Patent Publication Number: US-2023140233-A1

Title: Renewable energy system and electrical grid

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     The present application claims benefit of priority to U.S. Provisional Application No. 63/273,399, having a filing date of Oct. 29, 2021, which is incorporated herein by reference in its entirety. 
    
    
     STATEMENT OF PRIOR DISCLOSURE BY AN INVENTOR 
     Aspects of the present disclosure are described in Syed et al., “Moving Regression Filtering with Battery State of Charge Feedback Control for Solar PV Firming and Ramp Rate Curtailment,” IEEE Access, 18 Jan. 2021, DOI: 10.1109/ACCESS.2021.3052142, which is incorporated herein by reference in its entirety. 
     BACKGROUND OF THE INVENTION 
     Technical Field 
     The present disclosure is directed to a system and a method for solar photovoltaic (PV) variability reduction, reduced time delays and battery storage optimization using a moving linear regression based power firming filter combined with state of charge feedback control. 
     Description of Related Art 
     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 nor impliedly admitted as prior art against the present disclosure. 
     With the global population growth, demand for energy is also increasing to fulfill basic needs of the population. There are two sources of energy i.e., one is renewable energy sources and other one is non-renewable energy sources (such as fossil fuels). The use of non-renewable energy sources can cause problems such as environmental pollution and depletion of the source itself, therefore renewable energy sources can be preferable over the non-renewable energy sources. 
     Solar Photovoltaic (PV) power plants are a widespread choice for the renewable energy source, as solar energy is available and is comparatively inexpensive. The solar PV power plants comprise a large number of solar PV modules that convert solar energy into Direct Current (DC) electric power. Using a DC-Analog Current (AC) inverter, the generated DC power is converted into AC electric power. The inverted AC electric power is then injected into a power grid. 
     Due to the large-scale penetration of intermittent PV power modules, multiple variabilities occur in the power grid, such as frequency issues and voltage deviations. To counteract such issues, Battery Energy Storage System (BESS) can be integrated into the power grid as the BESS can assist in reducing the PV fluctuations and provide optimal operation. A storage system stores renewable energy or residual energy of the power grid in a battery, and supplies power to a load when needed. Storage systems are expensive and smoothing filters coupled with the BESS can provide cost reduction and power smoothing. 
     Existing solutions employ traditional filters such as Low Pass Filters (LPF), Moving Average (MA), and Moving Median (MM) for solar power smoothing. However, these filters have inadequate power tracking capabilities particularly with larger window sizes and time constants, which subsequently depreciates the storage system performance. To compensate for the delayed power tracking, larger energy storage systems are required, which in turn adds to the overall operational costs. 
     Accordingly, it is one object of the present disclosure to provide a system and a method for reducing solar PV variability and optimizing the battery storage in an accurate and cost-efficient manner. 
     SUMMARY 
     In an exemplary embodiment, a solar photovoltaic (PV) network is disclosed. The solar photovoltaic (PV) network includes a PV module, a Moving Regression (MR) filter, a State of Charge (SoC) feedback control, a Battery Energy Storage System (BESS), and an electrical grid. 
     In an exemplary embodiment, a system for solar PV variability reduction, reduced time delays and battery storage optimization is disclosed. The system includes a Moving Regression (MR) filter, a State of Charge (SoC) feedback control, and a Battery Energy Storage System (BESS). The PV module comprises a PV array that receives solar light signals and generates unsmoothed solar PV power output that is coupled to the MR filter and the SoC feedback control. Outputs of the MR filter, the SoC feedback control and the BESS are combined to generate a smoothed solar PV power output, wherein the smoothed solar PV power output is coupled to the electrical grid. The MR filter is a non-parametric smoother that utilizes a machine learning concept of linear regression to smooth out solar PV variations at every time step. 
     In another exemplary embodiment, a method of providing a solar photovoltaic (PV) power to an electrical grid is disclosed. The method includes receiving solar signals by a PV module. The method further includes generating, by the PV module, an unsmoothed solar PV power output power (P PV ), and coupling the P PV  to a MR filter and to a SoC feedback control. The method further includes generating a filter smooth output power (P PO ) by the MR filter. The method further includes generating a power reference output (P ref ) by combining P PO  and an output of the SoC feedback control and coupling P ref  to a Battery Energy Storage System (BESS), wherein BESS generates a BESS power output (P BESS ). The method further includes generating an output grid power (P grid ) by combining P BESS  and another output of the SoC feedback control. The method further includes receiving output grid power P grid  by the electrical grid. 
     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. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       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: 
         FIG.  1    illustrates a microgrid system, according to aspects of the present disclosure; 
         FIG.  2    illustrates a solar photovoltaic (PV) network, according to aspects of the present disclosure; 
         FIG.  3    illustrates a flow chart for a method for providing a solar PV power to an electrical grid, according to aspects of the present disclosure; 
         FIG.  4    illustrates a diagram of power smoothing with a State of Charge (SoC) feedback control and a low pass filter, according to aspects of the present disclosure; 
         FIG.  5    represents a double moving average filter logic scheme, according to aspects of the present disclosure; 
         FIG.  6    represents recorded data corresponding to solar power, according to aspects of the present disclosure; 
         FIG.  7 A  represents the PV power smoothing using a Low Pass Filter (LPF) with time constants 24 and 48 minutes, according to aspects of the present disclosure; 
         FIG.  7 B  is an enlarged view of the representation of  FIG.  7 A , according to aspects of the present disclosure; 
         FIG.  8 A  represents a smoothing performance of a moving regression (MR) filter with different window sizes, according to aspects of the present disclosure; 
         FIG.  8 B  is an enlarged view of the representation of  FIG.  8 A , according to aspects of the present disclosure; 
         FIG.  9    represents a comparison of the LPF against the MR filter based on charging/discharging power, according to aspects of the present disclosure; 
         FIG.  10    represents a battery charging/discharging power performance comparison of the MR filter with different window sizes, according to aspects of the present disclosure; 
         FIG.  11    represents a battery SoC comparison of the LPF against the MR filter, according to aspects of the present disclosure; 
         FIG.  12    represents a battery SoC performance comparison of the MR filter with different window sizes, according to aspects of the present disclosure; 
         FIG.  13 A  represents a comparison of smoothing performance of a Moving Average (MA) filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  13 B  is an enlarged view of the representation of  FIG.  13 A , according to aspects of the present disclosure; 
         FIG.  14    represents a battery SoC comparison of the MA filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  15    represents a battery charging/discharging power performance comparison of the MA filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  16 A  represents a comparison of power smoothing performance of a Double Moving Average (DMA) filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  16 B  is an enlarged view of the representation of  FIG.  16 A , according to aspects of the present disclosure; 
         FIG.  17    represents a battery SoC comparison of the DMA filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  18    represents a battery charging/discharging power performance comparison of the DMA filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  19 A  represents a comparison of power smoothing performance of a Moving Median (MM) filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  19 B  is an enlarged view of the representation of  FIG.  19 A , according to aspects of the present disclosure; 
