secondary feeds a high-impedance input, so its current is negligible.

Primary: $E_{fd} = R_1 i_1 + L_1\dfrac{di_1}{dt}$, so $I_1(s) = \dfrac{E_{fd}(s)}{R_1 + sL_1}$.

Secondary: $V_F = M\dfrac{di_1}{dt}$, so $V_F(s) = sM\,I_1(s)$.

$$\frac{\Delta V_F(s)}{\Delta E_{fd}(s)} = \frac{sM}{R_1 + sL_1} = \frac{sK_F}{1 + sT_F}, \qquad K_F = \frac{M}{R_1},\ T_F = \frac{L_1}{R_1}$$

It gives a derivative (rate) feedback: output appears only while $E_{fd}$ is changing and is zero in steady state. So it damps oscillations and allows a high amplifier gain without affecting steady-state accuracy.

Block diagram

 Vref +    Ve  +-------+ VR +-------+ Efd +-------+
 --->(+)------>|  KA   |--->|  KE   |--+->|  KG   |--+-> Vt
     ^-  ^-    |1+sTA  |    |1+sTE  |  |  |1+sTG  |  |
     |   |     +-------+    +-------+  |  +-------+  |
     |   |        VF  +---------+      |             |
     |   +------------|  sKF    |<-----+             |
     |                | 1+sTF   |  (stabilizing      |
     |                +---------+   transformer)     |
     |        Vs      +---------+                    |
     +----------------|  KR     |<-------------------+
                      | 1+sTR   |
                      +---------+

Overall transfer function

Let $G_A = \dfrac{K_A}{1+sT_A}$, $G_E = \dfrac{K_E}{1+sT_E}$, $G_G = \dfrac{K_G}{1+sT_G}$, $H_F = \dfrac{sK_F}{1+sT_F}$, $H_R = \dfrac{K_R}{1+sT_R}$.

Inner (stabilizing) loop:

$$\frac{\Delta E_{fd}}{\Delta V_{ref} - \Delta V_s} = \frac{G_A G_E}{1 + G_A G_E H_F}$$

Outer (voltage) loop:

$$\frac{\Delta V_t(s)}{\Delta V_{ref}(s)} = \frac{G_A G_E G_G}{1 + G_A G_E H_F + G_A G_E G_G H_R}$$

Steady state ($s \to 0$, $H_F(0) = 0$, $K_R = 1$):

$$\frac{\Delta V_t}{\Delta V_{ref}} = \frac{K_A K_E K_G}{1 + K_A K_E K_G}$$

So the stabilizing transformer does not change the steady-state gain; it only improves the transient behaviour.

Effect of the stabilizer: without it, a high gain $K_A$ (needed for small steady-state error) makes the third-order loop oscillatory or unstable. The rate feedback adds damping, so a fast, well-damped voltage response is obtained with a high gain.

The dynamic response of an excitation system is how fast and how smoothly the generator terminal voltage follows a change in reference voltage (or recovers after a load change). It depends on the gains and time constants of the amplifier, exciter and generator field, and on stabilizing feedback.

Mathematical model

Basic AVR loop (sensor time constant neglected, unity feedback):

 Vref +   Ve  +-------+   +-------+   +-------+
 --->(+)----->|  KA   |-->|  KE   |-->|  KG   |--+--> Vt
     ^ -      |1+sTA  |   |1+sTE  |   |1+sTG  |  |
     |        +-------+   +-------+   +-------+  |
     +-------------------------------------------+

Open-loop transfer function:

$$G(s) = \frac{K}{(1 + sT_A)(1 + sT_E)(1 + sT_G)}, \qquad K = K_A K_E K_G$$

Closed loop:

$$\frac{\Delta V_t(s)}{\Delta V_{ref}(s)} = \frac{G(s)}{1 + G(s)}$$

Steady-state response

For a step $\Delta V_{ref}$, using the final value theorem:

$$\begin{aligned} \Delta V_t(\infty) &= \frac{K}{1 + K}\,\Delta V_{ref} \\ e_{ss} &= \frac{1}{1 + K}\,\Delta V_{ref} \end{aligned}$$

To keep the voltage error small (e.g. below 1 %), $K$ must be large (about 100 or more).

