Download TEMPUS: VOLTAGE AND FREQUENCY DROOP CONTROL 1

TEMPUS: VOLTAGE AND FREQUENCY DROOP CONTROL
1: Introduction
1.1: Power balance
In an electrical grid, it is important the generated power equals the consumed power. Since the
consumed active power is varying with respect to time, the traditional approach adjusts the generated
power in order to maintain the power balance. In order to be able to maintain this power balance in
case
- the production of a power station stops (e.g. due to a technical failure),
- a (sudden) increase of the consumed power occurs,
sufficient back-up production capacity is required. More precisely, a number of power stations must
operate at partial load and/or a number of power stations must operate in stand-by to allow a fast
increase of the power production. Unfortunately, a power station operating at partial load normally has
a lower efficiency than the same power station operating at full load. Moreover, a power station in
stand-by consumes power (e.g. fossil fuels) without generating electrical power.
The rise of renewable energy accounts for an increased number of decentralized power production
units which generally have a time-varying production which is hard to predict. In case 𝑃𝐶 (𝑡) is the
consumed active power, 𝑃𝐷𝐺 (𝑡) is the decentralized generated power (photovoltaic cells, wind
turbines, wave energy, …) and 𝑃𝐶𝐺 (𝑡) is the centralized generated power (nuclear power stations,
fossil fuel power stations), the power balance is maintained if for all instants of time 𝑡
𝑃𝐶 (𝑡) = 𝑃𝐷𝐺 (𝑡) + 𝑃𝐶𝐺 (𝑡).
Since not only 𝑃𝐶 (𝑡) but also 𝑃𝐷𝐺 (𝑡) is varying with respect to time, larger controlled variations of
𝑃𝐶𝐺 (𝑡) are needed. The rise of renewable energy production (𝑃𝐷𝐺 (𝑡) and its variations become larger)
implies larger variations of 𝑃𝐶𝐺 (𝑡) are needed. This increases the need for a larger number of power
stations operating at partial load or stand-by which accounts for an additional consumption of e.g.
fossil fuels.
In case 𝑃𝐶𝐺 (𝑡) is not able to adapt to the variations of 𝑃𝐶 (𝑡) and 𝑃𝐷𝐺 (𝑡), the power balance is not
maintained and a black-out can occur which accounts for a large economical damage. In case the
reliability of the public grid is decreasing (possibly a larger number of black-outs), it can be useful to
develop microgrids.
1.2: Microgrids
A microgrid is a local low voltage grid containing decentralized power production units, consumers
(electrical loads) and energy storage devices. The microgrid can be connected with the public grid (the
most common situation) by the PCC which account for Point of Common Coupling. In this situation,
from the point of view of the public grid the microgrid behaves as one single prosumer. The microgrid
is able to consume power (active or reactive) or to inject power into the public grid. The energy
storage devices in the microgrid can help to maintain the power balance in the public grid. In case the
public grid needs power, power is extracted from the energy storage devices and injected in the public
grid. In case the public grid has an excess of power, the microgrid can consume power and store it in
the energy storage devices.
In case a black-out of the public grid occurs, the microgrid will be switched off from the public grid.
The microgrid (and its loads) will operate in island mode which increases the reliability of the power
supply. Also when operating in island mode, the power balance is needed. Here, 𝑃𝐺 (𝑡) is the generated
power in the microgrid (e.g. due to photovoltaic panels, micro wind turbines, micro CHP installations,
…), 𝑃𝐶 (𝑡) is the consumed power and 𝑃𝑆 (𝑡) is the power of the storage devices. A positive 𝑃𝑆 (𝑡)
1
means power is consumed from the microgrid and stored in the storage device. A negative 𝑃𝑆 (𝑡)
means power is injected in the microgrid which is extracted from the storage device. The power
balance is maintained if for all instants of time 𝑡
𝑃𝐺 (𝑡) = 𝑃𝐶 (𝑡) + 𝑃𝑆 (𝑡).
