Aspects of this technology are described in an article titled “Global sliding-mode control with fractional-order terms for the robust optimal operation of a hybrid renewable microgrid with battery energy storage” in J. Electronics on Dec. 28, 2021, incorporated herein by reference in its entirety.
The inventors acknowledge the support provided by the Center of Renewable Energy and Power Systems at King Fahd University of Petroleum & Minerals under Project No. INRE2106 and King Abdullah City for Atomic and Renewable Energy (K.A.CARE).
The present disclosure is directed to a non-linear control of a hybrid microgrid system, and more particularly relates to a system and method for controlling a hybrid microgrid system in accordance with global sliding mode control with fractional order terms.
The “background” description provided herein is for the purpose of generally presenting the context of the disclosure. Work of the presently named inventors, to the extent it is described in this background section, as well as aspects of the description which may not otherwise qualify as prior art at the time of filing, are neither expressly or impliedly admitted as prior art against the present invention.
Over past few decades consumption of energy in urban as well as in rural areas has increased due to industrialization, rapid economic development, and social progress. The energy has been generated mostly from fossil fuels. Generating the energy from fossil fuels has certain harmful effects, such as climate change, green house effect, and environmental pollution. On the other hand, over consumption of the fossil fuels depletes natural resources at a fast rate. Reflecting such circumstances, need for introducing and expanding use of renewable energy sources such as wind, solar, hydro, and geothermal as alternative energy sources has been well recognized, since they are freely and abundantly available, efficient, reliable, and have no adverse environmetal effects.
Advances in power electronics technology has led to incerase in energy conversion efficiencies and continuing fall in costs of photovoltaic (PV) systems and wind turbines (WT). Global capacities of installed solar and wind power were 586 GW and 623 GW, respectively at the beginning of 2020. See for example, Jamal, S. et al., A review of Energy Management and Power Management Systems for Microgrid and Nanogrid Applications, Sustainability 2021, 13, 10331.
The PV systems and WT systems have complementary properties. During the day output from the PV system is high, whereas power generation by WT is low. On the other hand, during the night, solar irradiance is unavailable; the PV system does not generate power, while output power of the WT has tendency to be high. Thus, the PV system and the wind turbines are often tied to a common DC bus or to a common AC bus via boost converters to form a hybrid micro grid system (HMS). See for examples, Alayi, R et al., Optimal Load Frequency Control of Island Microgrids via a PID Controller in the Presence of Wind Turbine and PV. Sustainability 2021, 13, 10728, and Al-Quraan, A. et al., Modeling, Design and Control of a Standalone Hybrid PV-Wind Micro-Grid System, Energies 2021, 14, 4849.
The HMS is generally a collection of interlinked renewable and conventional energy resources connected to users and controlled by systems to ensure efficient energy usage and storage. The HMS is also capable to operate in both a grid connected mode and an islanded mode. See for example, Akram, U. et al., An Innovative Hybrid Wind-Solar and Battery-Supercapacitor Microgrid System-Development and Optimization. IEEE Access 2017, 5, 25897-25912, and also https://en.wikipedia.org/wiki/Microgrid.
The HMS is configured to permit insufficiency and uncertainty of the renewale energy sources. However, sometimes the renewable energy produced by the HMS is larger than an energy demand by the users. In such situations, a battery energy storage system (BESS) is utilized to store an excess power. When the energy demand is large and grid connected systems are unable to fulfill the energy demands, the BESS fulfills the energy demand by discharging its stored energy.
A sliding mode control is known as a robust control method applicable to a system with dynamic behaviour. See for examples, V. Utkin, Sliding Mode Control Design Principles and Applications to Electric Drives, IEEE Trans. Industrial Electronics, Vol. 40, No. 1, February 1993, and https://en.wikipedia.org/wiki/Sliding_mode_control, entire contents of both is incorporated by reference herein.
Recently, a global sliding mode control with fractional order operators have been reported in applications to robot control and vehicle control. See, Wang, J. et al., Fractional order sliding mode control via disturbanc observer for a class of fractional order systems with mismatched disturbance, Mechatronics 2018, 53, 8-19, and Xie, Y. et al., Coupled fractional-order sliding mode control and obstacle avoidance of a four-wheeled steerable mobile robot, ISA Trans. 2021, 108, 282-294, entire content of both is incorporated by reference herein.
In an exemplary embodiment, a method for controlling a hybrid microgrid system (HMS) is disclosed. The HMS includes a renewable energy source, a grid side converter (GSC) configured to output a power to a point of common coupling (PCC), a DC-link configured to receive a power from the renewable energy source and to supply a power to the GSC, a rechargeable battery configured to exchange a power via the DC-link, a load configured to receive a power via the PCC, a utility grid configured to exchange power via the PCC; and a controller configured to control the HMS by executing a program and in accordance with the method defined as a global sliding mode control with fractional order terms (GSMCFO) method. The program comprises a definition set customized for the HMS and to be referred in applying the GSMCFO method to the HMS.
The method for controlling the HMS includes preparing the definition set including a characteristic element ci to be measured, and equations defining a desired value ci* of the characteristic element ci, a fractional order sliding mode (FOSM) surface ζi for the characteristic element ci, and a control law element uicnt of the characteristic element ci. The method further includes monitoring the characteristic element ci(t) and a related status of the HMS. The method further includes calculating at least one of the equations based on the characteristic element ci(t) and the related status of the HMS monitored. The method further includes controlling the HMS based on the control law element uicnt(t) calculated and in accordance with the GSMCFO method.
The equation defining the FOSM surface ζi(t) of the characteristic element ci(t) comprises a fractional time integral of a tracking error ei(t) and a fractional time derivative of the tracking error ei(t). The tracking error ei(t) for the characteristic element ci(t) is defined as,
ei(t)=ci(t)−ci*(t)
Also, the equation defining the control law element uicnt(t) is configured to satisfy a condition,
so far as ζi(t) is not zero.
In another exemplary embodiment, a hybrid microgrid system (HMS) is disclosed. The HMS includes a renewable energy source, a grid side converter (GSC) configured to output a power to a point of common coupling (PCC), a DC-link configured to receive a power from the renewable energy source and to supply a power to the GSC, a rechargeable battery configured to exchange a power via the DC-link; a load configured to receive a power via the PCC; a utility grid configured to exchange a power via the PCC, and a controller including a processor, a memory, a bus-line, and I/O port.
The controller is configured to control the HMS by executing a program installed in the memory and in accordance with a global sliding mode control with fractional order terms (GSMCFO) method. The program includes a definition set customized for the HMS system and to be referred in applying the GSMCFO method to the HMS. The definition set includes a characteristic element ci to be measured, and the equations defining a desired value ci* of the characteristic element ci, a fractional order sliding mode (FOSM) surface ζi of the characteristic element ci, and a control law element uicnt of the characteristic element ci.
The controller is further configured to monitor the characteristic element ci(t) and a related status of the HMS, calculate at least one of the equations based on the characteristic element ci(t) and the related status of the HMS monitored, and control the HMS based on the control law element uicnt(t) calculated and in accordance with the GSMCFO method.
