Advanced Process Control
Control of
Grid-Connected Generators
Francesco Casella
(francesco.casella@polimi.it)
Control of Power Generation Systems: 3 Scenarios
•
Scenario #1: one synchronous generator connected to local loads
•
Scenario #2: one synchronous generator connected to strong node
•
Scenario #3: many synchronous generators + transmission grid + loads
2
Scenario #1: one generator + local loads
•
Generalization of hydro power plant case, assuming some kind of
turbogenerator
•
Ga(s): transfer function of turbine valve actuator
•
Gp(s): transfer function between turbine valve opening and mechanical
power applied to the turbo-generator shaft (e.g. Ghyd(s) in hydro plant)
•
Ta: turbo-generator acceleration time constant
dPl°
dw°
dAv°
-
C(s)
Ga(s)
dAv
Gp(s)
dPt
-
1
sT a
dw
3
Scenario #2: one generator + strong node
•
The plant is connected to a strong node of a large transmission network
•
The total installed power on the transmission networks is much larger
than the generator power → the frequency is fixed by the strong node
( )
d J Ω2
=P t − P e
dt 2
P t =P e
ω =Ω N p
ω =ω 0 =const
•
In this case, it only makes sense to control the (net) plant electrical power
output, possibly considering local auxiliary loads
dPaux
dPnet°
dAv°
-
C(s)
Ga(s)
dAv
Gp(s)
dPt
-
dPnet
4
Discussion
•
Scenario #2 makes sense for small plants, e.g. distributed generation
driven by renewable energy
•
Scenario #2 cannot be generalized to the entire grid:
if each plant produces a certain amount of power independent of the loads
– Who guarantees that the production matches the consumption?
– Who guarantees that the grid frequency is close to 50 Hz?
•
We then need to understand (at least in principle) how to handle large
power grids with many synchronous generators driven by turbines
•
This is currently the state of the art in
continental Europe
•
The ENTSO-E Continental
Synchronous area connects the national
grids
– from Portugal to Turkey
– From Denmark to Algeria and
Tunisia
5
Scenario #3: Many generators + transmission + loads
•
A detailed discussion of this system goes beyond the scope of this
course → refer to the Automation of Energy Systems course
•
In this context, we only try to understand the fundamentals of the system
operation and control
•
We derive a simplified model under the following assumptions:
– Ideal synchronous generators: the machine voltage phasor angle is
the same as the machine electrical angle (physical angle times Np)
– Ideal generator: no losses (hel = 1)
– Ideal voltage control: the machine voltage magnitude V is fixed
– Purely inductive transmission lines, resistive losses neglected
– Dependency of loads onto the frequency neglected (a = 0)
6
Scenario #3: Many generators + transmission + loads
( )
2
d J k Ωk
=P t , k −P e ,k
dt
2
Mechanical energy balance
ω k =Ωk N p, k
Mechanical vs. electrical frequency
dθ k
= ω k −ω 0
dt
Machine angle w.r.t reference frame rotating
@ reference speed w0 = 2p50 Hz = 314 rad/s
P e , k =P l , k +∑ P x , kj
Local loads (active power) + exchange of active
power with other generators
j
2
V
P x ,kj =
sin (θ k −θ j )
X kj
Active power exchange between genk and genj
depends on voltage phasor angle difference
and on mutual reactance Xij between the two
generators (purely inductive transmission lines)
7
Scenario #3: System Model
dθ k
= ω k −ω 0
dt
(
)
Jkω k dω k
V
=P
−
P
+
sin (θ k −θ j )
t ,k
l ,k ∑
2
dt
X
N p, k
j
kj
2
System Model
ω̄ k =ω 0
2
V
P̄ t ,k = P̄ l ,k + ∑
sin (θ̄ k −θ̄ j )
X
j
kj
Steady State
∑ P̄ t ,k =∑ P̄ l , k +0
Exchange terms cancel out in pairs
because of anti-simmetric sin function
k
k
2
V
P̄ x ,kj =
sin (θ̄ k −θ̄ j )
X kj
2
V
P̄ x ,kj ,max =
X kj
θ̄ k −θ̄ j≪1 rad
Transmission lines should operate
far from their limit capacity
to avoid stability issues
