ELEC0047 - Power system dynamics, control and stability
Turbines and speed governors
Thierry Van Cutsem
t.vancutsem@ulg.ac.be
www.montefiore.ulg.ac.be/~vct
November 2015
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Turbines and speed governors
Steam turbines
Steam turbines
SG: speed governor
measures speed and adjusts steam valves accordingly
CV: control (or high pressure) valves
maneuvered by speed governor in normal operating conditions
IV: intercept valves
fully opened in normal operating conditions; closed in case of overspeed
MSV, RSV: main stop valve and reheater stop valve
used as back-up in case of emergency
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Turbines and speed governors
Steam turbines
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Turbines and speed governors
Steam turbines
Assumptions:
power developed in one turbine stage ÷ steam flow at exit of that stage
steam flow at entry of HP vessel ÷ valve opening z
÷ steam pressure pc
steam flow at exit of a vessel follows steam flow at entry with a time constant
Per unit system:
each variable is divided by the value it takes when the turbine operates
at its nominal power PN . Time constants are kept in seconds.
THP ' 0.1 − 0.4 s
fHP ' 0.3
TR ' 4 − 11 s
fMP ' 0.4
TLP ' 0.3 − 0.5 s
fHP + fMP + fLP = 1
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Turbines and speed governors
Steam turbines
Interactions between turbine and boiler
for large disturbances, the change in steam flow dHP results in an opposite
change in steam pressure pc
taking this into account requires to model the boiler and its controllers
we only mention the boiler and turbine control modes
“Boiler-following” regulation
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Turbines and speed governors
Steam turbines
“Turbine-following” regulation
“Coordinated” or “integrated” regulation
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Turbines and speed governors
Steam turbines
Example
Responses to a demand of large production increase: comparison of the three
regulations
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Turbines and speed governors
Speed governors of steam turbines
Speed governors of steam turbines
Servomotor modelling
z: opening of control valves (0 < z < 1 in per unit)
z o : valve opening setpoint (changed when power output of unit is changed)
σ: permanent speed droop (or statism)
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Turbines and speed governors
Speed governors of steam turbines
The non-windup integrator
ẋ
=
0
if x = xmax and u > 0
=
0
if x = xmin and u < 0
=
u
otherwise
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Turbines and speed governors
Speed governors of steam turbines
Equivalent block-diagram
Tsm = 1/(K σ)
servomotor time constant (' a few 10−1 s)
A little more detailed model
Tr : time constant of “speed relay” (additional amplifier) (' 0.1 s)
a transfer function (1 + sT1 )/(1 + sT2 ) may be used to improve dynamics
block 2 accounts for nonlinear variation of steam flow with valve opening
block 1 compensates block 2
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Turbines and speed governors
Speed governors of steam turbines
Steady-state characteristics
turbine:
⇒ Pm = z
pc = 1 pu
speed governor: assuming z is not limited:
ω−1
σ
z = zo −
and referring to the system frequency f (in Hz) with nominal value fN (in Hz):
z = zo −
f − fN
σfN
combining both:
Pm = z o −
f − fN
σfN
z o seen as a power setpoint, in pu on the turbine power.
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Turbines and speed governors
Hydraulic turbines
Hydraulic turbines
Action (or impulse-type) turbines
The potential energy of water is converted into pressure and then into kinetic
energy by passing through nozzles. The runner is at atmospheric pressure. The
high-velocity jets of water hit spoon-shaped buckets on the runner.
Pelton turbine
used for large water heights (300 m or more)
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Turbines and speed governors
Hydraulic turbines
Reaction turbines
The potential energy of water is partly converted into pressure. The water
supplies energy to the runner in both kinetic and pressure forms. Pressure within
the turbine is above atmospheric.
Require large water flows to produce significant powers.
Rotation speeds are lower than with impulse turbines.