The equation defining the FOSM surface ζi(t) for the characteristic element ci(t) comprises a fractional time integral of a tracking error ei(t) and a fractional time derivative of the tracking error ei(t) , with ei(t)=ci(t)−ci*(t) . Also, the equation defining the control law element uicnt(t) is configured to satisfy a condition:
so far as ζi(t) is not zero.
In another exemplary embodiment, a controller of a hybrid microgrid system (HMS) is disclosed. The HMS includes a renewable energy source, a grid side converter (GSC) configured to output a power to a point of common coupling (PCC), a DC-link configured to receive a power from the renewable energy source and to supply a power to the GSC, a rechargeable battery configured to exchange a power via the DC-link, a load configured to receive a power via the PCC, a utility grid configured to exchange a power via the PCC, and the controller including a processor, a memory, a bus-line, and I/O port. 20 The controller is configured to control the HMS by executing a program and in accordance with a global sliding mode control with fractional order terms (GSMCFO) method. The program includes a definition set customized for the HMS and to be referred in applying the GSMCFO method to the HMS. The definition set includes a characteristic element ci to be measured, and equations defining a desired value ci* of the characteristic element ci, a fractional order sliding mode (FOSM) surface ζi of the characteristic element ci and a control law element uicnt of the characteristic element ci. In one aspect, the controller is configured to monitor the characteristic element ci(t) and a related status of the HMS, calculate at least one of the equations defined in the definition set with the program and based on the characteristic element ci(t1) and the related status monitored, and control the HMS based on the control law element uicnt(t1) calculated, and in accordance with the GSMCFO.
The equation defining the FOSM surface ζi(t) of the characteristic element ci(t) includes a fractional time integral of a tracking error ei(t) given by ei(t)=ci(t)−ci*(t) and a fractional time derivative of the tracking error ei(t). Also, the equation defining the control law element uicnt(t) is configured to satisfy a condition:
so far as ζi(t) is not zero.
The foregoing general description of the illustrative embodiments and the following detailed description thereof are merely exemplary aspects of the teachings of this disclosure, and are not restrictive.
A more complete appreciation of this disclosure and many of the attendant advantages thereof will be readily obtained as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings, wherein:
In the drawings, like reference numerals designate identical or corresponding parts throughout the several views. Further, as used herein, the words “a,” “an” and the like generally carry a meaning of “one or more,” unless stated otherwise.
Furthermore, the terms “approximately,” “approximate,” “about,” and similar terms generally refer to ranges that include the identified value within a margin of 20%, 10%, or preferably 5%, and any values therebetween.
As described earlier, the HMS is configured to tackles the insufficiency and uncertainty of the renewable energy sources. In order to realize an efficient and cost-effective configuration of the HMS, numbers of power electronics components and power conversion stages have to be minimized. Further, it is also important to harness the maximum power from the PV system and the wind turbine even under astable operating conditions.
Some of the researches have proposed connecting the PV array directly to the common DC bus without using a boost converter in between, so as to minimize the numbers of the power conversion stages and the power electronics components. In such approaches where the boost converter was removed, usually a linear controller, utilizing a linear control mechanism such as proportional integral (PI) control was used for controlling switching operation of other converters, and for maintaining the power balance in the HMS. However, the linear controllers so far reported suffer from one or more drawbacks hindering their adoption.
For example, the linear control mechanism in such cases provides unsatisfactory transients and steady-state performances under circumstances of parametric uncertainties or sudden change in operating point due to fluctuating nature of renewable energy sources and load variations on consumption side. Also, those HMS take a long time to settle in presence of disturbances when the controller follows a linear control mechanism.
Accordingly, it is highly desired to realize an improved control method for the HMS with both a robustness and a short transient time under the circumstances of sudden changes in operating conditions.
A method of controlling the HMS and the HMS with improved efficiency and robustness against varying parameters are disclosed as detailed below. The proposed HMS is constructed with a cost competitive configuration. A controller of the HMS is configured to control the HMS with a global sliding-mode control method with fractional-order terms (GSMCFO) that improves both the transient time and robustness of the HMS, due to greater flexibility of fractional calculus (hereinafter interchangeably referred to as GSMCFO controller).
The GSMCFO controller provides a maximum power point tracking (MPPT) of the PV systems and wind power (WT) generators. Also, the GSMCFO controller provides a) excellent transient and robust steady-state performances of the HMS under parametric uncertainties, b) regulation of DC-link voltage irrespective of the operating conditions, c) extraction of maximum power from both the PV systems and the WT systems with a rotor side converter (RSC), d) active power transfer to the grid via a grid side converter (GSC), and e) maintaining the power balance of the HMS by effective charging/discharging of battery energy storage system (BESS).
The PMSG 104 includes a stator and a rotor, the latter is mechanically linked with the WT and is configured to be driven by the WT 102 and to convert a mechanical power generated by the WT 102 into an equivalent electrical energy. The RSC 106 is electrically connected with the PMSG 104 and configured to convert an AC current from the output of the PMSG 104 into an equivalent DC current.
The DC-link 132 is electrically connected with an output line of the RSC 106 at one end of the DC-link 132, and with an input line of the GSC 108 at the other end of the DC-link 132. The DC-link 132 is configured to provide a common wired connection, a DC-bus, from the RSC 106 to the GSC for a plurality of other resources. The DC-link 132 includes a capacitor Cdc 133 connected in between the positive terminal and the negative terminal of the DC-link 132.
The PV system 110 is connected to the DC-link 132 directly, that is, without any converters. The PV system 110 is configured to output a DC current that directly adds up to the input current of the DC-link 132. A diode 118 is connected in series with the PV system 110. A positive terminal (P side) of the diode 118 is connected to a positive output terminal of the PV arrays, whereas a negative terminal (N side) of the diode 118 is connected to a positive terminal of the DC-link 132. The diode is configured to prevent a reverse current from the DC-link 132 back to the PV system 110, thereby avoiding reverse-flow damages to the PV system 110. The diode 118 is selected from a group containing, but not limited to a PIN diode, an avalanche diode, a P-N junction diode, a Zener diode, etc. In other implementations, a switch is inserted instead of the diode, where the switch is configured to turn on or off by detecting a voltage difference between the positive output terminal of the PV system and the DC-link.
The BESS 114 is constituted by a rechargeable battery selected from a group containing, but not limited to Lithium-Ion (Li—ON), Nickel-Cadmium (NiCd), Nickel-Metal Hydride (NiMH), Lead-Acid batteries and such batteries. The BESS 114 is configured to supply an energy to the HMS 100 when a load demand, that is, “a power demanded at the load 126 at the PCC” is more than an energy required to be produced by the renewable energy sources of the HMS 100 due to intermittent conditions of the environment.
The BBBC 112 is electrically connected between the BESS 114 and the DC-link 132. In some implementations, for example, the BBBC 112 is a DC-to-DC converter. The BESS 114 is electrically connected to the DC-link 132 through the BBBC 112. The BBBC 112 is configured to operate either in a buck mode (step down) or a boost mode (Step-up). During charging of the BESS 114, the BBBC 112 operates in a buck mode whereas during discharging, the BBBC 112 operates in a boost mode. The buck mode is operated when the renewable energy sources of the HMS 100 generate more power than the load demand. When the HMS 100 generates more power than the load demand, an excess power is stored in the BESS 114 that is used as a reserved power. When the load fluctuates or some of the renewable resources are momentarily unable to supply power due to intermittent environmental condition, the BBBC 112 operates in boost mode to balance the load demand. In boost mode, stored power in the BESS 144 discharges, which is provided to the load in order to balance the load demand and avoid a load shedding.