8
Scenario #3: Linearized Model and Mechanical Equivalent
d Δθ k
=Δ ω k
dt
(
≈1
Jkω 0 d Δ ω k
V2
=Δ P t ,k − Δ P l , k + ∑
⏞
cos(θ̄ k −θ̄ j ) (Δ θ k −Δ θ j )
2
dt
X
N p, k
j
kj
~
Δ θ k ⇔θ k
Δ ω ⇔ω~
k
k
J kω 0
⇔~
Jk
N
2
p, k
)
~
dθ k ~
=ω k
dt
~
~ dωk
~ ~
Jk
=τ tk − τ lk +∑ K kj (θ k −θ j )
dt
j
(
)
Linearized
System Model
Mechanical
Equivalent
Δ P⇔ τ
2
V
cos (θ̄ k −θ̄ j )⇔ K kj
X kj
•
•
•
N rotating disks
All connected by elastic springs
Driven by turbine torques and
load torques
9
Interpretation of mechanical equivalent
•
The small motion of a synchronously connected grid with N synchronous
generator is equivalent to the motion of an N-d.o.f. rotational system with
elastic springs
•
Modal analysis:
– One rigid mode ↔ collective synchronous motion of all generators
– N-1 elastic modes ↔ relative oscillations between generators
•
Damping is added to the system by:
– Resistive losses in the transmission lines
– Resistive losses in the generators
– Damping cages in the generators
(→ parasite currents when the generator accelerates or decelerates)
– Frequency controllers (see later)
•
Weak springs (low Kkj) correspond to
– High reactance Xkj → long lines
– Low cos() term → high angles
2
V
cos (θ̄ k −θ̄ j )⇔ K kj
X kj
•
As a first approximation we can neglect the elastic modes and consider
an ideal synchronous system corresponding to the rigid mode
•
Basically one big rotor with the combined moment of inertia
10
A Very, Very, Very Big Rotor Indeed...
11
Other Synchronous Systems Worldwide
12
Grid Control: System Model
•
System model considering only the rigid mode
dPt1
P̄ 1
DPt1
DPl
-
1
2
J ω¯ s
dw
DPtN
dPtN
P̄ N
13
Grid Control: Naive Approach
•
A naive approach is to have all generators each trying to control the grid
frequency
•
Each generator includes integral action to achieve zero static error
dw°
-
dAv1°
C1(s)
G1(s)
dPt1
P̄ 1
DPt1
dPff1
-
Feed-forward
daily schedule
dPffN
dw°
-
•
CN(s)
DPl
1
2
J ω¯ s
dw
DPtN
dAvN°
GN(s)
dPtN
P̄ N
Feed-forward terms can be added to account for the daily planned
production schedule (day-ahead market)
14
Grid Control: Naive Approach – Equivalent Diagram
•
If all generators are of the same type, all Gk(s) are equal, otherwise we
consider an average Gav(s)
•
Reduced sensitivity to load changes compared to single generator
dw°
-
dAv1°
C1(s)
α1
dPt1
dPl
dPff1
Gav(s)
dPffN
dw°
-
T a=
dAvN°
CN(s)
∑ k J k Ω2k
P TOT
α k=
αN
P̄ 1
P TOT
dPt
-
δ P=
ΔP
P TOT
1
sT a
dw
dPtN
∑k α k =1
15
Grid Control: Naive Approach – Equivalent Diagram
•
Now we put in parallel all the controllers, introducing an equivalent
a-weighted average
dPl
dPff
dw°
-
Cav(s)
Gav(s)
dPt
-
1
sTa
dw
•
Apparently, the situation is exactly the same as in the case of a single
generator with local loads
•
Same design considerations
Normalized signals have the installed power of the whole grid at the
denominator
•
– Much smaller normalized perturbations
– Better performance (incentive at having very large grids!)
Is this actually
the case?
16
Grid Control: Naive Approach
•
The anwser is NO!
dPl
dPff
dw°
-
Cav(s)
Gav(s)
dPt
-
1
sTa
dw
•
In this equivalent diagram, we put N controllers with integral action
(a pole in the origin) in parallel, obtaining a single controller with a single
pole in the origin.
•
Thus, there are N-1 pole-zero cancellations in the diagram
•
The feedback moves one pole in the origin to around wc, but the other
ones, which are cancelled in the loop, remain where they are
N-1 not asymptotically stable eigenvalues in the NR/NO part
Individual turbine power outputs may drift w.r.t each other;
individual power plant behaviour is undetermined
17
Proper Design: Primary Frequency Control
•
Individual controllers must not have integral action!
•
One must use type-0 controllers, e.g.