Transient response and stability

Characteristic equation:

$$(1 + sT_A)(1 + sT_E)(1 + sT_G) + K = 0$$ $$T_A T_E T_G\, s^3 + (T_A T_E + T_E T_G + T_A T_G)\, s^2 + (T_A + T_E + T_G)\, s + (1 + K) = 0$$

By the Routh criterion, the system is stable only if

$$(T_A T_E + T_E T_G + T_A T_G)(T_A + T_E + T_G) > T_A T_E T_G\,(1 + K)$$

So a large $K$ (good accuracy) makes the response more oscillatory and can make it unstable. This is the basic conflict in AVR design.

 Vt
      high K: fast but oscillatory
    |    .-.
    |   /   '-.__.-------------
    |  /    ___.-------------- low K
    | / _.-'   (slower, larger error)
    |/.'
    +---------------------------> t

Improving the dynamic response

• Stabilizing (rate) feedback from exciter output to amplifier input, $\dfrac{sK_F}{1 + sT_F}$: adds damping only during transients, leaving steady-state gain unchanged. Closed loop becomes

$$\frac{\Delta V_t}{\Delta V_{ref}} = \frac{G_A G_E G_G}{1 + G_A G_E H_F + G_A G_E G_G}$$

• Fast exciters (static thyristor or brushless) with small $T_E$, and high ceiling voltage and response ratio, so the field current can be forced quickly.
• Lead-lag compensation or a power system stabilizer (PSS) for damping rotor oscillations.

A good excitation system gives a fast rise, small overshoot, short settling time and a small steady-state voltage error.

Function of excitation system

The excitation system supplies and controls the DC field current of a synchronous generator. Its functions are:

• Supply DC current to the rotor field winding to produce the main flux.
• Keep the terminal voltage constant as load changes (automatic voltage regulation).
• Control the reactive power (VAr) output and share it correctly among parallel generators.
• Improve transient and steady-state stability by fast field forcing during faults.
• Provide protective limits (over/under-excitation limiters, V/Hz limiter).

Transfer function with stabilizing transformer

The system has four blocks: amplifier, exciter, generator field and a stabilizing transformer that feeds back the rate of change of exciter voltage to damp oscillations.

Vref +  e   +   +-----------+    +---------+  Efd
 --->(Σ)-->(Σ)-->| Amplifier |--->| Exciter |--+--+
      ^-    ^-   +-----------+    +---------+  |  |
      |     |     +---------------+            |  |
      |     +-----| Stab. transf. |<-----------+  |
      |           +---------------+               |
      |   Vt      +-----------+                   |
      +-----------| Generator |<------------------+
                  +-----------+

1. Amplifier: $\dfrac{V_R(s)}{V_e(s)} = \dfrac{K_A}{1+sT_A}$

2. Exciter (field $R_e$, $L_e$; $E_{fd} = K i_e$): from $v_R = R_e i_e + L_e \frac{di_e}{dt}$,

$$\frac{E_{fd}(s)}{V_R(s)} = \frac{K/R_e}{1+s(L_e/R_e)} = \frac{K_E}{1+sT_E}$$

3. Generator field (open circuit): $\dfrac{V_t(s)}{E_{fd}(s)} = \dfrac{K_G}{1+sT_G}$

4. Stabilizing transformer: primary ($R_1$, $L_1$) is across the exciter output; secondary (mutual inductance $M$) is practically open-circuited by the high-impedance amplifier input.

$$\begin{aligned} E_{fd} &= R_1 i_1 + L_1 \frac{di_1}{dt} \;\Rightarrow\; I_1(s) = \frac{E_{fd}(s)}{R_1(1+sT_F)} \\ V_F &= M\frac{di_1}{dt} \;\Rightarrow\; \frac{V_F(s)}{E_{fd}(s)} = \frac{sK_F}{1+sT_F} \end{aligned}$$

where $K_F = M/R_1$ and $T_F = L_1/R_1$.