         FIG.  20    represents a battery SoC comparison of the MM filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  21    represents a battery charging/discharging power performance comparison of the MM filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  22 A  represents a comparison of power smoothing performance of a Savitsky-Golay (SG) filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  22 B  is an enlarged view of the representation of  FIG.  22 A , according to aspects of the present disclosure; 
         FIG.  23    represents a battery SoC comparison of the SG filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  24    represents a battery charging/discharging power performance comparison of the SG filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  25 A  represents a comparison of power smoothing performance of a gaussian filter (GF) against the MR filter, according to aspects of the present disclosure; 
         FIG.  25 B  is an enlarged view of the representation of  FIG.  25 A , according to aspects of the present disclosure; 
         FIG.  26    represents a battery SoC comparison of the GF against the MR filter, according to aspects of the present disclosure; 
         FIG.  27    represents a battery charging/discharging power performance comparison of the GF against the MR filter, according to aspects of the present disclosure; 
         FIG.  28 A  represents a solar power ramp rate comparison of the LPF against the MR filter, according to aspects of the present disclosure; 
         FIG.  28 B  is an enlarged view of the representation of  FIG.  28 A , according to aspects of the present disclosure; 
         FIG.  29 A  represents a solar power ramp rate comparison of the MA filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  29 B  is an enlarged view of the representation of  FIG.  29 A , according to aspects of the present disclosure; 
         FIG.  30 A  represents a solar power ramp rate comparison of the MM filter against the MR filter, according to aspects of the present disclosure; 
         FIG.  30 B  is an enlarged view of the representation of  FIG.  30 A , according to aspects of the present disclosure; 
         FIG.  31 A  represents a solar power ramp rate comparison of the SG filter against the MR filter, according to aspects of the present disclosure; and 
         FIG.  31 B  is an enlarged view of the representation of  FIG.  31 A , according to aspects of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     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. 
     Aspects of this disclosure are directed to a system and a method for reducing solar photovoltaic (PV) variability with reduced time delays and battery storage optimization. The disclosure employs a moving regression (MR) filter combined with a state of charge (SoC) feedback control and a Battery Energy Storage System (BESS) for reducing solar PV variability, time delays, ramp rates and decreasing battery charging and discharging power. The present disclosure discloses that the MR filter achieves better solar power smoothing without increasing the BESS capacity as compared with prior art solutions such as, but not limited to, Low Pass Filters (LPF), Moving Average (MA), and Moving Median (MM), double moving average (DMA), and Savitsky-Golay (SG) filters. Further, the performance of the MR filter is less affected with the increase in window sizes. 
     In various aspects of the disclosure, non-limiting definitions of one or more terms that will be used in the document are provided below. 
     A term “power output smoothing filter” or simply “smoothing filter” is equivalent to “power firming filter”. LPF, MA, MM, DMA, SG, and MR filters are power output smoothing filters. 
     A term “power tracking” refers to a feature where the PV panels are directed towards the sun. With power tracking, PV panels can change their orientation throughout the day to follow the sun&#39;s path to maximize energy capture. 
     A term “power lag” can be defined relative to a “power factor”. A power factor is a measure of the phase difference between the voltage and current in an AC power system. In purely resistive loads (such as an incandescent lightbulb or electric kettle), the current is in phase with the voltage and there is ‘unity’ power factor. Capacitive and inductive loads (such as a capacitor banks or inductive motor respectively) can cause the current to ‘lead’ or ‘lag’ the voltage, resulting in a ‘non-unity’ power factor. In some aspects, increasing a window size can improve the smoothing performance, but can also cause an increase in power lagging. 
     The term “ramp rate” of the PV power is important to determine the capacity of the energy storage for dispatching smoothed PV power to the grid. Ramping refers to a change in power flow (or power generation) from one time unit to the next. Ramping restrictions limit the allowed net flow variations on consecutive hours on specific lines. Ramping restrictions can include information about flow in the last hour from the previous day in the calculation. 
     The term “window size” may be described as follows: The selection of a window size for a filter can be useful to analyze data. For example, when using a moving average filter with noisy data, selection of a larger window size can obtain smoother data. However, the smoother data may be less realistic. 
     The term “power grid” or “electrical grid” refers to an interconnected network for electricity delivery from producers to consumers. The power grid consists of a multitude of complex interconnections and provides electricity generation, transmission and distribution. As used herein, “electrical grid” or “power grid” may be referred to as simply a “grid”. From a location point of view, in a traditional network, the grid may be referred to as a “main grid”. The term “microgrid” refers to a local energy grid with control capability, which can be disconnected from the main grid and operate autonomously. 
     Solar PV plants generate intermittent power due to variations in the sunlight, triggered by shifting clouds. The subsequent fluctuating power creates mechanical complications with the automatic voltage controller, affects the process of the low voltage grid, causes frequency deviations and voltage issues, results in tap changer and power system failures that leads to utility damages and even grid shutdowns. So, to improve the permeation of large-scale solar power into the grid, the PV power output needs to be flattened, i.e., smoothed, before it can be infused into the main grid. 
     Several smoothing filter methodologies combined with BESS and control systems have been described to smooth out solar and wind power fluctuations. As previously noted, some commonly used filtering techniques such as Low Pass Filters (LPF), Moving Average (MA), and Moving Median (MIND, double moving average (DMA), gaussian filter (GF) and Savitsky-Golay (SG) filters, with longer time constants and larger window sizes, are potential solutions for solar power smoothing, but at the cost of causing a significant time delay. This time delay results in high charging/discharging power and frequent SoC changes of the battery, thereby increasing the capacity of the battery required and additionally reducing the lifetime of the battery used. Moreover, filters such as GFs tend to over smooth the PV profile thereby losing the wave characteristics which must be compensated by requiring additional power from the battery which in turn negatively affects the battery by decreasing its life duration. 
     The objective of embodiments of the present disclosure can be stated as follows: To smooth the solar power output fluctuations with minimum time delay, to decrease battery charging/discharging power, to provide proper SoC management and optimal battery storage capacity subjected to practical physical system constrains and optimum battery operation. The resulting smoothed power output not only assists in dispatching power that complies with the grid code, but also raises the overall advantages of the PV power as it becomes more controllable.  FIG.  1    illustrates a block diagram of a microgrid system  100  with energy storage system  106  for providing uninterrupted power and balancing load demands for an area having varying power needs, according to one or more aspects of the present disclosure. 
     Referring to  FIG.  1   , the microgrid system  100  includes at least one electricity generation unit (e.g., solar power plant  102 , wind power turbine  104 ), an energy storage system  106 , a utility  108 , a load system  110 , and a control system  112 . The construction of microgrid system  100  is similar to that of a known microgrid, and thus the construction is not explained in detail for the sake of brevity. The at least one electricity generation unit (solar power plant  102 , wind power turbine  104 ) is configured to generate electricity and further configured to feed the generated electricity into a main grid for distribution. The electricity generation unit (solar power plant  102 , wind power turbine  104 ) may be a solar power plant, a wind power plant, a small hydro-power plant, a biomass-power plant, any renewable electricity generation unit, or a combination thereof. 