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Turbines and speed governors
Hydraulic turbines
Francis turbine
for water heights up to ' 360 m
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Turbines and speed governors
Hydraulic turbines
Kaplan turbine
For water heights up to ' 45 m
Variable-pitch blades can be used (angle adjusted to water flow to maximize
efficiency)
Mainly used in run-of-river hydro plants
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Turbines and speed governors
Hydraulic turbines
Bulb turbine
For small water heights
Mainly used in run-of-river hydro plants
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Turbines and speed governors
Hydraulic turbines
Simple model of a hydro turbine
Assumptions:
water assumed incompressible
pressure travelling waves (hammer effect) neglected
Hs : water height
ρ specific mass of water (kg/m3 )
g gravity acceleration (m/s2 )
Q water flow (m3 /s)
E energy provided by 1 m3 of water (J/m3 )
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Turbines and speed governors
Hydraulic turbines
Potential energy contained in 1 m3 of water in upper reservoir:
Epot = ρgHs
Total power provided by water (a part of which goes in losses):
P = ρgHs Q
Let’s define the head:
H=
E
ρg
(m)
where E is the energy delivered by 1 m3 of water.
Total power provided by water (a part of which goes in losses):
P = EQ = ρgHQ
in steady state : H = Hs
during transients : H 6= Hs
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Turbines and speed governors
Hydraulic turbines
Basic relationships:
1
mechanical power provided by turbine, taking into account losses in
conduites, etc.:
Pm = ρgH(Q − Qv ) < P
2
water flow:
√
Q = kQ z H
z : section of gate (0 ≤ z ≤ A)
3
acceleration of water column in conduite:
ρLA
dv
= ρgA(Hs − H)
dt
Q = Av ⇒
dQ
gA
=
(Hs − H)
dt
L
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Turbines and speed governors
Hydraulic turbines
Passing to per unit values
base of a variable = value taken by variable at nominal operating point of turbine:
mechanical power Pm = nominal power PN of turbine
head H = height Hs
gate opening z = A
water flow Q = nominal value QN
water speed v = QN /A
At nominal operating point:
PN = ρgHs (QN − Qv )
p
Q N = kQ A H s
Normalizing the power equation:
Pm pu =
with KP =
H Q − Qv
H
QN
Q − Qv
=
= KP Hpu (Qpu − Qv
Hs QN − Qv
Hs QN − Qv
QN
1
1 − Qv
pu )
pu
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Turbines and speed governors
Hydraulic turbines
Normalizing the flow equation:
Qpu = zpu
p
Hpu
Normalizing the water acceleration equation:
g AHs Hs − H
1
dQpu
=
=
(1 − Hpu )
dt
L QN
Hs
Tw
where Tw =
L vN
L QN
=
is the water starting time at nominal operating point.
g AHs
g Hs
Tw = time taken by water, starting from standstill, to reach nominal speed under
the effet of head Hs (0.5 - 4 s)
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Turbines and speed governors
Hydraulic turbines
Response of a hydro turbine to small disturbances
Small disturbances around operating point (z o , H o = 1, Q o ).
Transfer function between ∆z and ∆Pm ?
√
zo
∆Q =
H o ∆z + √ ∆H
2 Ho
sTw ∆Q = −∆H
∆Pm
= KP H o ∆Q + KP (Q o − Qv )∆H
Eliminating ∆Q and ∆H yields:
o 3/2
1−
∆Pm = KP (H )
(Q o −Qv ) 0
√
Tw s
zo Ho
∆z
0
T
1 + s 2w
0
zo
where Tw = Tw √
is the water starting time at the considered operating point.