The GSC 108 is configured to convert a DC voltage over the DC-link 132 to an AC output voltage/current. The output line of the GSC 108 is configured to supply the AC output current to a point of a common coupling (PCC) 128 via a filter including an inductor Lf 122 and a resistor Rf 124 connected in series.
A load 126 connected to the PCC is configured to receive power from the PCC. The PCC is also connected with a grid 130 and is configured to receive or transfer a power from/to the grid 130. A “power grid” generally means an interconnected local or wide area public network for electricity delivery. The grid 130 here represents an access point to the power grid. Optionally, the HMS is configured isolated from the grid to form an islanded HMS. The load 126 refers to a common load due to residential consumption, industry consumption and the like. The load in residential consumption includes lighting instruments, air conditioners, etc. The load in industry consumption includes machineries etc.
The GSMCFO controller 120 or simply the controller 120 is configured to control the HMS 100 in accordance with a global sliding-mode control utilizing a fractional-order terms (GSMCFO). The controller 120 is configured to achieve the MPPT of both the PV system 110 and the WT generator system 102 and 104, regulation of the voltage of the DC-link 132, and power balance of the HMS by controlling the battery current Ib with excellent transient and steady-state performances even under disturbances and uncertainties in the ambient environment.
The controller 120 includes a processor 120-1, a memory 120-2 with a preinstalled software, input ports 120-3 and output ports 120-5, and a bus line 120-4. The preinstalled software includes an instruction set and a definition set. The controller is configured to fetch the instruction set to the processor 120-1, execute them referring the definition set and based on a monitored status of the HMS.
The instruction set includes instructions for steps to be executed in controlling the HMO in accordance with the GSMCFO. The steps to be executed include monitoring a current status of the HMS, calculating equations defined in the definition set based on the current status monitored, and controlling the HMS based on calculation results.
The definition set includes a characteristic element ci to be monitored, a desired value ci* of the characteristic element ci, and equations defining a fractional order sliding mode (FOSM) surface ζi of the characteristic element ci , an equivalent control law element uieqv of the characteristic element ci , and a control law element uicnt of the characteristic element ci .
The instruction set also includes an instruction for performing iteration of those steps or part of those steps until the FOSM surface ζi or the tracking error ei(t) reduces to values equal to or smaller than each criterion predetermined.
The characteristic elements are selected based on modeling of each of constituents of the HMS. The characteristic elements selected for the WT generator system 102 and 104 are an angular frequency ωr(=c1) of the WT and a d axis stator current Ids(=c2) of the PMSG 104. The characteristics element selected for the DC-link 132 is a DC-link voltage Vdc(=c3). The characteristics element selected for the GSC output are a d-axis AC current Id(=c4) and a q-axis AC current Iq(=c5). The characteristic element selected for the BBBC 112 is a battery current Ib(=c6).
Preliminaries for GSMCFO Method
Definitions and properties of the fractional-order calculus are briefly reviewed here. The GSMCFO fully utilizes the fractional-order calculus.
A Riemann-Liouville fractional derivation of order μ(0<μ<1) of a continuous function ϕ(t) is defined by:
where, m is a positive integer number, Γ is a Gamma function defined as,
On the other hand, a Riemann-Liouville fractional integration of order μ(0<μ<1) of a continuous function ϕ(t) is described by:
Below three properties of the fractional-order calculus are noticeable.
Property 1: If 0<μ<1, then:(
Jtμϕ(t)=ϕ(t). (4)
Property 2: If 0<μ<1, then:(
Jtμϕ(t))=ϕ(t). (5)
Property 3: The fractional derivation or integration is a linear operation:(ϕ1(t)+ϕ2(t)=
ϕ1(t)+
ϕ2(t). (6)
Modeling of the WT Generator and Application of the GSMCFO Method
The mechanical power captured by the wind turbine (WT) 102 from the wind is given by:
where Pw is a mechanical power captured by the WT 102, VW is a wind speed, Cp is a power coefficient, ρ is an air density, R is a radius of the WT 102, ωr is an angular speed of the WT 102, a tip-speed ratio is represented by λ, βis a pitch angle, J is an inertia of a mechanical shaft, P is a number of pole pairs, and Te and Tm are an electrical and a mechanical torque, respectively. See for example, Housseini, B. et al., Robust Nonlinear Controller Design for On-Grid/Off-Grid Wind Energy Battery-Storage System. IEEE Trans. Smart Grid 2018, 9, 5588-5598, entire contents of which is incorporated herein by reference.
Now, assuming a desired tip speed ratio λ* gives a maximum value of the power coefficient Cp(λ, β) of Eq. (8) for any given wind speed Vw, a desired angular speed ωr* of the WT, namely the desired value of the characteristic element c1* is given from Eq. (4) as:
The voltage and the torque dynamics of the PMSG 104 in the d-q reference frames are given as:
where Vds and Vqs are the d-axis and q-axis stator voltages of the PMSG 104, respectively, Ld and Lq are the d-axis and q-axis self-inductances, respectively, Ids and Iqs are the d-axis and q-axis stator currents, respectively, Lq is the q-axis self-inductance, Λr is the rotor flux, and Rs is the stator resistance. As the PMSG 104 is a non-salient PMSG 104, Ld and Lq are equal, the electrical torque Te is given as:
To minimize the stator iron loss and maximize the efficiency of the PMSG 104, the d-axis stator current is preferably be zero. Thus, the desired value for the characteristic element c2* given by the desired d-axis stator current is set to zero:
c2*=Ids*=0. (17)
A tracking error e1(t) for the characteristic element c1, that is, the angular frequency ωr of the WT and a tracking error e2(t) for the characteristic element c2, that is the d-axis stator current Ids at a time t are defined as,
Further taking time derivations, following relations are obtained:
where,
H1 and H2 represent lumped disturbances in the system.
Now, fractional-order sliding-mode (FOSM) surfaces ζ1(t) and ζ2(t) are defined based on the tracking errors e1(t) and e2(t), for the angular speed ωr and for the d-axis stator current Ids, respectively, as below.
where μ∈(0, 1), and k1, k2 and σ1 are positive constants, denotes a Riemann-Liouville fractional integration,
and
denote Riemann-Liouville fractional derivations.
Taking the Riemann-Liouville fractional derivations of ζ1(t) and ζ2(t),
Replacing ë1(t) and ė2(t) in the above two equations by inserting (21) and (22), the equations (24) and (25) can be rewritten as,
Here, the equivalent control law elements u1eqv(t) and u2eqv(t) of the characteristic elements c1(=ωr) and c2(=Ids) are derived from conditions giving ζi(t)=0, (I=1, 2) in Eqs. (26) and (27), and chosen as q-axis stator voltage Vqs and d-axis stator voltage Vds, respectively, referring the two equations. The equations for u1eqv(t) and u2eqv(t) are obtained by further replacing Hi(i=1, 2) with RJt1−μδi (i=1, 2), respectively, as follows:
Here the lumped external disturbances and parametric perturbations Hi in the HMS, are assumed to satisfy below conditions:
∥Hi∥≤δi, and ∥
Hi∥≤δi, (tm 30)
where δi is a positive function.