Due to the installation of renewable energy resources, 𝑃𝐺 (𝑡) is often weather dependent and difficult
to predict. Controlling 𝑃𝐶 (𝑡) in an appropriate way helps to maintain the power balance. For instance,
it is useful to postpone the use of a refrigerator, a freezer, a boiler …. (which have sufficient thermal
inertia) to the moments where 𝑃𝐺 (𝑡) is large (Demand Side Management). Finally, 𝑃𝑆 (𝑡) is used to
maintain the needed power balance but notice an appropriate control of 𝑃𝐶 (𝑡) reduces the need for
𝑃𝑆 (𝑡) which reduces the needed energy storage capacity.
In general, data communication is frequently used to control the power consumption of the loads. It is
however also important to obtain the power balance without data communication i.e. by using droop
control. Using droop control not only an active power balance but also a reactive power balance can be
obtained.
2: Power flow in an electrical grid
̅1 = 𝑈1 at A and 𝑈
̅2 = 𝑈2 𝑒 −𝑗𝛿 =
Figure 1 visualizes a line of an electrical grid. Notice the voltages 𝑈
𝑈2 (𝑐𝑜𝑠 𝛿 − 𝑗 𝑠𝑖𝑛 𝛿) at B. The grid impedance between A and B equals 𝑍̅ = 𝑅 + 𝑗𝑋 = 𝑍(𝑐𝑜𝑠 𝜃 +
𝑗 𝑠𝑖𝑛 𝜃) = 𝑍 𝑒 𝑗𝜃 implying a current
𝐼 ̅ = 𝐼(𝑐𝑜𝑠 𝜙 − 𝑗 𝑠𝑖𝑛 𝜙) =
is flowing from A to B.
̅1 − 𝑈
̅2
𝑈
𝑍̅
Figure 1: Power flow through a line
The power flowing from A to B equals
̅1 𝐼 ∗̅ = 𝑈1 (
𝑆̅ = 𝑃 + 𝑗 𝑄 = 𝑈
̅2 ∗ 𝑈12
𝑈1 − 𝑈
𝑈1 𝑈2 𝑒 𝑗(𝛿+𝜃)
𝑒 𝑗𝜃 −
) =
𝑍
𝑍
𝑍̅
which means an active power
𝑃=
𝑈12
𝑈1 𝑈2
𝑐𝑜𝑠 𝜃 −
𝑐𝑜𝑠 (𝛿 + 𝜃)
𝑍
𝑍
𝑄=
𝑈12
𝑈1 𝑈2
𝑠𝑖𝑛 𝜃 −
𝑠𝑖𝑛 (𝛿 + 𝜃)
𝑍
𝑍
and a reactive power
is flowing from A to B. Since 𝑍̅ = 𝑅 + 𝑗𝑋 = 𝑍(𝑐𝑜𝑠 𝜃 + 𝑗 𝑠𝑖𝑛 𝜃), one obtains that
𝑐𝑜𝑠 𝜃
𝑅
𝑅
= 2= 2
𝑍
𝑍
𝑅 + 𝑋2
2
𝑠𝑖𝑛 𝜃
𝑋
𝑋
= 2= 2
.
𝑍
𝑍
𝑅 + 𝑋2
Since 𝑐𝑜𝑠(𝛿 + 𝜃) = 𝑐𝑜𝑠 𝛿 𝑐𝑜𝑠 𝜃 − 𝑠𝑖𝑛 𝛿 𝑠𝑖𝑛 𝜃 and 𝑠𝑖𝑛(𝛿 + 𝜃) = 𝑠𝑖𝑛 𝛿 𝑐𝑜𝑠 𝜃 + 𝑐𝑜𝑠 𝛿 𝑠𝑖𝑛 𝜃,
𝑃 = 𝑈12
𝑅
𝑅 𝑐𝑜𝑠 𝛿
𝑋 𝑠𝑖𝑛 𝛿
𝑈1
(𝑅(𝑈1 − 𝑈2 𝑐𝑜𝑠 𝛿) + 𝑋𝑈2 𝑠𝑖𝑛 𝛿)
− 𝑈1 𝑈2 ( 2
−
)= 2
𝑅2 + 𝑋2
𝑅 + 𝑋2 𝑅2 + 𝑋2
𝑅 + 𝑋2
and
𝑄 = 𝑈12
𝑅2
𝑋
𝑅 𝑠𝑖𝑛 𝛿
𝑋 𝑐𝑜𝑠 𝛿
𝑈1
− 𝑈1 𝑈2 ( 2
+ 2
)= 2
(−𝑅𝑈2 𝑠𝑖𝑛 𝛿 + 𝑋(𝑈1 − 𝑈2 𝑐𝑜𝑠 𝛿)).