5. Closed loop. Let $G_1 = \dfrac{K_A K_E}{(1+sT_A)(1+sT_E)}$, $H_F = \dfrac{sK_F}{1+sT_F}$, $G_2 = \dfrac{K_G}{1+sT_G}$. The minor loop gives $\dfrac{G_1}{1+G_1H_F}$; with unity voltage feedback:

$$\frac{V_t(s)}{V_{ref}(s)} = \frac{G_1G_2}{1+G_1H_F+G_1G_2}$$

Substituting and clearing fractions:

$$\frac{V_t(s)}{V_{ref}(s)} = \frac{K_AK_EK_G(1+sT_F)}{(1+sT_A)(1+sT_E)(1+sT_G)(1+sT_F) + sK_FK_AK_E(1+sT_G) + K_AK_EK_G(1+sT_F)}$$

The $sK_F$ term adds damping, so a high loop gain (small steady-state error) can be used without oscillation.

The excitation system is the set of equipment (exciter, regulator, control and protection) that supplies the DC field current to a synchronous generator and controls it to hold terminal voltage and reactive power at the set values.

Functional block diagram

          +-----------+  +---------+  +-----------+
Vref ->(Σ)| Regulator |->| Exciter |->| Generator |-> Vt
       ^ ^|(amplifier)|  |         |  | + system  |
       | |+-----------+  +----+----+  +-----+-----+
       | |  +------------+    |             |
       | +--| Stabilizer |<---+             |
       |    +------------+                  |
       |    +--------------------------+    |
       +----| Voltage transducer and   |<---+
            | load compensator         |
            +--------------------------+
 Limiters/protection act on the regulator;
 PSS (optional) adds a signal at the summing point

Role of each block

Transfer functions

$$G_R = \frac{K_R}{1+sT_R},\quad G_A = \frac{K_A}{1+sT_A},\quad G_E = \frac{K_E}{1+sT_E},\quad G_G = \frac{K_G}{1+sT_G},\quad H_F = \frac{sK_F}{1+sT_F}$$

($G_R$: sensor, $G_A$: amplifier, $G_E$: exciter, $G_G$: generator, $H_F$: stabilizer.)

The exciter TF comes from its field circuit: $v_R = R_e i_e + L_e\,di_e/dt$ and $E_{fd} = Ki_e$, so $E_{fd}/V_R = K_E/(1+sT_E)$ with $T_E = L_e/R_e$. The stabilizing transformer gives $V_F = M\,di_1/dt$ with $i_1 = E_{fd}/(R_1+sL_1)$, hence $H_F = sK_F/(1+sT_F)$, $K_F = M/R_1$.

Minor (stabilizing) loop around amplifier and exciter:

$$G_1(s) = \frac{G_AG_E}{1+G_AG_EH_F}$$

Complete closed loop (sensor in feedback path):

$$\frac{V_t(s)}{V_{ref}(s)} = \frac{G_AG_EG_G}{1+G_AG_EH_F+G_AG_EG_GG_R}$$

With unity sensor ($K_R=1$, $T_R \approx 0$):

$$\frac{V_t}{V_{ref}} = \frac{K_AK_EK_G(1+sT_F)}{(1+sT_A)(1+sT_E)(1+sT_G)(1+sT_F)+sK_FK_AK_E(1+sT_G)+K_AK_EK_G(1+sT_F)}$$

Steady-state ($s\to 0$): $\dfrac{V_t}{V_{ref}} = \dfrac{K}{1+K}$, with $K = K_AK_EK_G$, so the steady error is $\dfrac{1}{1+K}$ and falls as the loop gain rises.