Ho
If Qv is neglected:
o 3/2 1
∆Pm = KP (H )
0
− sTw
1+s
Tw0
2
∆z
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Turbines and speed governors
Hydraulic turbines
non-minimum phase system: zero in right half complex plane
initial reaction opposite to final reaction
Example: response ∆Pm to step change in gate opening of magnitude ∆Z :
o 3/2
lim ∆Pm (t) = lim sKP (H )
t→0
s→∞
1−
0
(Q o −Qv )
√
sTw
zo Ho
T0
1 + s 2w
∆Z
(Q o − Qv )
= −2KP H o
∆Z
s
zo
initial behaviour: inertia of water ⇒ speed v and flow Q do not change ⇒
head H decreases ⇒ mechanical power Pm decreases
after some time: Q increases and H comes back to 1 ⇒ Pm increases
non-minimum phase systems may bring instability when embedded in
feedback system (one branch of the root locus ends up on the zero)
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Turbines and speed governors
Hydraulic turbines
Speed governors of hydro turbines
Presence of a pilot servomotor: Tp ' 0.05 s
K ' 3 − 5 pu/pu
with σ ' 0.04 − 0.05, the turbine and speed governor would be unstable
when the hydro plant is in isolated mode or in a system with a high
proportion of hydro plants
first solution: increase σ
⇒ the power plant will participate less to frequency control : not desirable
other solution: add a compensator that temporarily increases the value of σ
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Turbines and speed governors
Hydraulic turbines
In the very first moment after a disturbance:
lim
σ+
s→∞
sδTr
=σ+δ
1 + sTr
σ = 0.04, δ ' 0.2 − 1.0, temporary statism = 6 - 26 × permanent statism
In steady state:
lim σ +
s→0
sδTr
=σ
1 + sTr
Tr : “reset time”: ' 2.5 − 25s
characterizes the time to come back to steady-state statism.
In some speed governors, the transfer function
K
1 + sTr
1 + s(δ/σ)Tr
is used in the feed-forward branch of the speed governor
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Turbines and speed governors
Case study. Frequency regulation in an isolated system
Case study. Frequency regulation in an isolated system
Hydro plant:
generator: 300 MVA, 3 rotor winding model
turbine: 285 MW, Tw = 1.5 s
Qv = 0.1
automatic voltage regulator: static gain G = 150
exciter: time constant Te = 0.5 s
speed governor: σ = 0.04
mechanical-hydraulic : K = 4
PI controller: see slide 29
ż min = −0.02
ż max = 0.02 pu/s
Tp = 0
Load:
behaves as constant impedance, insensitive to frequency
5 % step increase of admittance at t = 1 s
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Turbines and speed governors
Case study. Frequency regulation in an isolated system
Mechanical-hydraulic speed governor with compensation: δ = 0.5
Tr = 5 s
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Turbines and speed governors
Case study. Frequency regulation in an isolated system
Mechanical-hydraulic speed governor without compensation (δ = 0.)
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Turbines and speed governors
Case study. Frequency regulation in an isolated system
Speed governor with PI control
servomotor: K = 4 ż min = −0.02 pu/s
PI controller: Tm = 1.9 s
Kp = 2
ż max = 0.02 pu/s
Ki = 0.4
Tp = 0
σ = 0.04
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Turbines and speed governors
Case study. Frequency regulation in an isolated system
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Turbines and speed governors
Case study. Primary and secondary frequency regulation
Case study. Primary and secondary frequency regulation
Primary frequency control:
left area: generator 2 (PN = 850 MW, σ = 0.05) → β left = 283.3 MW.s
right area: generator 4 (PN = 850 MW, σ = 0.05) → β right = 283.3 MW.s
Secondary frequency control:
left area: generator 1 (PN = 850 MW)
right area: generator 3 (PN = 850 MW)
o
regulates P7−8 , the active power flow in tie-lines 7-8, to P7−8
= 400 MW
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Turbines and speed governors
Case study. Primary and secondary frequency regulation
5 % step increase of load at bus 9. Primary frequency control only
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Turbines and speed governors
Case study. Primary and secondary frequency regulation
f1 (resp. f3 ) : frequency in left (resp. right) area (Hz)
obtained from rotor speed of gen. 1 (resp. 3) (available in pu)
ACE left (resp. ACE right ) : Area Control Error of left (resp. right) area (MW)
∆Pc1 (resp. ∆Pc3 ) : power setpoint correction sent to gen. 1 (resp. 3) (MW)
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Turbines and speed governors
Case study. Primary and secondary frequency regulation
Same load increase. Primary and secondary frequency control
(λleft = 283.3 λright = 283.3 Kileft = Kiright = −0.02)
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Turbines and speed governors
Case study. Primary and secondary frequency regulation
Same load increase.
Adjustment of power of gener. 1 by secondary frequency controller of left area, for
various values of λleft
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Turbines and speed governors
Case study. Primary and secondary frequency regulation
At t = 1 s, the areas decide to decrease their power exchange to 300 MW.
o
is set to 300 MW.
Load demand is unchanged.
P7−8
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