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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 ππΆ (π‘) is the consumed active power, ππ·πΊ (π‘) is the decentralized generated power (photovoltaic cells, wind turbines, wave energy, β¦) and ππΆπΊ (π‘) is the centralized generated power (nuclear power stations, fossil fuel power stations), the power balance is maintained if for all instants of time π‘ ππΆ (π‘) = ππ·πΊ (π‘) + ππΆπΊ (π‘). Since not only ππΆ (π‘) but also ππ·πΊ (π‘) is varying with respect to time, larger controlled variations of ππΆπΊ (π‘) are needed. The rise of renewable energy production (ππ·πΊ (π‘) and its variations become larger) implies larger variations of ππΆπΊ (π‘) 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 ππΆπΊ (π‘) is not able to adapt to the variations of ππΆ (π‘) and ππ·πΊ (π‘), 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, ππΊ (π‘) is the generated power in the microgrid (e.g. due to photovoltaic panels, micro wind turbines, micro CHP installations, β¦), ππΆ (π‘) is the consumed power and ππ (π‘) is the power of the storage devices. A positive ππ (π‘) 1 means power is consumed from the microgrid and stored in the storage device. A negative ππ (π‘) 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 π‘ ππΊ (π‘) = ππΆ (π‘) + ππ (π‘). Due to the installation of renewable energy resources, ππΊ (π‘) is often weather dependent and difficult to predict. Controlling ππΆ (π‘) 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 β¦. (which have sufficient thermal inertia) to the moments where ππΊ (π‘) is large (Demand Side Management). Finally, ππ (π‘) is used to maintain the needed power balance but notice an appropriate control of ππΆ (π‘) reduces the need for ππ (π‘) 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 Μ 1 = π1 at A and π Μ 2 = π2 π βππΏ = Figure 1 visualizes a line of an electrical grid. Notice the voltages π π2 (πππ πΏ β π π ππ πΏ) at B. The grid impedance between A and B equals πΜ = π + ππ = π(πππ π + π π ππ π) = π π ππ implying a current πΌ Μ = πΌ(πππ π β π π ππ π) = is flowing from A to B. Μ 1 β π Μ 2 π πΜ Figure 1: Power flow through a line The power flowing from A to B equals Μ 1 πΌ βΜ = π1 ( πΜ = π + π π = π Μ 2 β π12 π1 β π π1 π2 π π(πΏ+π) π ππ β ) = π π πΜ which means an active power π= π12 π1 π2 πππ π β πππ (πΏ + π) π π π= π12 π1 π2 π ππ π β π ππ (πΏ + π) π π and a reactive power is flowing from A to B. Since πΜ = π + ππ = π(πππ π + π π ππ π), one obtains that πππ π π π = 2= 2 π π π + π2 2 π ππ π π π = 2= 2 . π π π + π2 Since πππ (πΏ + π) = πππ πΏ πππ π β π ππ πΏ π ππ π and π ππ(πΏ + π) = π ππ πΏ πππ π + πππ πΏ π ππ π, π = π12 π π πππ πΏ π π ππ πΏ π1 (π (π1 β π2 πππ πΏ) + ππ2 π ππ πΏ) β π1 π2 ( 2 β )= 2 π 2 + π2 π + π2 π 2 + π2 π + π2 and π = π12 π 2 π π π ππ πΏ π πππ πΏ π1 β π1 π2 ( 2 + 2 )= 2 (βπ π2 π ππ πΏ + π(π1 β π2 πππ πΏ)). 2 2 2 +π π +π π +π π + π2 Notice π2 π ππ πΏ and π1 β π2 πππ πΏ in the expressions for π and π, and notice also ππ β π π = π1 π2 π ππ πΏ implying π2 π ππ πΏ = ππ β π π . π1 Notice also π π + ππ = (π1 βπ2 πππ πΏ)π1 implying π1 β π2 πππ πΏ = π π + ππ . π1 3: Inductive grid impedance 3.1: A high voltage grid When considering a high voltage grid, the impedance πΜ is mainly inductive i.e. π β« π . This implies π2 π ππ πΏ β ππ π1 and ππ . π1 Μ 1 and π Μ 2 ) is small, π ππ πΏ β πΏ which implies Since πΏ (the phase difference between the voltages π π1 β π2 πππ πΏ β πΏβ ππ . π1 π2 This means the phase difference πΏ mainly depends on π and can be controlled by controlling π. Alternatively, by controlling Ξ΄ it is possible to control π. 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 πΏ is small, πππ πΏ β 1 which implies ππ . π1 This means the voltage difference π1 β π2 mainly depends on Q and π1 β π2 can be controlled by controlling Q. Alternatively, by controlling π1 β π2 it is possible to control Q. π1 β π2 β 3 3.2: Droop control in a high voltage grid Summarizing, by adjusting π and π 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 π β π0 = βππ (π β π0 ) π β π0 = βππ (π β π0 ). Here, π0 is the nominal grid frequency and π0 is the nominal grid voltage. In case π = π0, the generating unit generates an active power π = π0 and in case π = π0 , the generating unit generates a reactive power π = π0 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 π 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 π 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 π = π0 and π = π0 . By a parallel shift of the droop curves in Figure 2, the same power exchange can occur at a steady state π = π0 and π = π0 (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. π βͺ π ). This implies π2 π ππ πΏ = reduces to (since πΏ is small, π ππ πΏ β πΏ) πΏβ ππ β π π . π1 ππ β π π π π ββ . π1 π2 π1 π2 Since πΏ is small, πππ πΏ β 1 which implies 4 π1 β π2 πππ πΏ = π π + ππ . π1 reduces to π1 β π2 β π π + ππ π π β . π1 π1 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, π β π0 = βππ (π β π0 ). Using the grid voltage, droop regulation allows to control the active power needed to maintain the active power balance. More precisely, π β π0 = βππ (π β π0 ). 