Need of excitation system

A synchronous generator needs DC field current to build up its flux. The excitation system supplies this current and controls it so that:

• terminal voltage stays constant from no load to full load;
• reactive power is shared properly among parallel machines;
• the field is "forced" quickly during faults to keep the machine in step (better transient stability);
• field current stays inside the machine's capability limits.

Basic loop and its limitation

Vref    +--------+  +--------+ Efd +--------+  Vt
 -->(Σ)->|  K_A   |->|  K_E   |---->|  K_G   |--+-->
    ^-   |1+sT_A  |  |1+sT_E  |     |1+sT_G  |  |
    |    +--------+  +--------+     +--------+  |
    +-------------------------------------------+

With loop gain $K = K_AK_EK_G$, the steady-state error for a step in $V_{ref}$ is

$$e_{ss} = \frac{1}{1+K}$$

Raising $K$ lowers the error but, with three lags in the loop, the response becomes oscillatory and finally unstable. The exciter time constant $T_E$ (about 0.5 to 1 s for a DC exciter) is the main cause of slow response.

Methods to improve the dynamic response

1. Rate (derivative) feedback via a stabilizing transformer. Feed back $H_F = \dfrac{sK_F}{1+sT_F}$ from $E_{fd}$ to the amplifier input. The closed loop becomes

$$\frac{V_t}{V_{ref}} = \frac{G_AG_EG_G}{1+G_AG_EH_F+G_AG_EG_G}$$

The extra term $G_AG_EH_F$ contains $s$, so it adds damping only during transients; at steady state ($s\to0$) it vanishes and the low error is kept. Hence high gain and good damping are obtained together.

2. Reducing the effective exciter time constant by feedback. Put a proportional feedback $K_f$ around the exciter:

$$\begin{aligned} \frac{E_{fd}}{V_R} &= \frac{\dfrac{K_E}{1+sT_E}}{1+\dfrac{K_EK_f}{1+sT_E}} = \frac{K_E}{1+K_EK_f+sT_E} \\ &= \frac{K_E/(1+K_EK_f)}{1+s\,\dfrac{T_E}{1+K_EK_f}} \end{aligned}$$

The time constant falls from $T_E$ to $T_E/(1+K_EK_f)$: the exciter responds $(1+K_EK_f)$ times faster. The lost gain is made up in the amplifier.

3. Fast exciters. Replace rotating DC exciters by static (thyristor) or brushless high-initial-response exciters with small time constants and high ceiling voltage (2 to 3 times rated), so field current rises quickly.

4. Power system stabilizer (PSS). Adds a signal from rotor speed or electrical power through lead–lag blocks, giving positive damping to low-frequency rotor oscillations that a fast AVR may otherwise make worse.

5. Lead compensation in the regulator to cancel the largest lag.

Together these give a fast rise, small overshoot and small steady-state error.

An excitation system supplies and controls the DC current in the rotor field winding of a synchronous generator, so as to hold terminal voltage and control reactive power.

Brush (DC) excitation system

In a brush excitation system the field current is produced outside the rotor (by a DC exciter or by rectifiers) and is fed to the rotating field winding through brushes and slip rings. The classic form uses a shaft-mounted DC generator (main exciter), often with a smaller pilot exciter.

 Pilot     Main        Main alternator
 exciter   exciter    +-------------------+
 +----+    +----+     | rotor field       |
 | PE |--->| ME |=B/S=>| winding    stator |--> 3-ph
 +----+    +----+     |            wndg   |    out
   |   field  ^       +-------------------+
   |   rheo.  |                 |
   +----------+--<-- AVR <-- PT-+
 <====== common shaft (turbine) ======>
 B/S = brushes and slip rings

Components and working

1. Pilot exciter: small self-excited DC shunt generator on the same shaft. It supplies the field of the main exciter.
2. Main exciter: separately excited DC generator on the shaft. Its armature output (a few hundred volts DC) feeds the alternator field.
3. Brushes and slip rings: carry the main exciter output to the rotating field winding of the alternator.
4. Automatic voltage regulator (AVR): senses terminal voltage through a PT, compares it with the reference and adjusts the main exciter field current (through a rheostat, amplidyne or magnetic amplifier).
5. Field breaker and discharge resistor: isolate and de-excite the field quickly during internal faults.