Now, the control law elements u1cnt(t) and u2cnt(t) of the characteristic elements c1(=ωr) and c2(=Ids) are defined as below based on the equivalent control law elements Eqs. (28) and (29):
where, α∈(0, 1), and γi(I=1, 2) are positive constants, and SG (ζi(t)) (i=1, 2) represent signum function sgn(ζi(t)) and its smooth approximations, for example,
with ϑ>0.
It is noted that the FOSM surfaces ζi(t), (i=1, 2) and time derivatives of the FOSM
(i=1, 2) satisfy below relation Eq. (33) when the control law element uicnt(t), (i=1, 2) are applied in controlling the WT and the PMSG.
As described above, the GSMCFO method includes calculations of the FOSM surface ζi(t) and the control law elements uicnt(t) including a fractional time integral and a fractional time differentiation of the tracking errors e1 and e2. The controller is configured to monitor a status of the HMS, including the characteristics elements c1(=ωr) and the rotor current c2 (=Ids), and configured to control the RSC 106 by applying the control law elements uicnt(t), (i=1, 2) obtained by fractional calculus, when disturbances are perceived. The control law elements guarantee the characteristics elements ωr and Ids to effectively arrive and stay on the sliding mode surfaces ζ1(t) and ζ2(t), in the presence of external disturbances, parametric perturbation, and unknown dynamics.
When the controller 120 applied the control law elements uicnt(t), (i=1, 2), a condition:
is satisfied, as far as ζi(t) is nonzero. Whenever the condition was satisfied, it is an indication that the angular frequency ωr and the d-axis stator rotor current Ids are approaching the FOSM surface ζi(t), and sliding along the FOSM surface toward a global stabilized point satisfying
(i=1, 2).
However, when there is a momentary reduction in the operating frequency ωr of the rotor or the rotor current Ids, the error function e(t) is detected as non-zero. This indicates a variation of the operating frequency ωr of the rotor or the current Ids through the rotor are different compared to the desired rotational frequency ωr or the rotor current Ids. The momentarily 10 reduction may be due to intermittent noise in the atmosphere such as wind speed reduction for a certain duration. At this time, the program in the definition set in the controller 120 executes multiple equations containing the error function, FOSM surface calculation based on the error function as well as control law element based on the equations of the FOSM surface. Based upon the value of the control law element u1cnt(t), the controller 120 signals to the RSC 106 to modify the pulse width modulation signals to track the next optimum operating point of the rotor to harness the maximum wind power again. Since the control law element u1cnt(t) allows ωr and Ids to effectively arrive and stay on the sliding mode surfaces ζ1(t) and ζ2(t), the modified pulse width modulation signals are supplied to the RSC 106, such that the RSC 106 may generate an equivalent proportional current or voltage at its output, even in the presence of atmospheric disturbances, parametric perturbation, or any unknown dynamics. For example, if initially the WT 102 was generating 50 V DC voltage at the output of the RSC 106 at the specific ωr and Ids, the WT 102 is now generating a reduced voltage (for example, 47 V) in presence of atmospheric disturbances. However, since the FOSM surfaces are highly convergent, the system quickly attains the new stable points, that is, 47 V, without any glitch or overshoot at the output of the RSC 106 because of the fractional sliding mode surfaces ζ1(t) and ζ2(t).
Modeling of the PV System
The PV system 110 is formed by an arrangement of PV cells (consisting of n- and p-type semiconductor materials) in a framework. The PV cells generate a DC voltage from solar radiation. An equivalent circuit of a PV cell includes a current source, a diode, a series resistor, and a parallel resistor. An output current of the PV system 110 is given by,
where,
where Iph is a photo-generated current, Ipv is the PV output current, IS is the reverse saturation current, Ns and Np are the number of series-and parallel-connected PV cells, q is the electron charge, Vdio is a diode voltage, A represents a p-n junction factor, kB represents the Boltzmann constant, T is a cell temperature, and Rse and Rsh are equivalent series and shunt resistances, respectively.
MPPT Operation of the PV System and Regulating the DC Link Voltage
The controller 120 is configured to perform the maximum power point tracking (MPPT) operation of the PV system 110 at different irradiance levels using an advanced multi-sector perturb and observe approach, and configured to compute corresponding MPPT voltage VpvMPPT and current IpvMPPT, simultaneously. See, for example, Youssef, A. R. et al., Advanced multi-sector P&O maximum power point tracking technique for wind energy conversion system. Int. J. Electr. Power Energy Syst. 2019, 107, 89-97, entire contents of which is incorporated herein by reference. The MPPT voltage, VpvMPPT is set as the reference voltage for the DC-link Vdc*. When the irradiance level is very low (at night), a VpvMPPT is also low (below the minimum allowable DC-link voltage), then the controller 120 is configured to set the Vdc* as the nominal DC-link voltage Vdc*. On the other hand, when the irradiance level is too high, a VpvMPPT is also high. If the VpvMPPT was greater than the allowable maximum DC-link voltage, the controller is configured to set the Vdc* as the nominal DC voltage. The HMS can be configured to control the MPPT operation of the controller via the control of the GSC 108.
Modeling of the DC-Link and Application of GSMCFO Method
The power generated from the renewable energy sources in the HMS 100 is transferred to the GSC through the DC-link 132. A dynamic model of the DC-link voltage Vdc(=c3) of the DC-link 132 is derived as follows:
where Cdc is the capacitance of the DC-bus, Isdc is the output DC current from the RSC 106, Ipv is the output current from the PV system, Ib is the battery current from the BESS 114, D is the duty cycle of the BBBC 112 for the BESS 114, and Igdc is a DC current input to the GSC 108.
Assuming that the RSC 106 and GSC 108 are ideal, the following relations are valid:
where Id and Vd are the d-axis AC current and AC voltage of the AC output of the GSC 108, respectively. Using the above equation, the DC voltage dynamic equation can be rewritten as:
The characteristics element c3 is selected as the DC-link voltage Vdc. Thus, the controller 120 is configured to monitor the DC-link voltage Vdc. A tracking errors e3(t) for the DC-link voltage is given by:
e3(t)=Vdc(t)−Vdc*(t), (40)
where, Vdc*(t) represents a desired value for the DC-link voltage Vdc.
The desired value Vdc* defined for the DC-link voltage Vdc is given as,
wherein, VpvMPPT represents an output voltage of the PV system under a maximum power point tracking (MPPT) operation, Vdcmin, Vdcmax and Vdcnom represent a minimum allowable value, a maximum allowable value, and a nominal value of the DC-link voltages, each predetermined respectively.
Thus, the tracking error e3(t) is defined as a difference between a monitored voltage of the DC-link 132 and a desired voltage of the DC-Link 132.
A time derivative of the tracking error e3(t) is given using Eq. (39) as below:
where H3 represents the additive lumped disturbance in the HMS 100.
The FOSM surface ζ3(t) for the DC-link voltage Vdc is defined as:
ζ3(t)=k3ec(t)+
e3(t), (42)
where μ(0<μ<1) and k3 are positive constants, denotes a Riemann-Liouville fractional integration,
denotes a Riemann-Liouville fractional derivations.