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 π and it is not possible to neglect π), a straightforward independent control of the active and the reactive powers is not possible. In order to obtain an independent control mechanism, the βmodified active powerβ πβ² and the βmodified reactive powerβ πβ² are defined as (π is the angle of the grid impedance i.e. πΜ = π(πππ π + π π ππ π)) π π β πβ² π ππ π βπππ π π π ] [π ] . [ ]=[ ] [ ] = [π π π π π πβ² πππ π π ππ π π π This definition of the modified active power πβ² implies ππ β π π = ππβ² and π2 π ππ πΏ = ππ β π π ππβ² = . π1 π1 Since πΏ is small, πΏ β π πππΏ = ππβ² π1 π2 which results in a droop regulation mechanism π β π0 = βππβ² (πβ² β π0β² ) visualized in Figure 3. 5 Figure 3: Droop regulation of the modified active power This definition of the modified reactive power πβ² implies π π + ππ = ππβ² and π1 β π2 πππ πΏ = π π + ππ ππβ² = . π1 π1 Since πΏ is small, πππ πΏ β 1 giving π1 β π2 β π1 β π2 πππ πΏ = ππβ² π1 which results in a droop regulation mechanism π β π0 = βππβ² (πβ² β π0β² ) 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 π and π. Based on π and π, the modified powers πβ² and πβ² are known (the grid impedance is assumed to be ohmic inductive). More precisely, a power generating unit compares π and π with the nominal π0 and π0 which allows to calculate the required πβ² and πβ² (using constants ππβ² and ππβ² ). For instance, in case the frequency π = π0 no modified active power is exchanged i.e. πβ² = πβ²0 = 0. In case π < π0, modified active power πβ² > 0 is injected into the grid. In case π > π0, modified active power is consumed from the grid (πβ² < 0). In case the voltage level π = π0 no modified reactive power is exchanged i.e. πβ² = πβ²0 = 0. In case π < π0 , modified reactive power πβ² > 0 is injected into the grid. In case π > π0 , modified reactive power is consumed from the grid (πβ² < 0). Since the angle π of the grid impedance is known, the real powers π and π are also known by the expression 6 π π ππ π [ ]=[ π πππ π βπππ π β1 πβ² ] [ ]. πβ² π ππ π 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 πΏππππ£ in each phase (giving a reactance πππππ£ = ππΏππππ£ ). In order to inject a positive active power π into the grid, the phase of the voltage obtained by the inverter must lead an angle πππππ£ π πΏβ π2 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 π + βπ (with βπ > 0) in case the grid voltage has an RMS value π with βπ β πππππ£ π . π 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 π‘ ππΊ (π‘) = ππΆ (π‘) with a nominal steady state frequency π = π0 (due to the secondary control) and a nominal steady state voltage level π = π0 . 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 πΎ(π) to produce an active power π. Suppose this cost to produce π can be expressed as a quadratic expression πΎ(π) = πΆβ²1 + πΆβ²2 π + πΆβ²3 π2 implying a marginal cost πΆ(π) = ππΎ(π) = πΆβ²2 + 2πΆβ²3 π. ππ 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 π by the first generating unit requires a cost πΎ1 (π) = πΆβ²11 + πΆβ²21 π + πΆβ²31 π2 implying a marginal cost πΆ1 (π) = πΆβ²21 + 2πΆβ²31 π. Producing π by the second generating unit requires a cost πΎ2 (π) = πΆβ²12 + πΆβ²22 π + πΆβ²32 π2 implying a marginal cost πΆ2 (π) = πΆβ²22 + 2πΆβ²32 π. Suppose the first generating unit produces a power π1 and the second generating unit produces a power π2 in order to obtain the required total power ππ‘ = π1 + π2 . The first generating unit has a marginal cost πΆ1 (π1 ) and the second generating unit has a marginal cost πΆ2 (π2 ). It is possible to change the produced power of the first generating unit to π1 + βπ1 and the power produced by the second generating unit to π2 + βπ2 . In case βπ1 = ββπ2, the same total active power ππ‘ is produced. Figure 7: Economical optimization of the power production Figure 7 visualizes the marginal costs of the generating units and Figure 7 assumes πΆ1 (π1 ) > πΆ2 (π2 ). Since πΆ1 (π1 ) > πΆ2 (π2 ), it is useful to reduce the power production of generating unit 1 and to increase the production of generating unit 2 giving a βπ1 < 0 and a βπ2 > 0. This implies the working points giving πΆ1 (π1 ) and πΆ2 (π2 ) are replaced by working points giving πΆ1 (π1 + βπ1 ) and πΆ2 (π2 + βπ2 ). It is important 8 πΆ1 (π1 + βπ1 ) = πΆ2 (π2 + βπ2 ) = πΆ3 . By generating π1 + βπ1 and π2 + βπ2 instead of π1 and π2 , the total production cost is reduced by πΎ1 (π1 ) β πΎ1 (π1 + βπ1 ) + πΎ2 (π2 ) β πΎ2 (π2 + βπ2 ) which equals π1 β« π1 +βπ1 π2 +βπ2 πΆ1 (π) ππ β β« πΆ2 (π) ππ . π2 This expression is positive since πΆ1 (π) > πΆ2 (π) > 0 and βπ2 = ββπ1 > 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