1
C (s)= σ
•
•
1 1+sT 1
C (s)= σ
1+sT 2
In this way, there are no cancellation of non-asymptotically stable poles
in the parallel connection and the system is well-behaved
Price to pay: static error in response to changes of power consumption
Δ P e (t)=Δ P e step(t )
δ ω =σ grid
Δ Pe
P TOT
σ grid =
P TOT
∑k P̄ k
σk
1
δ P tk = σ δ ω
k
P̄
Δ P tk = σ k Δωω
k
r
Each plant contributes more
or less based on droop
18
Primary Frequency Control
•
Primary frequency control is compulsory for all plants over a certain
minimum power
•
The goal of primary frequency control is to stabilize the grid frequency
over short time intervals (15 seconds), avoiding short-term excessive
frequency deviations
•
The grid frequency deviation dw is the coordinating signal for the plant
controllers. No separate communication channel required
Some other mechanism is required to recover the reference frequency in
the medium term
•
•
Primary frequency control also provides damping to the elastic modes of
the synchronously connected grid: assuming very fast power response
δ P t (t )=− δσω
P̄
Δ P t (t )=− σ ωt Δ ω
r
•
~
⇔ τ =− D ω
primary frequency control provides damping in the mech. equivalent
In practice there is some phase delay, but as long as it is much less than
-90°, the damping effect is still good
19
Secondary Frequency Control
•
Main ideas of Secondary Frequency Control
– Cascaded control strategy, relying on Primary Frequency control as
inner loop
– Only some plants, who are will to provide the reserve for this service,
participate to it, and are paid for the service
– Use the feed-forward power offsets as manipulated variables
– Only one centralized controller for the entire synchronous grid
dw°
-
dAv1°
C1(s)
G1(s)
dPt1
P̄ 1
DPt1
dPff1
DPl
-
dPffN
dw°
-
CN(s)
1
2
J ω¯ s
dw
DPtN
dAvN°
GN(s)
dPtN
P̄ N
20
Secondary Frequency Control
β1
DPs1
DPs
βM
∑k β k =1
DPsM
Grid including
primary frequency
control
δ ω ≈ σ grid
Δ P s P TOT
dw
1
s
1+ ω
cp
•
After an increase of load (or the loss of a production plant), the frequency
falls, so that the primary controllers can provide the additional power
imbalance
•
The grid frequency can be restored by adding extra feed-forward power
to the system, relieving the primary frequency controllers from the burden
of providing the power imbalance
•
The static gain is determined by sgrid. The response time is the response
time of the primary controllers (assumed all equal for simplicity)
21
Secondary Frequency Control: Cascaded Control Design
β1
dw°=0
-
DPs
Cs(s)
βM
C (s)=
DPs1
K Is
s
∑k β k =1
K σ
L(s)= Is grid
s P TOT
DPsM
δ ω ≈ σ grid
Δ P s P TOT
|L(jw)|dB
1
s
1+ ω
cp
K Is σ grid
P TOT
ω cs ≪ω cp
ω cs =
φ m≈90 °
5
T set ≈ ω
Grid including
primary frequency
control
dw
1
s
1+ ω
cp
wcs
wcp
w
cs
22
Grid Codes
•
Regulations concerning primary and secondary frequency control are
dictated by grid codes
•
Important role played by nonlinear effects, e.g. saturations
•
Primary control is usually mandatory above a certain size
– Maximum droop required
– Provide required extra power within 15 s of frequency drop, maintain
for 15 min (while secondary control takes over)
•
Secondary control is usually a paid service. Spinning reserve is needed,
and gets paid even if not actually used
•
The big picture is a lot more complex, as it involves many other factors
– Day ahead scheduling with market bids
– Control of power flow across national borders
– Congestion management
– ...
23
Frequency Control: Current & Future Trends
•
Increasing penetration of intermittent renewable distributed generation
– more need of primary frequency control
– possible need of partial curtailing of renewable energy to provide
primary frequency control margin
•
Traditional large power plants with synchronous generators directly
connected to the grid gradually replaced by generators connected via
power electronics (AC-AC converters)
– Less inertia on the grid
– Grid behaviour more and more depending on software rather than
physical laws
– Grid following controllers vs. grid forming controllers
•
DC links allow to move power over longer distances
– Traditionally used for undersea cables, now considered for longdistance power transmission over land
– no limitation due to V2/X factor
– very fast reaction times, depending on software
•
Demand-side management: loads that can be modulated help
maintaining the grid at equilibrium (Virtual Power Plants - VPPs)
24
Frequency Control: Current & Future Trends
•
Role of storage increasingly important towards full decarbonization
•
Batteries:
– Expensive, short storage time (1 hr)
– Could be used for primary frequency control
•
Thermal loads as virtual storage
– Buildings heated by heat pumps and industrial fridges can store heat
for several hours, allowing to shift the electrical consumption by
several hours
– Steel mills and other heavy industry can also shift loads on demand,
providing virtual storage
•
Long-term storage with hydrogen
– Electrical power (in particular from intermittent renewable sources)
can be used in electrolyzers to produce hydrogen
– Hydrogen can be used in fuel cells or thermal power plant to produce
energy
– The overall process is currently too expensive, but there are plans to
make it viable to support the energy transition
Smart Grid
25