When terminal voltage falls (more load or lagging pf), the AVR raises the main exciter field current, $E_{fd}$ rises, alternator field current rises and $V_t$ returns to the set value.

A modern variant of brush excitation is the static exciter, where thyristor rectifiers fed from the generator terminals supply the field through slip rings.

Merits and demerits

A static excitation system has no rotating exciter. The field current is supplied by thyristor (SCR) rectifiers fed from the generator terminals through an excitation transformer, and is given to the rotor through slip rings and brushes.

 Gen terminals --+--------------------------> to grid
                 |
            +----+----+
            | Excit.  |   step-down
            | transf. |
            +----+----+
                 | 3-ph AC
            +----+----+     firing   +-----+
            | Thyristor|<-----------| AVR |<-- PT, CT
            | bridge   |            +-----+
            +----+----+
                 | DC (field breaker)
              slip rings
                 |
            Rotor field winding

Working

1. A part of the generator output is stepped down by the excitation transformer.
2. A fully controlled 3-phase thyristor bridge rectifies it to DC.
3. The AVR compares terminal voltage (from PT) with the reference and changes the firing angle of the thyristors. A smaller firing angle gives higher field voltage.
4. DC is fed to the field through slip rings.
5. Field flashing: at start-up there is no terminal voltage, so the field is first energised from the station battery or residual magnetism until the voltage builds up.
6. A field breaker and a crowbar/discharge resistor de-excite the field on faults. The bridge can also invert (negative field voltage) for fast de-excitation.

Types

• Potential-source (bus-fed) static exciter: supply from a terminal transformer only.
• Compound-source static exciter: supply from both a PT and a series CT, so excitation is kept during a close-in fault.

Advantages

• Very fast response (time constant a few ms) and high ceiling voltage; improves transient stability.
• No rotating exciter, so shorter shaft and less maintenance.
• Fast field suppression by inverting the bridge.

Disadvantages

• Slip rings and brushes are still needed.
• Field supply depends on terminal voltage; a close-in fault reduces excitation (unless compounded).
• Thyristors produce harmonics and need cooling.

An excitation system supplies the DC field current of a synchronous generator and controls it to hold the terminal voltage and the reactive power output at the desired values.

On the basis of their power source and performance, excitation systems are of three types.

1. DC excitation system

A shaft-driven DC generator (main exciter), often with a pilot exciter, supplies the field through slip rings.

 Pilot exc --> Main DC exc ==slip rings==> Gen field
                   ^                          |
                   +------ AVR <---- PT <-----+

• Slow response (exciter time constant 0.5 to 1 s); commutator and brushes need maintenance.
• Used in old and small units.

2. AC excitation system

An AC exciter (alternator) is on the same shaft and its output is rectified.

(a) Stationary rectifier: rectifiers are static, output reaches the field through slip rings.

(b) Rotating rectifier (brushless): the exciter has stationary field and rotating armature; diodes mounted on the shaft feed the field directly, so no brushes or slip rings.

 Stationary   |  Rotating parts on shaft          |
 AVR -> exc   |  exciter    rotating   gen field  |
 field ======>|  armature -> diodes --> winding   |
 (PMG supply) |  (3-ph AC)   (DC)                 |

• Medium response; brushless type is almost maintenance-free and suits large turbo-generators and hazardous areas.
• Field current cannot be measured or de-excited directly.

3. Static excitation system

Power is taken from the generator terminals through an excitation transformer and rectified by a thyristor bridge; DC goes to the field through slip rings. The AVR controls the firing angle.

 Gen bus -> Exc. transformer -> Thyristor bridge
                                    | (AVR firing)
                                    v
                         slip rings -> Gen field

• Very fast response and high ceiling voltage; best for transient stability.
• Needs field flashing at start; supply falls during close-in faults.

Comparison of performance