     In an aspect, the solar power plant  102  includes a plurality of PV modules and a DC/DC voltage stabilizing module, such that the solar energy is converted into electric energy and transmitted to the direct current bus using the DC/DC voltage stabilizing module. Each of the PV modules includes a PV array and a boost converter. 
     In an aspect, the wind turbines  104  is configured to convert the wind energy into electric energy. The wind turbines  104  includes a synchronous generator, an AD/DC rectifier and a DC/DC voltage stabilizing module which are sequentially connected to convert the wind energy into the electric energy. 
     The energy storage system  106  includes an energy storage battery pack and a variable flow control unit for storing electric energy provided by the solar power plant  102  and the wind turbines  104  and is further configured to provide the electric energy for the direct current bus. The energy storage system  106  is configured to increase energy efficiency by storing generated electric energy by each connected system including the electricity generation unit (solar power plant  102 , wind power turbine  104 ), substations, and transmission lines. Further, the energy storage system  106  is configured to selectively and efficiently use the stored electric energy when the electric energy is needed. In an aspect, the energy storage system  106  may include a lead-acid battery, a redox flow battery, a sodium-sulfur battery, a lithium-ion battery, an ultracapacitor, etc. 
     In an operative embodiment, the utility  108  is a commercial electric power distribution system. The utility  108  is configured to receive generated electric energy (electricity) from at least one electricity generation unit (solar power plant  102 , wind power turbine  104 ), and the energy storage system  106 , and is further configured to transmit the received electricity over a certain distance via transmission lines. Further, the utility  108  is configured to distribute the electricity to the consumer through a distribution system. In some embodiments, utility  108  may be referred to as the main grid. 
     End points of the utility  108  are consumer locations when electricity is used to turn on various equipment such as the lights, television, dishwasher or such equipment&#39;s (acting as a load for the utility). The load system  110  is the total demand for power in the grid by various consumers. In an aspect, the load system  110  includes a commercial load, a household load, a factory load, etc. In another aspect, the load system  110  varies by hour, day, and season. For example, during summer, air conditioners are heavily used, thereby increasing load demand of the load system  110 . The pattern of living also contributes to a varying demand for electricity on the load system  110 . 
     In an aspect, the microgrid system  100  includes the control system  112  for operating one or more modules/units/subsystems of the microgrid system  100  in a safe and an effective way. The utility  108  serves millions of consumers and has a considerable amount of load that may change in rapidly and uncontrolled way. Therefore, the control system  112  is configured for proper monitoring and operations of the various units of the microgrid system  100 . In an aspect, the control system  112  may include multiple controllers and sensors distributed over in the area of the microgrid system  100 . In an aspect, the control system  112  may include a Supervisory Control and Data Acquisition (SCADA) system that collects data and distributes the instructions accordingly. The control system  112  may be configured to maximize the use of renewable energy sources (taking demand, for example, due to weather in the consideration and other factors), minimizing the dependencies on fossil fuel, and maintaining the reliability of all the units of the microgrid system  100 , while matching the load requirements. In an aspect, the control system  112  is configured to monitor load demand and accordingly manage the distribution of the electricity over a network among all the consumers. In an aspect, the control system  112  includes network health monitoring capabilities and adaptability to compensate for element failures. If the main grid, i.e., utility  108 , fails, the microgrid system  100  is employed to fulfill the power supply requirements of the consumers, attached with the main grid. As the electric power generated by the microgrid system  100  is limited, the control system  112  is configured to utilize the capacity of the microgrid system  100  such that the power supply requirements in the system are not affected. The control system  112  is configured to improve the stability of the microgrid system  100  by coordinating with all the energy generation units of the microgrid system  100 . 
     In an operative aspect, the present disclosure discloses a system for solar PV variability reduction with reduced time delays and battery storage optimization. The system includes the PV module, an energy storage module having a battery and a DC-DC converter. The present system also contains a DC-AC inverter to dispatch the power into a grid. The SoC feedback controller is connected with the battery system and a smoothing algorithm. The purpose of the smoothing algorithm and the battery system is to smooth the PV output power as solar irradiance and ambient temperature are uncontrolled in nature. The difference between the real PV power and the output from the smoothing topology will be responsible for charging and discharging the battery system. The resultant power of real PV and the battery output represents the smoothed power, which can be injected into the grid. The system is configured to smooth the fluctuated PV power, whereas controlling the battery ramp rate using the filter. Smoothing of the fluctuated power not only helps to dispatch a power that complies with the grid standard but maximizes the total benefits of the PV power as it becomes more controllable. 
       FIG.  2    illustrates a block diagram of a solar PV network  200  (hereinafter interchangeably referred to as “the network 200”), according to aspects of the present disclosure. Referring to  FIG.  2   , the network  200  includes a PV module  201 , a moving regression (MR) filter  206 , a State of Charge (SoC) feedback control  208 , a summation unit  210 , a Battery Energy Storage System (BESS)  212 , a combiner  214 , and an electrical grid  216 . 
     The PV module  201  is configured to receive solar light signals and to generate unsmoothed solar PV power output (P PV ). To achieve a required voltage and current, a group of PV modules (also called PV panels) are wired into an array called as a PV array  202 . Several PV modules  201  can be wired together in series and/or parallel to deliver voltage and current according to the system requirements. In an aspect, the PV module  201  includes the PV array  202 , and a boost converter  204 . 
     The boost converter  204  is configured to boost the power produced by the PV module  201  and transfer the boosted power to the MR filter  206 , and the SoC feedback control  208 . In an aspect, the boost converter  204  (for example, step-up converter) is a DC-to-DC power converter that steps up voltage (while stepping down current) from its input (supply) to its output (load). 
     The MR filter  206  and the SoC feedback control  208  are coupled to the PV module  201  to receive unsmoothed solar PV power output power (P PV ). The MR filter  206  is configured to reduce ripple from the received unsmoothed solar PV power output power (P PV ) and to generate a filter smooth output power (P PO ). 
     The MR filter  206  is a non-parametric smoother that is configured to employ at least one machine learning method to smooth out the PV variations at every time step. In an aspect, at least one machine learning method is a linear regression. 
     In an aspect, the MR filter  206  is coupled to circuitry (not shown) including a memory storing program instructions and at least one processor configured to perform the program instructions. In an aspect, the memory is configured to store a linear regression model and a predefined dataset for training the linear regression model. The stored program instructions include a program that implements a method for using machine-learning methods to perform linear regression on an unsmoothed power in accordance with embodiments of the present disclosure and may implement other embodiments described in this specification. 
     According to an aspect of the present disclosure, at least one processor may be implemented as microprocessors, microcomputers, microcontrollers, digital signal processors, central processing units, state machines, logic circuitries, and/or any devices that manipulate signals based on operational instructions. Among other capabilities, at least one processor may be configured to fetch and execute computer-readable instructions stored in the memory. 