2
2
2
+𝑋
𝑅 +𝑋
𝑅 +𝑋
𝑅 + 𝑋2
Notice 𝑈2 𝑠𝑖𝑛 𝛿 and 𝑈1 − 𝑈2 𝑐𝑜𝑠 𝛿 in the expressions for 𝑃 and 𝑄, and notice also
𝑋𝑃 − 𝑅𝑄 = 𝑈1 𝑈2 𝑠𝑖𝑛 𝛿
implying
𝑈2 𝑠𝑖𝑛 𝛿 =
𝑋𝑃 − 𝑅𝑄
.
𝑈1
Notice also
𝑅𝑃 + 𝑋𝑄 = (𝑈1 −𝑈2 𝑐𝑜𝑠𝛿)𝑈1
implying
𝑈1 − 𝑈2 𝑐𝑜𝑠 𝛿 =
𝑅𝑃 + 𝑋𝑄
.
𝑈1
3: Inductive grid impedance
3.1: A high voltage grid
When considering a high voltage grid, the impedance 𝑍̅ is mainly inductive i.e. 𝑋 ≫ 𝑅. This implies
𝑈2 𝑠𝑖𝑛 𝛿 ≈
𝑋𝑃
𝑈1
and
𝑋𝑄
.
𝑈1
̅1 and 𝑈
̅2 ) is small, 𝑠𝑖𝑛 𝛿 ≈ 𝛿 which implies
Since 𝛿 (the phase difference between the voltages 𝑈
𝑈1 − 𝑈2 𝑐𝑜𝑠 𝛿 ≈
𝛿≈
𝑋𝑃
.
𝑈1 𝑈2
This means the phase difference 𝛿 mainly depends on 𝑃 and can be controlled by controlling 𝑃.
Alternatively, by controlling δ it is possible to control 𝑃. Notice this expression formulates the close
relationship between the active power balance and the grid frequency. In case the generated active
power is larger than the consumed power, the excess of energy is stored as kinetic energy in the
rotating machines (synchronous and asynchronous) connected with the grid. This implies the grid
frequency increases i.e. the phase of the grid voltage increases faster.
Since 𝛿 is small, 𝑐𝑜𝑠 𝛿 ≈ 1 which implies
𝑋𝑄
.
𝑈1
This means the voltage difference 𝑈1 − 𝑈2 mainly depends on Q and 𝑈1 − 𝑈2 can be controlled by
controlling Q. Alternatively, by controlling 𝑈1 − 𝑈2 it is possible to control Q.
𝑈1 − 𝑈2 ≈
3
3.2: Droop control in a high voltage grid
Summarizing, by adjusting 𝑃 and 𝑄 independently, the frequency and the amplitude of the grid
voltage are determined. These conclusions allow to formulate the frequency droop regulation and the
voltage droop regulation as
𝑓 − 𝑓0 = −𝑘𝑃 (𝑃 − 𝑃0 )
𝑈 − 𝑈0 = −𝑘𝑄 (𝑄 − 𝑄0 ).
Here, 𝑓0 is the nominal grid frequency and 𝑈0 is the nominal grid voltage. In case 𝑓 = 𝑓0, the
generating unit generates an active power 𝑃 = 𝑃0 and in case 𝑈 = 𝑈0 , the generating unit generates a
reactive power 𝑄 = 𝑄0 as visualized in Figure 2. Figure 2 also visualizes the entire frequency droop
and the entire voltage droop regulation.
Figure 2: Frequency droop and voltage droop regulation
As the grid frequency increases, the generated active power 𝑃 decreases which helps to maintain the
active power balance (when the grid frequency decreases, the generated active power increases). As
the grid voltage increases, the generated reactive power 𝑄 decreases which helps to maintain the
reactive power balance.