Taking a Riemann-Liouville fractional derivation of ζ3(t) gives:ζ3(t)=k3e3(t)+ėc(t) (43)
Inserting Eq. (41) for ė3(t) into the above equation gives:
Here, the equivalent control law elements u3eqv(t) of the characteristic elements c3(=Vdc) is derived from the conditions giving ζi(t)=0, (i=3) in Eq. (44), and is selected as the d-axis AC current Id of the AC output from the GSC 108, referring the equation. The equations for the equivalent control law elements u3eqv(t) is obtained by further replacing the external lumped disturbance Hi(i=3) with
δi(i=3), as follows:
where, δ3 represents external perturbations with a positive function δ3, and assumed to satisfy the condition given by Eq. (30).
Now, the control law elements u3cnt(t) of the characteristic elements c3(=Id) is defined based on the equivalent control law element u3eqv(t) in Eq. (45) as:
where, α∈(0, 1), and γ3 are positive constants, and SG(ζi(t))(i=3) represents a signum function sgn(ζi(t)) and its smooth approximations, for example,
with ϑ>0.
As described above, the GSMCFO method applied in controlling the DC-link 132 of the HMS 100 includes calculations of the FOSM surface ζ3(t) and the control law elements u3cnt(t) including a fractional time integral and a fractional time derivative of the tracking error e3. The controller is configured to monitor a status of the HMS, including the characteristics element c3, that is, the DC-link voltage Vdc, and configured to control the GSC 108 by applying the control law element u3cnt(t)(=Idcnt(t)) defined for the d-axis AC current Id of the GSC 108 and obtained by the fractional calculus, when disturbances are perceived. The control law element u3cnt(t)(=Idcnt(t)) guarantees the characteristic element c3(=Vdc) effectively arrive and stay on the FOSM surfaces ζ3(t) in the presence of external disturbances, parametric perturbation, and unknown dynamics.
Applying the GSMCFO method and the control law elements u3cnt(t) by the controller 120 realizes a condition:
as far as ζ3(t) is nonzero, as discussed earlier. Whenever the condition is satisfied, it is an indication that the characteristics element c3(=Vdc) of the GSC 108 is approaching the FOSM surface ζ3(t), and sliding along the FOSM surface toward a global stabilized point satisfying
It is clear from the above equations that the controller 120 includes the FOSM surface having integration and a differentiation terms of the errors e3. In other words, the equation defining the FOSM surface ζ3(t) of the characteristic element c3(t) comprises a fractional time integral of the tracking error e3(t) given by e3(t)=c3(t)−c3*(t) and a fractional time derivative of the tracking error e3(t).
The equivalent control law element and the control law element u1cnt(t) are the controlling signals to control the voltage of the DC-link 132 when the disturbances are perceived in the DC-link 132 due to any unknown reason. The control law element and the equivalent control law element makes e3 reach and remain on the surface ζ3(t)=0 in the presence of external disturbances, parametric perturbation, and unknown dynamics.
The controller 120 monitors the Vdc through the DC-link 132. The controller 120 also executes the programmatic instructions containing equations of FOSM and the equivalent control law and the control law element whenever, the control law element uicnt(t) satisfies a condition
so far as ζi(t) is nonzero. Whenever the condition satisfies, it is an indication to the controller 120 that the value of Vdc is optimum and no correction in the Pulse width modulation signals to track the optimum operating point of the PV system 110 to harness the maximum solar power is needed. At this moment, the error function e(t) is detected as zero.
However, when there is a momentarily reduction in the Vdc, the error function e3(t) is detected as non-zero. This indicates a variation of the Vdc compared to the desired voltage Vdc* of the DC-link 132. The momentarily reduction may be due to intermittent noise in the atmosphere such as cloudy environment for a certain duration. At this time, the program in the definition set in the controller 120 executes multiple equations containing the error function, FOSM surface calculation based on the error function as well as control law element based on the equations of the FOSM surface. Based upon the value of the control law element u3cnt(t), the controller 120 signals to the GSC 108 to modify the pulse width modulation signals to track the next optimum operating point of the PV system to harness the maximum solar power again. Since the control law element u3cnt(t) allows Vdc to effectively arrive and stay on the sliding mode surface ζ3cnt(t), the modified pulse width modulation signals are supplied to the GSC 108, such that the GSC 108 may generate an equivalent proportional current or voltage at its output, even in the presence of atmospheric disturbances, parametric perturbation, or any unknown dynamics. For example, initially the PV system 110 was generating 45V DC voltage to the DC-link 132, is it now generating a reduced voltage (for example, 39 V) in presence of atmospheric disturbances. However, since the FOSM surfaces are highly convergent, the system quickly attains the new stable points, without any glitch or overshoot at the input side of the GSC 108 because of the fractional sliding mode surface ζ3(t).
Modeling of the GSC and Application of GSMCFO Method
The GSC 108 of the HMS 100 is configured to (a) regulate the DC-link voltage, (b) achieve the MPPT performance of the solar PV system, (c) ensure power transfer from the DC side to the AC grid side, and (d) to regulate the active and reactive power at the AC grid side. A cascade control structure is adopted in certain implementations.
Dynamic equations of the GSC in the d-q reference frames can be expressed as follows:
where Id and Iq represent the d-axis and the q-axis AC currents of an AC output of the GSC 108, respectively, and Vd and Vq, the d-axis and the q-axis AC voltages of the output of the GSC 108, respectively, Ud and Uq are the d-axis and q-axis voltages at the PCC 128, Lf and Rf are an inductance 122 and a resistance 124 of the grid side filter, respectively and ωg is an electrical angular frequency of the output of the GSC 108.
The AC currents of the AC output the GSC are controlled for proper power transfer to the grid side. The active power is controlled by the d-axis AC current. Thus, the desired d-axis current Id* is given by Id. The reactive power is desired to be very close to zero by controlling the q-axis current and setting the desired q-axis current as Iq*=0.
Thus, the characteristics elements ci(t) (i=4, 5) are defined for the GSC are the d-axis AC current Id(=c4(t)) of the AC output from the GSC 108 for i=4, and the q-axis AC current Iq(=c5(t)) of the AC output from the GSC 108 for i=5. The controller 120 is configured to monitor the characteristic elements c4(t) (=Id) and c5(t) (=Iq).
The d-axis AC current Id giving the desired value Id* is given by the power balance condition at the PCC and a power transferred by the GSC to the PCC as below,
where Pdem and Pug represent a power demand at the load and a power exchanged between the PCC and the utility grid, respectively, where Pug>0, when provided from the utility grid to the PCC, Pug<0, when provided from the PCC side to the utility grid, Vd represent a d-axis AC voltage of the AC output from the GSC.
Thus, the desired values defined for the d-axis and the q-axis AC current are given respectively as,
The tracking errors e4(t) for the d-axis AC current Id and e5(t) for the q-axis AC current Iq are given as:
Differentiating the tracking error e4(t) and e5(t) gives:
where H4 and H5 are the lumped disturbances in the HMS 100.
FOSM surfaces ζ4(t) and ζ5(t) for the characteristics elements c4(t)(=Id) and c5(t) (=Iq) are defined respectively as:
where μ∈(0, 1), k4, and k5 are positive constants, and
each denotes a Riemann-Liouville fractional integral and a Riemann-Liouville fractional derivation, respectively.