     The memory may be coupled to the processor and may include any computer-readable medium known in the art including, for example, any type of disk, including floppy disk, optical disk, CD-ROM and magneto-optical disk, read only memory (ROM), random access memory (RAM), EPROM, EEPROM, magnetic or Optical card, any type of medium suitable for storing electronic instructions, but is not limited to these. 
     In the MR filter  206 , the window size may be varied according to requirements. Based on the window size of the MR filter  206 , the neighboring points k of the target value are used as the training values for a linear regression algorithm. In some examples, larger window sizes mean that more neighboring data k may be used for training the linear regression model in the MR filter  206 , and therefore higher the accuracy of the predicted smoothed value. In the linear regression algorithm, priority weights may be assigned to the neighboring values based on its distance from the target value. The MR filter  206  may also use the window size as an input parameter. The MR filter  206  has exceptional power tracking capabilities even with larger window sizes compared with other filter alternatives, such as LPF, MA filter, MM filter, GF, DMA filter, GF and SG filter. Therefore, the time delay caused by the MR filter  206  is less dependent on the window size of the MR filter  206 . In an aspect, the MR filter  206  assists in achieving higher degrees of PV power smoothing and power tracking, reducing time delays, and decreasing battery charging and discharging power, as compared with other filter alternatives. 
     The SoC feedback control  208  is configured to receive the unsmoothed solar PV power output power (P PV ) from the PV module  201 . The state of charge (SoC) is a measurement of the amount of energy available in a battery at a specific point in time expressed as a percentage. For example, the SoC reading for a battery might read 85% full or 20% full. The SoC provides information of how much longer the battery can provide power/energy before it is needed to be charged or replaced. Based on the SoC, the remaining capacity of the battery can be used in a more effective and controlled manner. The SoC feedback control  208  is configured to prolong the lifetime of the battery. Over-charging and over-discharging are two of the prime causes of reducing lifetime of the battery. The SoC information is configured to determine the end of the charging and discharging cycles, thereby preventing the battery from over-charging and over-discharging and enhancing the lifetime of the battery. Further, accurate SoC information is configured to keep the battery operating within the required, safe limits. The SoC is configured to allow the battery to discharge up to a fixed level, and when the power of the battery is reached to the fixed level, the SoC is configured to prevent the battery from over-discharging by communicating SoC information to a control unit, configured to take precautionary action. 
     In an aspect, the MR filter  206 , the SoC feedback control  208  and BESS  212  operate to reduce a power lag and a ramp rate for a PV system. In a further aspect, the MR filter  206  and the SoC feedback control  208  provide battery storage capacity optimization by managing battery charging and discharging state. In an operative aspect, the output power of the PV module  201  acts as the control signal, which is smoothed by the MR filter  206 . A difference between the photovoltaic power and the smoothed photovoltaic power is compensated by charging or discharging power of the BESS  212 . Further, the SoC feedback control  208  along with the MR filter  206  and the BESS  212  is configured to provide smoothing of solar PV variabilities. 
     As shown in  FIG.  2   , the output of the SoC feedback control  208  is further coupled to the summation unit  210  and the combiner  214 . 
     The summation unit  210  is commutatively coupled to the MR filter  206  and the SoC feedback control  208  to receive the filter smooth output power (P PO ) and an output of the SoC feedback control  208 , respectively. During the summation, the output received from the SoC feedback control  208  acts as negative feedback, and by applying the negative feedback stability of the signals is achieved. The summation unit  210  is configured to generate a power reference output (P ref ) by combining P PO  and the output of the SoC feedback control  208 . 
     The BESS  212  is coupled with the summation unit  210  and receives the power reference output (P ref ). The BESS  212  is further configured to generate a BESS power output (P BESS ). 
     The combiner  214  is configured to receive the smoothed output of the SoC feedback control  208  and the BESS power output (P BESS ) from the BESS  212 . In an aspect, the outputs of the SoC feedback control  208  and the BESS  212  are combined to generate a smoother solar PV power output that is coupled to the electrical grid  216 . The combiner  214  is configured to generate an output grid power (P grid ) by combining P BESS  and another output of the SoC feedback control  208 . The electrical grid  216  is configured to receive the output grid power P grid . FIG.  3    illustrates a method  300  for providing a solar PV power to the electrical grid, according to aspects of the present disclosure. 
     Step  302  includes receiving solar signals by a PV module  201 . According to an aspect of the present disclosure, the PV module  201  comprises the PV array  202  and the boost converter  204 . 
     Step  304  includes generating the unsmoothed solar PV power output power (P PV ) by the PV module  201  and coupling the P PV  to the MR filter  206  and to the SoC feedback control  208 . According to aspects of the present disclosure, the MR filter  206  and SoC feedback control  208  receive the unsmoothed solar PV power output from the PV module  201 . 
     Step  306  includes generating the filter smooth output power (P PO ) by the MR filter  206 . According to an aspect of the present disclosure, the MR filter  206  is a non-parametric smoother that utilizes a machine learning concept of linear regression to smooth out solar PV variations at every time step. In an aspect of the disclosure, a window size of the MR filter  206  determines training values of the linear regression. Further, a larger window size of the MR filter  206  causes a higher degree of PV power smoothing. 
     Step  308  includes generating the power reference output (P ref ) by summing P PO  and an output of the SoC feedback control  208 . Further, P ref  is coupled to a Battery Energy Storage System (BESS)  212 , wherein the BESS  212  generates a BESS power output (P BESS ). 
     Step  310  includes generating the output grid power (P grid ) by combining P BESS  and another output of the SoC feedback control  208 . 
     Step  312  includes receiving output grid power P grid  by the electrical grid  216 . 
     Low Pass Filter 
     A smoothing methodology employing SoC feedback control combined with a LPF is illustrated in  FIG.  4   .  FIG.  4    illustrates a block diagram of smoothing of the unsmoothed solar PV power output P PV  generated from a PV module (not shown) with the SoC feedback control  408  and LPF  402 , according to aspects of the present disclosure. 
     As shown in  FIG.  4   , the SoC feedback control  408  is coupled with a LPF  402 . The SoC feedback control  408  is configured to prolong the lifetime of the battery. The function of the LPF  402  is to block the high-frequency components while allowing the low-frequency components to pass through it. The unsmoothed solar PV power output power P PV  functions as a control signal that should be flattened through the low pass filter. The P PV  represents an unsmoothed PV signal, while P PO  represents the filter smoothed output power. Charging and discharging power signal P ref  of the BESS  412  is the difference between the smoothed signal P PO  and the unsmoothed signal P PV . The output of the BESS  412  is denoted by P BESS  The BESS  412  charges when P BESS  is positive (P BESS &gt;0) and discharges (P BESS &lt;0) with negative P BESS  The power to be infused into the utility network P grid  is the summation of the P BESS  and the PV power P PV . The time constant (T f ) of the LPF  402  is selected such that the T f  accounts for the solar irradiance aberrations and does not violate the ramp rate of the grid by approximately 10%. The degree of PV firming achieved is directly proportional to the T f  of the LPF  402 . Larger values of the T f  lead to smoother power output, but at the cost of causing a significant time delay. The following equations realize a model shown in  FIG.  4    that establishes the relationships governing the LPF  402  integrated with SoC feedback control  408 . 
     The LPF  402  is based on the transfer function: 
     