Using the primary control visualized in Figure 2, an active and a reactive power balance is obtained
but this power balance does not guarantee a steady state 𝑓 = 𝑓0 and 𝑈 = 𝑈0 . By a parallel shift of the
droop curves in Figure 2, the same power exchange can occur at a steady state 𝑓 = 𝑓0 and 𝑈 = 𝑈0
(secondary control).
4: Ohmic grid impedance
4.1: A low voltage grid
Notice the droop regulation approach of Figure 2 is valid in case the grid impedance is mainly
inductive (as it is the case in a high voltage grid). When considering low voltage grids (and
micogrids), the grid impedance is mainly ohmic (i.e. 𝑋 ≪ 𝑅). This implies
𝑈2 𝑠𝑖𝑛 𝛿 =
reduces to (since 𝛿 is small, 𝑠𝑖𝑛 𝛿 ≈ 𝛿)
𝛿≈
𝑋𝑃 − 𝑅𝑄
.
𝑈1
𝑋𝑃 − 𝑅𝑄
𝑅𝑄
≈−
.
𝑈1 𝑈2
𝑈1 𝑈2
Since 𝛿 is small, 𝑐𝑜𝑠 𝛿 ≈ 1 which implies
4
𝑈1 − 𝑈2 𝑐𝑜𝑠 𝛿 =
𝑅𝑃 + 𝑋𝑄
.
𝑈1
reduces to
𝑈1 − 𝑈2 ≈
𝑅𝑃 + 𝑋𝑄
𝑅𝑃
≈
.
𝑈1
𝑈1
4.2: Droop control in a low voltage grid
Using the grid frequency, droop regulation allows to control the reactive power needed to maintain the
reactive power balance. More precisely,
𝑓 − 𝑓0 = −𝑘𝑄 (𝑄 − 𝑄0 ).
Using the grid voltage, droop regulation allows to control the active power needed to maintain the
active power balance. More precisely,
𝑈 − 𝑈0 = −𝑘𝑃 (𝑃 − 𝑃0 ).
5: Ohmic inductive grid impedance
When considering a high voltage grid (inductive grid impedance), the active power balance is
controlled using the grid frequency and the reactive power balance is controlled using the voltage
level. The active and reactive powers can be controlled independently. When considering a low
voltage grid (ohmic grid impedance), the active power balance is controlled using the voltage level
and the reactive power is controlled using the grid frequency. The active and reactive power can be
controlled independently.
5.1: Droop control based on modified active and reactive power
However, in case the grid impedance is ohmic inductive (it is not possible to neglect 𝑅 and it is not
possible to neglect 𝑋), a straightforward independent control of the active and the reactive powers is
not possible. In order to obtain an independent control mechanism, the “modified active power” 𝑃′ and
the “modified reactive power” 𝑄′ are defined as (𝜃 is the angle of the grid impedance i.e. 𝑍̅ =
𝑍(𝑐𝑜𝑠 𝜃 + 𝑗 𝑠𝑖𝑛 𝜃))
𝑋
𝑅
−
𝑃′
𝑠𝑖𝑛 𝜃 −𝑐𝑜𝑠 𝜃 𝑃
𝑍 ] [𝑃 ] .
[ ]=[
] [ ] = [𝑍
𝑅
𝑋
𝑄
𝑄
𝑄′
𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜃
𝑍
𝑍
This definition of the modified active power 𝑃′ implies 𝑋𝑃 − 𝑅𝑄 = 𝑍𝑃′ and
𝑈2 𝑠𝑖𝑛 𝛿 =
𝑋𝑃 − 𝑅𝑄
𝑍𝑃′
=
.
𝑈1
𝑈1
Since 𝛿 is small,
𝛿 ≈ 𝑠𝑖𝑛𝛿 =
𝑍𝑃′
𝑈1 𝑈2
which results in a droop regulation mechanism
𝑓 − 𝑓0 = −𝑘𝑃′ (𝑃′ − 𝑃0′ )
visualized in Figure 3.
5
Figure 3: Droop regulation of the modified active power
This definition of the modified reactive power 𝑄′ implies 𝑅𝑃 + 𝑋𝑄 = 𝑍𝑄′ and
𝑈1 − 𝑈2 𝑐𝑜𝑠 𝛿 =
𝑅𝑃 + 𝑋𝑄
𝑍𝑄′
=
.