Riemann-Liouville fractional derivatives of the FOSM surfaces ζ4(t) and ζ5(t) are given respectively as:
Substituting Eqs. (51) and (52) for ė4 and ė5 of the above equations respectively, gives
Here, the equivalent control law elements u4eqv(t) and u5eqv(t) of the characteristic elements c4(=Id) and c5(=Iq) are derived from conditions giving ζi(t)=0, (i=4, 5) in Eqs. (57) and (58), respectively. Referring the two equations, the equivalent control law elements u4eqv(t) and u5eqv(t) are selected as Vd and Vq, the d-axis and the q-axis AC voltages of the output of the GSC 108, respectively.
The equations for the equivalent control law elements u4eqv(t) and u5eqv(t) are obtained by further replacing Hi(i=4, 5) with δi(i=4, 5), respectively, as follows:
where, the lumped disturbances Hi in the HMS, are assumed to be bounded and satisfy below conditions:
∥Hi∥≤δi, and ∥
Hi∥≤δi(i=4, 5) (65)
where δi, (i=4, 5) represent external disturbances and parametric perturbations with δi is a positive function.
Now, the control law elements u4cnt(t) and u5cnt(t) of the characteristic elements c4(=Id) and c5(=Iq) are defined based on the equivalent control law elements Eqs. (59) and (60), respectively,
where, α∈(0, 1), and γi(i=4, 5) are positive constants, and SG (ζi(t))(i=4, 5) represent signum function sgn(ζi(t)) and its smooth approximations, for example,
with ϑ>0.
As described above, the GSMCFO method applied to the GSC control of the HMS 100 includes calculations of the FOSM surfaces ζ4(t) and ζ5(t), the control law elements u4cnt(t) and utcnt(t) including a fractional time integral and a fractional time differentiation of the tracking errors e4 and e5, respectively. The controller is configured to i) monitor a status of the HMS, including the characteristics elements c4 and c5, defined and specified in the definition set as the d-axis AC current Id and the q-axis AC current Iq of the AC output of the GSC 108, respectively, ii) calculate the control law elements u4cnt(t)=Vdcnt(t), and u5cnt(t)=Vqcnt(t), defined with specific equations for the d-axis and the q-axis AC voltages of the output of the GSC 108, respectively, referring the definition set in the preinstalled software and based on monitored status of the HMS, and iii) control the GSC 108 by applying the control law elements u4cnt(t)=Vdcnt(t), and u5cnt(t)=Vqcnt(t) calculated, when the tracking errors are perceived. The control law elements guarantee the characteristic elements, the d-axis AC current Id and the q-axis AC current Iq of the GSC 108 effectively arrive and stay on the FOSM surfaces ζi(t)(i=4, 5) in the presence of external disturbances, parametric perturbation, and unknown dynamics.
Applying the GSMCFO method and the control law elements uicnt(t), (i=4, 5) by the controller 120 realizes a condition:
(I=4, 5) as far as ζi(t) is nonzero, as discussed earlier. Whenever the condition is satisfied, it is an indication that the characteristics element ci(i=4, 5), the d-axis AC current Id and the q-axis AC current Iq of the GSC 108 are approaching the FOSM surface ζi(t), (i=4, 5) and sliding along the FOSM surface toward a global stabilized point satisfying
(i=4, 5).
Modeling of the BBBC with the Rechargeable Battery and Application of GSMCFO Method
The bidirectional buck/boost converter (BBBC) 112 is configured to connect the rechargeable battery (BESS) 114 with the DC-link 132 as illustrated in
The BBBC 112 is configured to behave as a buck converter when charging the BESS 114 and as a boost converter when supplying the load 126 or discharging. In other words, when an excess power is generated compared to a power demand, the controller 120 is configured to store the excess power by recharging the BESS 114 via the BBBC 112 that behaves as a buck converter. On the other side, when the demand power is larger than a combined power from the WT, the PV system and the grid 130, the BESS 114 is configured to discharge via the BBBC 112 that behaves as a boost converter during discharging the BESS 114. Detailed mathematical model of the BBBC 112 and electrically connected BESS 114 is described below.
A dynamic model of the BESS 114 is given for charging operation by the BBBC as:
where Lb is the battery inductance, Ib is the battery current, Vb is the battery voltage, Vdc is the voltage of the DC-link 132, and D is the duty cycle of the BBBC 112. Dynamics of a state-of-charge (SOC) of the rechargeable battery is given by:
where Qb and ηb stand for the battery capacity and efficiency, respectively.
To maintain the efficiency, longevity and reliability, the SOC is constrained by the minimum SOC (SOCmin) and the maximum SOC (SOCmin) as provided below:
SOCmin<SOC<SOCmin. (70)
A desired battery current Ib* is determined based on the power balancing condition in the HMS 100, as below. The battery charging and discharging power Pb of the BESS 114 is controlled in accordance with a power balancing condition given as,
Pre+Pug=Pdem+Pb, (71)
where, Pre(≥0) is a sum of powers from renewable energy sources, Pug, a power provided from the utility grid, and Pdem(≥0), a load demand by the load 126. In Eq. (67), the battery power Pb is positive when the BESS is charged. Thus, Pb and a desired value of the battery current Ib* are given respectively as,
When a battery current monitored was Ib(t) at time t monitored, a tracking error e6(t) is given by:
e6(t)=Ib(t)−Ib*(t). (74)
Taking a time derivation of the tracking error e6(t) and substituting Eq. (64) give,
where H6 represents a lumped disturbance in the HMS 100.
A fractional-order sliding-mode (FOSM) surface ζ6(t) is defined as:
ζ6(t)=k6+
e6, (76)
where μ∈(0, 1) and k6 are positive constants, denotes a Riemann-Liouville fractional integration,
denotes a Riemann-Liouville fractional derivation.
Riemann-Liouville fractional derivatives of the FOSM surfaces ζ6(t) is given as:ζ6(t)=k6e6+ė6(t). (77)
Substituting Eq. (71) for ė6 of the above equations gives
Here, the equivalent control law elements u6eqv(t) of the characteristic elements c6(=Ib) is derived from a condition giving ζ6(t)=0 in Eq. (73). Referring Eq. (73), the equivalent control law elements u6eqv(t) is selected as the duty ratio D of the BBBC 112.
The equation for the equivalent control law elements u6eqv(t)=Deqv(t) is obtained by further replacing H6 with δ6:
where, the lumped disturbances H6 in the HMS, is assumed to be bounded and satisfies below conditions:
∥H6∥≤δ6, and
H6∥≤δ6, (80)
where δ6 represents external disturbances and parametric perturbations with δ6 a positive function.
The control law element u6cnt(t)=Dcnt(t) is defined as:
where, α∈(0, 1), and γ6 are positive constants, and SG(ζi(t))(i=6) represents a signum function sgn(ζi(t)) and its smooth approximations, for example,
with ϑ>0.