       
         
           
             
               
                 
                   
                     
                       H 
                       ⁡ 
                       ( 
                       s 
                       ) 
                     
                     = 
                     
                       1 
                       
                         sT 
                         
                           f 
                           + 
                           1 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               where 
               ⁢ 
                   
               
                 T 
                 f 
               
             
             = 
             
               RC 
               . 
             
           
         
       
     
     The P ref  after the LPF filtering is as follows: 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       ref 
                     
                     ( 
                     s 
                     ) 
                   
                   = 
                   
                     
                       
                         - 
                         
                           sT 
                           f 
                         
                       
                       
                         sT 
                         
                           f 
                           + 
                           1 
                         
                       
                     
                     . 
                     
                       
                         P 
                         PV 
                       
                       ( 
                       s 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     The battery SoC is given in equation (3), where E BESS  is the battery capacity. 
     
       
         
           
             
               
                 
                   
                     SoC 
                     ⁡ 
                     ( 
                     s 
                     ) 
                   
                   = 
                   
                     
                       - 
                       
                         P 
                         
                           BESS 
                           ⁡ 
                           ( 
                           s 
                           ) 
                         
                       
                     
                     
                       S 
                       . 
                       
                         E 
                         BESS 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     The battery SoC represents the level of charge of an electric battery relative to its capacity. 
     The battery thermal limitations may depend on the battery capacity. Higher capacity indicates that the PV power may be handled without violating the upper and lower battery thermal limitations. Following equation relates the battery capacity with the battery SoC and P PO . 
     
       
         
           
             
               
                 
                   
                     E 
                     BESS 
                   
                   = 
                   
                     
                       Tf 
                       · 
                       
                         
                           P 
                           PO 
                         
                         ( 
                         s 
                         ) 
                       
                     
                     
                       SoC 
                       ⁡ 
                       ( 
                       s 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     To overcome the issue of battery overcharging and deep discharging, a zoom coefficient K is introduced. The battery capacity is limited by setting a positive value for K. The average unsmoothed PV power is denoted by  P   PV . 
         KT   f   · P     PV   ≤E   BESS   (5)
 
     The coefficient K is optimized in equation (6) using the upper and lower SoC limits denoted by SoC mh  and SoC ml , respectively. 
       ( SoC   mh   +SoC   ml )· E   BESS =( E   BESS   −KT   f )·   P     PV   (6)
 
     The E BESS  is regulated by feedback control of the battery&#39;s SoC as shown in equation (7), thus the battery output is the summation of charging/discharging power, Storage capacity margins, and the smoothed PV power. 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       BESS 
                       ′ 
                     
                     . 
                     
                       T 
                       f 
                     
                   
                   = 
                   
                     
                       
                         SoC 
                         ⁡ 
                         ( 
                         s 
                         ) 
                       
                       . 
                       
                         E 
                         BESS 
                       
                     
                     - 
                     
                       ( 
                       
                         
                           
                             KT 
                             f 
                           
                           
                             sT 
                             
                               f 
                               + 
                               1 
                             
                           
                         
                         . 
                         
                           
                             P 
                             PV 
                           
                           ( 
                           s 
                           ) 
                         
                       
                       ) 
                     
                     - 
                     
                       ( 
                       
                         
                           E 
                           BESS 
                         
                         - 
                         
                           
                             KT 
                             f 
                           
                           . 
                           
                             
                               
                                 P 
                                 _ 
                               
                               PV 
                             
                             ( 
                             s 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     The dispatchable grid power is given by (8), which is the summation of the PV power P PV  and P BESS . 
         P   grid ( S )= P   BESS   +P   PV ( S )  (8)
 
     With (P BESS &lt;0), the battery is being injected with power (charging) and (P BESS &gt;0) suggests that the battery system is providing power to the grid (discharging). 
     Moving Average Filter 
     The following is a description of the smoothing methodology of using a moving average (MA) filter in a solar PV system. The fluctuations of the PV power can be smoothed out through the use of a MA filter. The MA filter operates by calculating the average value of the recorded PV power across a sliding window. The window size of the MA filter and the real photovoltaic power data are the required inputs to an algorithm. The magnitude of the moving average window size directly determines the extent of solar power firming. The amount of flatness can be altered by adjusting the value of the filter window size. The difference between the MA smoothing algorithm power output and the real PV power data can result in the charging and discharging power of the battery. The scale of power lagging increases as the window size increases, however, it also results in a considerably smoother power output. The algorithms shown in (9) and (10) are used to model the moving average filter. 
     
       
         
           
             
               
                 
                   
                     Y 
                     i 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       j 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                   
                                     S 
                                     
                                       i 
                                       + 
                                       j 
                                     
                                   
                                 
                                 M 
                               
                               , 
                             
                           
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 i 
                               
                               &gt; 
                               
                                 0 
                                 ⁢ 
                                     
                                 and 
                                 ⁢ 
                                     
                                 i 
                               
                               &lt; 
                               
                                 N 
                                 - 
                                 
                                   ( 
                                   
                                     M 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               0 
                               , 
                             
                           
                           
                             otherwise 
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Y 
                     i 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       
                                         - 
                                         
                                           ( 
                                           
                                             M 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       2 
                                     
                                     
                                       
                                         ( 
                                         
                                           M 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   
                                     S 
                                     
                                       i 
                                       + 
                                       j 
                                     
                                   
                                 
                                 M 
                               
                               , 
                             
                           
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 i 
                               
                               &gt; 
                               
                                 
                                   
                                     ( 
                                     
                                       M 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                     
                                 and 
                                 ⁢ 
                                     
                                 i 
                               
                               &lt; 
                               
                                 N 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       M 
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                         
                         
                           
                             
                               0 
                               , 
                             
                           
                           
                             otherwise 
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     where the total number of data points is represented by N. Whereas M denotes the mean across a specified time interval. The input for the MA algorithm is presented by S i+j  and the result of the algorithm is symbolized by Y i . Equation (10) is utilized when the number of data points to be used by the MA algorithm is odd. 
     Double Moving Average Filter 
     Another methodology for solar power smoothing is based on using a double moving average filter (DMA).  FIG.  5    represents the double moving average (DMA) filter  500 , according to aspects of the present disclosure. The DMA filter  500  can be established by merging two MA filters  502  in sequence. The block diagram shown in  FIG.  5    demonstrates the construction of a DMA filter  500 . Thus, the DMA filter  500  includes two window sizes, one for each MA filter  502 . The DMA filter  500  involves moving average calculation of the data using a first MA filter, and then a second MA filter computes using the averages from the first moving average as observations. 
     The DMA filter  500  has a most suitable smoothing, power tracking, battery state of charge management, and charging/discharging power performance when both the window sizes of the DMA filter  500  are chosen to be equal. As compared to the MA filter  206 , the DMA filter  500  has an improved overall performance neglecting the cost of implementation. Mathematically, by doubling the equations (9) and (10), the DMA filter  500  may be realized. As per equations (9) and (10), the output of the first MA filter Y i  is considered as an input to the second MA filter as shown in  FIG.  5   . 
     Equations (9) and (10), as previously noted, are used to model the MA filter  502   
     Y i =Moving Median Filter 
     The moving median (MM) filter also employs a window of a fixed length for power smoothing. However, instead of calculating the average across the window, the MM filter computes a median value of a vector of length given by window size. The MM filter is beneficial over the averaging algorithms as the short term and expeditious instabilities can be minimized. The MM filter is highly efficient when there are additional outliers in the enclosed data. That is, if the output solar power dataset within the specified window consists of numerous outlying data points, then the MINI filter will more efficiently handle this data in contrast to the averaging MA and DMA filters. When there are not enough elements for the window, the window size is automatically shortened. Nonetheless, the MINI filter is a non-separable filter, and the MM filter also results in less levelness for window sizes like the ones used in the MA or DMA filtering techniques. The MINI filter is used in existing studies for solar output power smoothing and assesses its performance against the LPF and the MA filter. 
     The median value y(n) is acquired over the slide window across its neighboring values as: 
         Y ( n )= f ( x ( n )), x ( n− 1), . . .  x ( n−m ),( n− 1), y ( n− 2), . . .  y ( n−m ),  (11)
 
         x ( n )=[ x ( n )), x ( n −1), . . .  x ( n−m )] T ,  (12)
 