𝑈1
𝑈1
Since 𝛿 is small, 𝑐𝑜𝑠 𝛿 ≈ 1 giving
𝑈1 − 𝑈2 ≈ 𝑈1 − 𝑈2 𝑐𝑜𝑠 𝛿 =
𝑍𝑄′
𝑈1
which results in a droop regulation mechanism
𝑈 − 𝑈0 = −𝑘𝑄′ (𝑄′ − 𝑄0′ )
visualized in Figure 4.
Figure 4: Droop regulation of the modified reactive power
5.2: Power balance in a microgrid
In order to maintain the power balance in a microgrid operating in island mode, all power generating
units have to inject an appropriate 𝑃 and 𝑄. Based on 𝑓 and 𝑈, the modified powers 𝑃′ and 𝑄′ are
known (the grid impedance is assumed to be ohmic inductive). More precisely, a power generating
unit compares 𝑓 and 𝑈 with the nominal 𝑓0 and 𝑈0 which allows to calculate the required 𝑃′ and 𝑄′
(using constants 𝑘𝑃′ and 𝑘𝑄′ ).
For instance, in case the frequency 𝑓 = 𝑓0 no modified active power is exchanged i.e. 𝑃′ = 𝑃′0 = 0.
In case 𝑓 < 𝑓0, modified active power 𝑃′ > 0 is injected into the grid. In case 𝑓 > 𝑓0, modified active
power is consumed from the grid (𝑃′ < 0). In case the voltage level 𝑈 = 𝑈0 no modified reactive
power is exchanged i.e. 𝑄′ = 𝑄′0 = 0. In case 𝑈 < 𝑈0 , modified reactive power 𝑄′ > 0 is injected
into the grid. In case 𝑈 > 𝑈0 , modified reactive power is consumed from the grid (𝑄′ < 0).
Since the angle 𝜃 of the grid impedance is known, the real powers 𝑃 and 𝑄 are also known by the
expression
6
𝑃
𝑠𝑖𝑛 𝜃
[ ]=[
𝑄
𝑐𝑜𝑠 𝜃
−𝑐𝑜𝑠 𝜃 −1 𝑃′
] [ ].
𝑄′
𝑠𝑖𝑛 𝜃
Consider e.g. a battery and using a converter, this battery is connected with the grid as visualized in
Figure 5. The converter contains an inverter and inductors 𝐿𝑐𝑜𝑛𝑣 in each phase (giving a reactance
𝑋𝑐𝑜𝑛𝑣 = 𝜔𝐿𝑐𝑜𝑛𝑣 ). In order to inject a positive active power 𝑃 into the grid, the phase of the voltage
obtained by the inverter must lead an angle
𝑋𝑐𝑜𝑛𝑣 𝑃
𝛿≈
𝑈2
with respect to the grid voltage. In order to inject a reactive power into the grid, the voltage obtained
by the inverter must have an RMS value 𝑈 + ∆𝑈 (with ∆𝑈 > 0) in case the grid voltage has an RMS
value 𝑈 with
∆𝑈 ≈
𝑋𝑐𝑜𝑛𝑣 𝑄
.
𝑈
Figure 5: Battery energy storage system
6: Primary, secondary and tertiary control
Using the droop control mechanism, the active power balance is maintained (primary control)
implying for all instants of time 𝑡
𝑃𝐺 (𝑡) = 𝑃𝐶 (𝑡)
with a nominal steady state frequency 𝑓 = 𝑓0 (due to the secondary control) and a nominal steady state
voltage level 𝑈 = 𝑈0 . Due to the tertiary control, the power is generated in the most economical
(cheapest) way.
6.1: Power production cost and marginal cost
When considering a generating unit, it costs money to generate the electrical power. Figure 6
visualizes the cost 𝐾(𝑃) to produce an active power 𝑃. Suppose this cost to produce 𝑃 can be
expressed as a quadratic expression
𝐾(𝑃) = 𝐶′1 + 𝐶′2 𝑃 + 𝐶′3 𝑃2
implying a marginal cost
𝐶(𝑃) =
𝜕𝐾(𝑃)
= 𝐶′2 + 2𝐶′3 𝑃.