As described above, the GSMCFO method applied to the GSC control of the HMS 100 includes calculation of the FOSM surfaces ζ6(t), the control law elements u6cnt(t) including a fractional time integral and a fractional time differentiation of the tracking errors e6. The controller is configured to i) monitor a status of the HMS, including the characteristics element c6 defined and specified in the definition set as the battery current Ib, ii) calculate the control law elements u6cnt(t)=Dcnt(t), defined with the specific equation for the duty ratio of the BBBC 112, referring the definition set in the preinstalled software and based on monitored status of the HMS, and iii) control the BBBC 112 by applying the control law element u6cnt(t)=Dcnt(t) calculated, when the tracking error is perceived. The control law element guarantees the characteristic element c6, that is, the battery current Ib effectively arrive and stay on the FOSM surfaces ζ6(t) in the presence of external disturbances, parametric perturbation, and unknown dynamics.
Applying the GSMCFO method and the control law elements u6cnt(t) by the controller 120 realizes a condition:
as far as ζ6(t) is nonzero, as discussed earlier. Whenever the condition is satisfied, it is an indication that the characteristics element c6, that is, the battery current Ib is approaching the FOSM surface ζ6(t) and sliding along the FOSM surface toward a global stabilized point satisfying
The control law element guarantees Ib to effectively arrive and stay on the sliding mode surface ζ6(t) in the presence of external disturbances, parametric perturbation, and unknown dynamics.
In some implementations according to certain embodiments, the signum functions SG(ζi(t)) in the control law elements uicnt(t) are selected as hyperbolic tangent functions tanh(ζi/θ), where θ(>0) is a constant. This is effective to avoid a chattering that may be caused due to a high frequency switching of the discontinuous function sgn(ζi(i)). In such case, the modified control law elements would be as below:
Here, a mathematical proof is described on a point that the GSMCFO method described above guarantees all states' trajectories converge to the sliding-mode surfaces ζi=0, (i=1,2, . . . , 6) within a finite time.
A Lyapunov function candidate is selected as:
Differentiating L with respect to time gives:
Substituting (77)-(82) into (84), one obtains:
Based on assumptions, ∥Hi∥≤δi, (i=1, 6) we have:
Therefore, we obtain:
Consequently, it can be concluded that all the states' trajectories will converge to their respective sliding-mode surfaces ζi=0, (i=1, 2, . . . , 6).
Further a convergence time of the states' trajectories is finite as described below. From (87) and with a definition of norm,
one has:
Rearranging (88) yields:
Integrating both sides of (89) from zero to tr yields:
Eq. (90) indicates that the states' 'rajectories arrive at their respective sliding manifolds ζi=0, (i=1, 2, . . . , 6) within a finite time Tr given by:
Power balancing Operations of the HMS
As noted above, the controller of the HMS 100 is configured to satisfy the power balance condition given by Eq. (67),
Pre+Pug=Pdem+Pb, (97)
where, Pre(≥0) is a sum of powers from renewable energy sources, Pug, a power provided from the utility grid, Pdem(≥0), a load demand, and a battery power Pb.
As a measure for the power balance, a power imbalance ΔP of the HMS 100 is defined as,
ΔP=Pre+Pug−Pdem−Pb. (98)
Thus, the controller is configured to monitor each on the components of the power imbalance and to realize and maintain the balancing condition, alternatively given as,
ΔP=0. (99)
Here, the power Pre from the renewable energy sources and the load demand Pdem are deemed not adjustable in ordinary cases. The WT generator and the PV system are operated under the MPPT modes in normal situations. The load demand P dem is also assumed non-negotiable. On the other hand, the battery power Pb and the grid power Pug are assumed adjustable in both directions, that is, either providing to the load to offset a shortage, or receiving an excess from the HMS.
The controller of the HMS is configured to monitor each of the components of the power imbalance equation and a value of the power imbalance ΔP. When the power Pre from the renewable energy sources fluctuates as conventionally occurs, the controller is configured to realize and maintain the power balance condition ΔP=0, by adjusting the battery power Pb and the grid power Pug.
In controlling the battery power, the controller is configured to satisfy a restriction condition on a state of charge (SOC) of the BESS for a reliability purpose, given as,
SOCmin≤SOC≤SOCmax. (100)
When the SOC reaches the maximum value while maintaining the power balance condition by providing the Pb to the BESS, a power imbalance ΔP is caused when the charging is stopped to be Pb=0. In that instance, the imbalance, that is, an excess energy can be absorbed by the grid power Pug. When the utility grid was supplying a larger amount than the excess energy ΔP, the grid power is reduced to absorb the excess energy. The utility grid can receive the excess energy without supplying the grid power (Pug<0). Some examples for the power balancing operations of the controller are described below.
Case 1: When the power of the PV system 110 is decreased whereas the power due to the WT 102 is constant.
Suppose initially the PV system 110 is generating the 100 KW of power, the WT generator 102 is generating 70 KW of power and the grid 130 is providing 200 KW of power at a load demand at 350 KW, where the battery power is zero.
The power imbalance is calculated
ΔP=Pre+Pug−Pdem−Pb=170+200−350−0=20 (KW)
Hence, the surplus 20KW is used to charge the BESS 114 via the BBBC 112. Accordingly, the controller 120 signals the BBBC 112 to start operating in the buck mode. (Pb=20 KW) By this operation, the imbalance is reduced to zero. (ΔP=0)
Suppose the PV system 110 is now producing smaller amount of power, for example only 80 KW for 10 seconds. Decrease in the power may be due to cloudy environment. During this period, the power imbalance turns to ΔP=−20 KW. At this initial instance, the controller immediately stops the battery power 20 KW providing to charge the BESS via the BBBC.
After 10th second, when the PV system 110 again starts generating 100 KW of power, the controller 120 again signals the BBBC 112 to start charging the BESS at 20 KW in the boost mode. (Pb=20 KW)
Case 2: When the power of the PV system 110 is decreased whereas the power due to the WT generator 102 is constant and the grid 130 injects power
Suppose initially same situation as the Case 1; ΔP=Pre+Pug−Pdem−Pb=170 +200 −350−0=20 (KW), the power imbalance ΔP=20 (KW) is balanced by charging the BESS 114 via the BBBC 112. (Pb=20 KW) Now, suppose the PV system 110 reduces its generation to a half (50 KW) for 10 seconds. Decrease in the power may be due to cloudy environment. The reduction causes the power imbalance of ΔP=−50 KW.
At this time, the controller immediately signals the BBBC 112 to stop charging (Pb: 20 KW to 0) and start discharging the BESS 114 Pb=−30 KW to realize the power balance condition. (ΔP=−50+20+30=0)
Suppose further, the SOC of the BESS 114 reaches to the SOCmin within 7 seconds, the controller 120 signals the BBBC 112 to stop discharging. At this moment, the controller 120 signals the grid 130 to start supplying an additional 30 KW, a total of 230 KW power to the load in the HMS 100 to meet the power balance requirement. The grid 130 starts supplying the 15 power.
After 10th second the PV system 110 again starts generating 100 KW of power. At this moment, the controller signals the grid 130 to stop injecting the additional 30 KW of power to the HMS 100. Also, the controller 120 again signals the BBBC 112 to start charging the BESS at 20 KW.
Case 3: The SOC of the BESS reaches to the maximum value SOCmax, while the power of the PV system 110 as well as one of the WT generator 102 are unchanged, then the grid absorbs the excess power
Suppose initially the same situation as of the Case 1; ΔP=Pre+Pug−Pdem−Pb=170+200−350−0=20 (KW), the power imbalance ΔP=20 (KW) is balanced by charging the BESS 114 via the BBBC 112. (Pb=20 KW) .