     For the given vector x(n), the result Y(n) represents the middle value of the vector. 
         y ( n )= med{x ( i )}, i=n,n− 1, n−m.   (13)
 
     When the window size is chosen to be odd, the window is positioned around the element in the current position. In case of even a window size, the window is placed around the previous and current data points. However, for higher-dimensional data the specified window may incorporate all the entries inside the ellipsoidal region. 
     Savitsky-Golay Filter 
     In contrast to the MA filter which uses the averaging algorithms mentioned in equations (9) and (10), a Savitsky-Golay (SG) filter utilizes the technique of least square of polynomial fitting over a sliding window. Therefore, the SG filter is also called a least squares polynomial filter. The filter coefficients may be obtained through an unweighted linear least square fit employing a polynomial of a given degree. A key advantage that the SG filter has against the MA filter is that the SG filter manages to pertain the main features of the data such as the height, width and peak of the signal which otherwise is weakened by the MA filter. Also, the MA filter tends to filter out a substantial amount of the signal&#39;s high frequency components and conserves the lesser moments of the peak. The overall performance of the SG filter may tend to improve as the polynomial order of the filter is increased. That is, at higher orders, it is possible to achieve a high level of smoothing without attenuating the data features. However, the SG filter may be less effective at rejecting noise as compared to the MA filter and the SG filter may have odd numbers for window sizes. Additionally, the SG filter may be implemented for the purpose of photovoltaic power smoothing. 
     Applying the SG filter to a set of data given as follows 
         y ( n )= x ( n )+ w ( n ),  (14)
 
     where x(n) is an estimated time series data, and w(n) is a noise signal on y(n). 
         {circumflex over (x)} ( n )=Σ k=−M   M   h ( n ) y ( n−k ),  (15)
 
     where M denotes the SG filter parameter, h(n) is the impulse response of the filter over (|n|≤M). The coefficient of polynomial k that best suits y(n) over (|n|≤M) is defined as {circumflex over (x)}(n) at (n=0). The requirement (0&lt;k≤2M) for the polynomial order k should be met due to the symmetric nature of the impulse response at (n=0). Consequently, the impulse response h(n) is associated with the transfer function of the SG filter as follows: 
         H ( z )=Σ n   h ( n ) z   −n .  (16)
 
     The error between smoothed and the unsmoothed signal is calculated using the squared error formula as given in equation (17). 
         E=Σ   n=−M   M ( y ( n )− p ( n )) 2 .  (17)
 
     where p(n) denotes the polynomial in (18). 
         p ( n )=Σ k=0   C   k   n   k .  (18)
 
     Gaussian Filter 
     The Gaussian filter (GF) is a smoothing filter whose impulse response is an approximation of the Gaussian function shown in equation (19). 
     
       
         
           
             
               
                 
                   
                     f 
                     ⁡ 
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     a 
                     . 
                     
                       exp 
                       ( 
                       
                         
                           - 
                           
                             
                               ( 
                               
                                 x 
                                 - 
                                 b 
                               
                               ) 
                             
                             2 
                           
                         
                         
                           2 
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                       
                       ) 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     where a is the curve peak height, b is the center peak position, and c is the standard deviation or the Gaussian root mean square width. 
     The GF has no overshooting properties while minimizing a rise and fall time which results in least feasible group delay. In many existing solutions, the GF smoothing is used to firm the generation of renewable energy sources. It also demonstrates the Gaussian filter&#39;s ability in decreasing the drawbacks of the MA filter such as inadequate smoothing and power tracking. Similar to the MA filter, the GF uses a sliding window, but the degree of smoothing is determined by the standard deviation of the gaussian instead of an averaging algorithm. The GF results in a bell-shaped distribution. Subsequently, the GF occasionally tends to over-smooth the PV data resulting in the loss of the signal&#39;s characteristics. 
     In one dimensional, the GF can be realized as follows: 
     
       
         
           
             
               
                 
                   
                     
                       G 
                       ⁡ 
                       ( 
                       
                         x 
                         , 
                         σ 
                       
                       ) 
                     
                     = 
                     
                       
                         1 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                       
                       ⁢ 
                       
                         exp 
                         ( 
                         
                           
                             - 
                             
                               x 
                               2 
                             
                           
                           
                             2 
                             ⁢ 
                             
                               σ 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     where σ represents the standard deviation of the GF. 
     Moving Regression Filter 
     The MR filter  206  is a non-parametric smoother that utilizes the machine learning concept of linear regression to smooth out the PV variations at every time step. Based on the window size of the MR filter, the neighboring points k of the target value are used as the training values for the linear regression algorithm. Hence, larger window sizes mean that more neighboring data k will be used for training the linear regression model in the MR filter and hence higher the accuracy of the predicted smoothed value. Priority weights are assigned to the neighboring values based on its distance from the target value. Similar to the moving average and median filters, the MR filter also uses the window size as its input parameter. However, the MR filter has exceptional power tracking capabilities even with larger window sizes. That is, the time delay caused is less dependent on the window size for a MR filter. 
     A Tri-cubic function is used for calculating the distance weights w(x) for each k neighbors of x according to its distance x′: 
     
       
         
           
             
               
                 
                   
                     w 
                     ⁡ 
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       
                                         
                                           ❘ 
                                           &#34;\[LeftBracketingBar]&#34; 
                                         
                                         x 
                                         
                                           ❘ 
                                           &#34;\[RightBracketingBar]&#34; 
                                         
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                                 3 
                               
                               , 
                             
                           
                           
                             
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 x 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                               &lt; 
                               1 
                             
                           
                         
                         
                           
                             
                               0 
                               , 
                             
                           
                           
                             
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 x 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                               ≥ 
                               1 
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Normalizing equation (21) so that larger distances are associated with lower weights: 
     
       
         
           
             
               
                 
                   
                     w 
                     ⁡ 
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       
                                         
                                           ❘ 
                                           &#34;\[LeftBracketingBar]&#34; 
                                         
                                         
                                           
                                             d 
                                             ⁡ 
                                             ( 
                                             
                                               x 
                                               , 
                                               
                                                 x 
                                                 ′ 
                                               
                                             
                                             ) 
                                           