𝜕𝑃
7
Figure 6: Production cost K(p) of a generating unit
6.2: Parallel production units
Suppose there are two generating units connected in parallel with the grid. Producing 𝑃 by the first
generating unit requires a cost
𝐾1 (𝑃) = 𝐶′11 + 𝐶′21 𝑃 + 𝐶′31 𝑃2
implying a marginal cost
𝐶1 (𝑃) = 𝐶′21 + 2𝐶′31 𝑃.
Producing 𝑃 by the second generating unit requires a cost
𝐾2 (𝑃) = 𝐶′12 + 𝐶′22 𝑃 + 𝐶′32 𝑃2
implying a marginal cost
𝐶2 (𝑃) = 𝐶′22 + 2𝐶′32 𝑃.
Suppose the first generating unit produces a power 𝑃1 and the second generating unit produces a
power 𝑃2 in order to obtain the required total power 𝑃𝑡 = 𝑃1 + 𝑃2 . The first generating unit has a
marginal cost 𝐶1 (𝑃1 ) and the second generating unit has a marginal cost 𝐶2 (𝑃2 ). It is possible to
change the produced power of the first generating unit to 𝑃1 + ∆𝑃1 and the power produced by the
second generating unit to 𝑃2 + ∆𝑃2 . In case ∆𝑃1 = −∆𝑃2, the same total active power 𝑃𝑡 is produced.
Figure 7: Economical optimization of the power production
Figure 7 visualizes the marginal costs of the generating units and Figure 7 assumes 𝐶1 (𝑃1 ) > 𝐶2 (𝑃2 ).
Since 𝐶1 (𝑃1 ) > 𝐶2 (𝑃2 ), it is useful to reduce the power production of generating unit 1 and to increase
the production of generating unit 2 giving a ∆𝑃1 < 0 and a ∆𝑃2 > 0. This implies the working points
giving 𝐶1 (𝑃1 ) and 𝐶2 (𝑃2 ) are replaced by working points giving 𝐶1 (𝑃1 + ∆𝑃1 ) and 𝐶2 (𝑃2 + ∆𝑃2 ). It is
important
8
𝐶1 (𝑃1 + ∆𝑃1 ) = 𝐶2 (𝑃2 + ∆𝑃2 ) = 𝐶3 .
By generating 𝑃1 + ∆𝑃1 and 𝑃2 + ∆𝑃2 instead of 𝑃1 and 𝑃2 , the total production cost is reduced by
𝐾1 (𝑃1 ) − 𝐾1 (𝑃1 + ∆𝑃1 ) + 𝐾2 (𝑃2 ) − 𝐾2 (𝑃2 + ∆𝑃2 )
which equals
𝑃1
∫
𝑃1 +∆𝑃1
𝑃2 +∆𝑃2
𝐶1 (𝑃) 𝑑𝑃 − ∫
𝐶2 (𝑃) 𝑑𝑃 .
𝑃2
This expression is positive since 𝐶1 (𝑃) > 𝐶2 (𝑃) > 0 and ∆𝑃2 = −∆𝑃1 > 0 which implies a reduction
of the total production cost.
References
De Brabandere K., Bolsens B., Van den Keybus J., Woyte A., Driesen J. and Belmans R., A Voltage
and Frequency Droop Control Method for Parallel Inverters, IEEE Transactions on Power Electronics,
vol. 22, no. 4, July 2007.
De Brabandere K., Vanthournout K., Driesen J., Deconinck G. and Belmans R., Control of
Microgrids,
Deconinck G., An evaluation of two-way communication means for advanced metering in Flanders
(Belgium), IEEE International Instrumentation and Measurement Technology Conference, Victoria,
Vancouver Island, Canada, 12-15 May, 2008.
Vanthournout K., De Brabandere K., Haesen E., Van den Keybus J., Deconinck G. and Belmans R.,
Agora: Distributed Tertiary Control of Distributed Resources, 15th PSCC, Liege, 22-26 August, 2005.
9