Suppose then at the 11th second, the BESS reaches the condition SOC=SOCmax. Then, the controller immediately stops charging the BESS, and signals the grid 130 to start absorbing the excess power. Accordingly, the grid 130 starts absorbing the excess power, that is, 20 KW, thus reduces the grid power from 200 KW to 180 KW. The power is balanced in the HMS 100.
Tracking Performances: Simulation Results
Tracking performances of the HMS 100 controlled by the GSMCFO method under circumstance changes are described below. The configuration of the HMS 100 above described was implemented in MATLAB/SIMULINK. Design parameters and their values assumed for components including the WT 102, the PMSG 104, the PV system 110, the BESS 114 and the grid 130 are summarized in Tables 1 and 2. Control parameters for the GSMCFO-method are provided in Table 3. Simulation results obtained for four scenarios are described.
Under Scenario 1, the wind speed is assumed below the cut-in speed and the wind power is not large enough to overcome the mechanical resistance of the wind turbine. On the other hand, irradiance level varies depending on time. Therefore, the renewable power is generated by solar PV alone.
Here, complementary property of the solar and wind power sources was examined.
Solid line 402 and dashed line 404 each represents a solar irradiance level (in kW/m2) and a wind speed (in 10 m/s), respectively. For a time period 0≤t (s)≤5, the solar irradiance level 402 is still high, and the wind speed is above cut-off. From t=5 s, the solar irradiance starts decreasing. While, the wind speed starts increasing from t=10 s.
A variation of the solid line 422 and negative value of the power indicates the BBBC 112 operated in the buck mode to charge an excess power the load demand. When an excess power is generated, the BESS 114 is configured to store the excess power, so far as the SOC of the BESS allows charging. On the other hand, when the SOC reached its maximum value SOCmax at t (s)=32.3, the BESS 114 stopped charging and the grid 130 started to absorb the excess power as illustrated in
Here, performances of the HMS 100 with the controller 120 are compared with those of a HMS with a linear PI controller in the known art. The superiority the controller 120 is demonstrated. For performing the comparison, the GSMCFO controller 120 in the HMS 100 under certain embodiments was replaced, at the same parameters, with the PI controller utilized in a following reference: Chishti, F. et al, LMMN-Based Adaptive Control for Power Quality Improvement of Grid Intertie Wind—PV System. IEEE Trans. Ind. Inform. 2019, 15, 4900-4912, entire content is incorporated herein by reference. Same power variations as illustrated
Dashed-two dotted line 502 represents an observed rotor speed for the HMS with the PI controller, where overshoots and damping oscillations were observed. Peak values of the overshoots exhibited about 30 to 50% larger values than the reference values, while the damping oscillation continued for about three seconds.
Solid line 516 represents a desired value (reference) of the battery current Ib*. Dashed line 518 represents an observed battery current for the HMS with the GSMCFO Controller 120. The dashed line 518 demonstrated the exact tracking performance again with minor transition delays less than a second and without overshoots at transitions. Dashed-two dotted line 514 represents an observed battery current for the HMS with the PI controller. Here, the dashed-two dotted line 514 exhibited transition delays longer than a second, overshoots reaching about 10% larger or smaller Ib values than the reference, and overshoot damping continuing more than a few seconds.
Solid line 522 represents an observed power deviation ΔP for the HMS with the GSMCFO Controller 120. Dashed-line 520 represents an observed power deviation ΔP for the HMS with the PI controller. Both lines 520 and 522 exhibited transition spikes with almost same peak values less than about 5% of switched power values. However, the GSMCFO controller settled within a second without overshooting, on the other hand, the PI controller struggled to maintain ΔP=0, due to overshooting and longer damping time.
Based on the performance comparison above described, it was confirmed that the GSMCFO controller 120 provides superior transient and steady-state performances to the PI controller. Moreover, the GSMCFO controller 120 demonstrated capability to keep the power balance (ΔP=0) of the HMS 100 despite the lumped disturbances in the system and sudden changes of the load demand and power generations.
The settling time and the overshoot comparison between the GSMCFO Controller 120 and the PI controller are presented in Tables 4 and 5, as below.
The above tables 4 and 5 further corroborate superior tracking performances of the GSMCFO to the PI controller in the presence of parametric uncertainties and external disturbances.
The method 600 includes steps illustrated by blocks in
At step 610, a definition set to be referred in performing the GSMCFO method in controlling a HMS system, for example, the HMS 100 according to certain embodiments, is prepared and customized for the HMS 100. The definition set includes: a characteristic element ci to be monitored, equations defining a desired value ci* of the characteristic element ci, a fractional order sliding mode (FOSM) surface ζi for the characteristic element ci, and a control law element uicnt of the characteristic element ci. The equation defining the FOSM surface ζi(t) of the characteristic element ci(t) include a fractional time integral of a tracking error ei(t) and a fractional time derivative of the tracking error ei(t). The tracking error ei(t) for the characteristic element ci(t) is given by,
ei(t)=ci(t)−ci*(t). (101)
At step 620, current status of the HMS is monitored and monitored information is gathered and stored in the controller. The current status to be monitored includes the characteristic element ci(t) and a related status of the HMS appeared in the definition set.
At step 630, calculations of the equations defined in the definition set are performed by the controller 120, based on the characteristic element ci(t) monitored and the related status of the HMS monitored.
At step 640, the controller 120 controls the HMS based on the control law element uicnt(t) calculated and in accordance with the GSMCFO method.
Above steps or part of those steps are iterated until the FOSM surface ζi(t) or the tracking error ei(t) reduces to values equal to or smaller than each criterion predetermined.
Based upon numerous examples and the experimental observations, an efficient and cost-effective hybrid microgrid system configuration was achieved by linking the PV system directly, that is without any converters, with the DC-link. This reduces the number of electronic converters and thus enhances efficiency and reduces cost of the HMS.
A controlled operation of the HMS was achieved using a GSMCFO controller 120. Four scenarios were provided to demonstrate the performance of the GSMCFO controller 120 under various power generation and load conditions. In each of these scenarios, the GSMCFO controller 120 demonstrated i) excellent transient and steady-state performances, ii) global stability of the HMS subjected to parametric uncertainties, variations of renewable power sources and load demand, iii) regulation of DC-link voltage irrespective of the operating conditions, iv) extracting maximum power from both the wind and PV systems by the RSC and GSC, respectively, and v) active power transfer to the grid, and maintaining the power balance of the hybrid microgrid by effective charging/discharging of the BESS.
Due to the robustness and fast convergence of the GSMCFO controller 120, the overshoots in ωr, Vdc, Ib, and ΔP were 0.0%, and all settled at 0.7 s, 0.2 s, 0.8 s, and 0.8 s, respectively. The overshoot and settling time of the GSMCFO controller 120 was found to be highly satisfactory compared to the PI controller. Therefore, the GSMCFO controller 120 can improve the efficiency, the cost and the power balance of the hybrid microgrid.
Numerous modifications and variations of the present disclosure are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the invention may be practiced otherwise than as specifically described herein.
Number | Date | Country |
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2021101279 | May 2021 | AU |
114172398 | Mar 2022 | CN |
202041048086 | Nov 2020 | IN |
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