                                           
                                             
                                               max 
                                               i 
                                             
                                             
                                               d 
                                               ⁡ 
                                               ( 
                                               
                                                 
                                                   x 
                                                   i 
                                                 
                                                 , 
                                                 
                                                   x 
                                                   ′ 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         
                                           ❘ 
                                           &#34;\[RightBracketingBar]&#34; 
                                         
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                                 3 
                               
                               , 
                             
                           
                           
                             
                               
                                 
                                   
                                     ❘ 
                                     &#34;\[LeftBracketingBar]&#34; 
                                   
                                   x 
                                   
                                     ❘ 
                                     &#34;\[RightBracketingBar]&#34; 
                                   
                                 
                                 &lt; 
                                 1 
                               
                               , 
                               
                                 
                                   x 
                                   i 
                                 
                                 ⁢ 
                                 ϵ 
                                 ⁢ 
                                 D 
                               
                             
                           
                         
                         
                           
                             
                               0 
                               , 
                             
                           
                           
                             
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 x 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                               ≥ 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     where d(x, x′) is the distance between one of the nearest neighbors k and x′. Locality is achieved by assigning higher priority to data closest to x′, and the data point farthest from x′ is given the lowest priority. Hence, points corresponding to a maximum distance from x′ will have zero weight, whereas the point closest to x′ will have the maximum possible weight of one. 
     A local regression model uses a weighted first-degree linear regression method to calculate the corresponding output estimate y′ by using the sampled values of x and y as inputs to the local regression algorithm. 
       β=( X   T   WX ) −1   X   T   WY,   (23)
 
     where β is the vector of linear parameters, W is the matrix with all the calculated weights, X and Y are the matrices containing all x and y observations, respectively. 
     The X matrix with n dimensions and m observations: 
     
       
         
           
             
               
                 
                   X 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             
                               x 
                               1 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                           
                             
                               x 
                               2 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               x 
                               
                                 n 
                                 - 
                                 1 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                         
                           
                             1 
                           
                           
                             
                               x 
                               1 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                           
                             
                               x 
                               2 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               x 
                               
                                 n 
                                 - 
                                 1 
                               
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                         
                         
                           
                             … 
                           
                           
                             … 
                           
                           
                             … 
                           
                           
                             … 
                           
                           
                             … 
                           
                         
                         
                           
                             1 
                           
                           
                             
                               x 
                               1 
                               
                                 ( 
                                 m 
                                 ) 
                               
                             
                           
                           
                             
                               x 
                               2 
                               
                                 ( 
                                 m 
                                 ) 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               x 
                               
                                 n 
                                 - 
                                 1 
                               
                               
                                 ( 
                                 m 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Similarly, the Y matrix with n dimensions and m observations: 
     
       
         
           
             
               
                 
                   Y 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             
                               y 
                               1 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                           
                             
                               y 
                               2 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               y 
                               
                                 n 
                                 - 
                                 1 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                         
                           
                             1 
                           
                           
                             
                               y 
                               1 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                           
                             
                               y 
                               2 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               y 
                               
                                 n 
                                 - 
                                 1 
                               
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                         
                         
                           
                             … 
                           
                           
                             … 
                           
                           
                             … 
                           
                           
                             … 
                           
                           
                             … 
                           
                         
                         
                           
                             1 
                           
                           
                             
                               y 
                               1 
                               
                                 ( 
                                 m 
                                 ) 
                               
                             
                           
                           
                             
                               y 
                               2 
                               
                                 ( 
                                 m 
                                 ) 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               y 
                               
                                 n 
                                 - 
                                 1 
                               
                               
                                 ( 
                                 m 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     The new values of y for smoother solar PV output are calculated by using the following equation: 
         y′=β   T   X,   (26)
 
     where β T  is the transpose matrix of equation (23). 
     Experimental Data and Analysis 
     The following examples of experimental data and analysis are provided to illustrate further and to facilitate the understanding of the present disclosure. 
     To examine the performance, according to aspects of the present disclosure, of the proposed filtering methodology with the BESS, a real solar PV profile was imported to MATLAB for carrying out the required simulations. A performance-based comparison among the MR filters with varying window sizes, followed by MR filters assessment against the LPF, MA, MM, DMA, GF and SG filters were conducted. 
     A comparison of the proposed methodology with other existing smoothing techniques is performed to examine various parameters such as solar power smoothing execution of the MR filter, battery charging/discharging power, and SoC management capabilities. Additionally, an assessment of the battery ramp rate before and after filtering is also observed to further examine the performance of the proposed technique with the existing techniques. 
     The experimental results of various filters for power smoothening is described below: 
       FIG.  6    represents a recorded data corresponding to solar power. In  FIG.  6   , signal  601  indicates the PV power generated by the PV module. 
       FIG.  7 A  represents a PV power smoothing using a LPF with time constants (T.C) 24 and 48 minutes, according to aspects of the present disclosure.  FIG.  7 B  is an enlarged view of the representation of  FIG.  7 A . As shown in  FIG.  7 A  and  FIG.  7 B , signal  701  indicates the PV power. It is evident from  FIG.  7 A  that the value of the LPF time constant affects both the degree of smoothing and power tracking of the output power. The LPF (T.C=24 mins) (signal  703 ) has better power tracking than LPF (T.C=48 mins) (signal  704 ) but it has a poorer smoothing capability than LPF (T.C=48 mins) (signal  704 ). Consequently, the LPF (T.C=48 mins) (signal  704 ) has better smoothing performance but at the cost of severe power lagging. This indicates that larger time constants result in better smoothing with increased time delay. The firming performance of the proposed MR filter (signal  702 ) with a window size of 45 minutes is also shown in  FIG.  7 A  and  FIG.  7 B . Although the window size of the MR filter is greater than the time constant value of LPF (T.C=24 mins) (signal  703 ), it still manages to produce a significantly smoother power output with excellent power tracking. Also, in comparison to the LPF (T.C=48 mins) (signal  704 ) which has a size similar to MR (W.S=45) (signal  702 ), the MR filter still has both better power firming and tracking abilities. 
       FIG.  8 A  represents a smoothing performance of the MR filter with different window sizes, according to aspects of the present disclosure.  FIG.  8 B  is an enlarged view of the representation of  FIG.  8 A . As shown in  FIG.  8 A  and  FIG.  8 B , signal  801  indicates the PV output generated by the PV module. It can be noted from  FIGS.  8 A and  8 B  the smoothing performance of the MR filter is less affected by the increase in window size, as illustrated by signal  802  (window size=45), signal  803  (window size=63), and signal  804  (window size (W.S)=72). The MR filter with increasing window sizes however causes a slight time delay in the output power. But, in comparison to the time delay caused by the LPF filter as in  FIG.  7 A  and  FIG.  7 B , it is far greater than the MR filter time delay. Also, the MR filter with a higher window size than the LPF has a significantly better overall power flattening and trailing performance. 
       FIG.  9    represents a comparison of the LPF against the MR filter based on charging/discharging power. The time delays result in higher charging and discharging power of the battery, as validated from the battery charging and discharging power performance shown in  FIG.  9   . The LPF (T.C=48 mins) (as shown by signal  903 ) has a considerably higher charging/discharging battery than LPF (T.C=24 mins) (shown by signal  901 ) which is a direct result of the time delay caused. Additionally, it can also be observed from  FIG.  9    that the MR filter (W.S=45) (signal  902 ) has drastically reduced the battery charging/discharging power even with a substantially larger window size. 
       FIG.  10    represents a battery charging/discharging power performance comparison of the MR filter with different window sizes (as shown by signals  1001 , and  1002 ). As shown in  FIG.  10   , the increase in the window size of the MR filter increases the battery charging/discharging power but the increase is much lower as compared to the LPF&#39;s effect on the battery. Higher charging and discharging power of the battery means that larger batteries will be required and having large batteries is economically unfeasible. Thus, using the MR filter is a better solution than the LPF for optimizing the battery size while simultaneously achieving higher degrees of the PV power smoothing. 
       FIG.  11    represents a battery State of Charge (SoC) comparison of the LPF against the MR filter. Finally, the effect of time delay on the battery state of charge by the LPF filter is shown in  FIG.  11   . It is clearly shown that the time delay caused due to the LPF having higher LPF time constants (signals  1101 , and  1103 ) greatly affects the battery SoC and to get better smoothing performance more charging and discharging rates are the result. It is also noticeable from  FIG.  11    that the MR filter has the most proper battery SoC management (as illustrated by signal  1102 ) even with a relatively large window size of 45 minutes. 
       FIG.  12    represents a battery SoC performance comparison of the MR filter with different window sizes (as shown by signals  1201 ,  1202 , and  1203 ), according to aspects of the present disclosure. The increase in window size for the MR filter has relatively (in comparison to LPFs) lower impact on the SoC performance as demonstrated in  FIG.  12   . The LPF (T.C=48 mins) (signal  1103 ) reaches a maximum value of approximately 68% SoC (in  FIG.  11   ), whereas the MR filter (W.S=45) with a larger window has a maximum SoC of approximately 50.1% (in  FIG.  12   ) (as shown by signal  1203 ) while achieving a higher degree of solar power smoothing with the least time delay. Signal  1201  shows the battery SoC performance of the MR filter (W.S=72), whereas Signal  1202  shows the battery SoC performance of the MR filter (W.S=60). Overcharging and deep discharging of the battery affects the battery lifetime, and thus by employing an LPF filter the battery lifetime will potentially be reduced greatly. The MR filter aims to increase the battery lifetime. 
       FIG.  13 A  represents a comparison of the PV smoothing performance of the MA filter (as displayed by signals  1303  and  1304 ) against the MR filter (shown by signal  1302 ). Further,  FIG.  13 B  is an enlarged view of the representation of  FIG.  13 A . As shown in  FIG.  13 A  and  FIG.  13 B , signal  1301  indicates the PV output generated by the PV module. It can be observed that the MA filter also suffers from poor power tracking as the window size is increased. The MA with W.S=48 (signal  1303 ) has better smoothing performance than the MA (W.S=24) (signal  1304 ) but relatively worse power tracking ability. From  FIG.  13 A  and  FIG.  13 B , it is evident that the MR filter (shown by signal  1302 ) outperforms both the MA filters in terms of power firming and tracking competencies. 
       FIG.  14    represents a battery SoC comparison of the MA filter against the MR filter. Although both the MA filters (shown in  FIG.  14   ) (as displayed by signals  1401  and  1403 ) have poor SoC management performance, and the MA filter (W.S=48) (as shown by signal  1403 ) tends to deeply discharge the battery. This indicates that the battery life will greatly be reduced through the implementation of the MR filter. On the other hand, it can also be noted from  FIG.  14    that the MR (W.S=45) (signal  1402 ) has proper SoC management while having a larger window size. 
       FIG.  15    represents a battery charging/discharging power performance comparison of the MA filter (signal  1502 ) against the MR filter (signal  1501 ). From  FIG.  15    it can also be remarked that the MR filter (W.S=45) (signal  1501 ) has a lower charging and discharging power than the MA (W.S=48) (signal  1502 ). Thus, the MR filter improves the overall performance of the battery in contrast to the MA filter. 
       FIG.  16 A  represents a comparison of power smoothing performance of the DMA filter against the MR filter. Further,  FIG.  16 B  is an enlarged view of the representation of  FIG.  16 A . As shown in  FIG.  16 A  and  FIG.  16 B , signal  1601  indicates the PV output generated by the PV module. The flattening performance of the DMA filter shown in  FIG.  16 A  and  FIG.  16 B , suggests that it has a better smoothing and tracking performance than the regular MA filter in  FIG.  13   . However,  FIG.  16 A  and  FIG.  16 B , also imply that although the DMA filter (W.S=48) (signal  1604 ) has a better power firming performance than the DMA filter (W.S=24) (signal  1603 ), there is still a lag produced in the output power by the DMA filter (W.S=48) (signal  1604 ). The MR filter output (signal  1602 ) in  FIG.  16 A  and  FIG.  16 B , has a window size similar to the DMA filter with a windows size of 48 minutes, but it still operates to have greater smoothing performance with particularly good power trailing. 
       FIG.  17    represents a battery SoC comparison of the DMA filter against the MR filter. Both the DMA filters (shown in  FIG.  17   ) (as displayed by signals  1701  and  1703 ) have poor SoC management performance. On the other hand, it can also be noted from  FIG.  17    that the MR (W.S=45) (signal  1702 ) has proper SoC management while having a larger window size. 
     Further,  FIG.  18    represents a battery charging/discharging power performance comparison of the DMA against the MR filter (as shown by signal  1802 ). Because of the MR filters time delay reduction capacity, the MR filter exceeds the DMA filters with reduced charging and discharging power and proper SoC management as shown in  FIG.  17    and  FIG.  18   , respectively. As shown in  FIG.  18   , the DMA (W.S=48) (signal  1801 ) tends to deeply discharge the battery at around 200 minutes and overcharges at around the 700 minutes mark. 
       FIG.  19 A  represents a comparison of power smoothing performance of the MM filter against MR filter.  FIG.  19 B  is an enlarged view of the representation of  FIG.  19 A . As shown in  FIG.  19 A  and  FIG.  19 B , signal  1901  indicates the PV output generated by the PV module. It can be seen from  FIG.  19 A  and  FIG.  19 B  that the MM filter (signals  1903 , and  1904 ) has better power tracking than both the MA and DMA filters. However, the MM filter lacks behind the MA and DMA filters in terms of smoothing ability. The MR filter (signal  1902 ) has improved power smoothing and trailing than both the MM filters of window sizes 24 and 48 minutes. 
       FIG.  20    represents a battery SoC comparison of the MM filter against the MR filter. Also, since the MM filters have poor smoothing performances, it results in poor SoC management which can be shown by signal  2001 , and  2003  from  FIG.  20   . The MM (W.S=48) (signal  2003 ) in particular tends to overcharge the associated battery. In contrast, the MR filter (signal  2002 ) maintains the battery state of charge. 
       FIG.  21    represents a battery charging/discharging power performance comparison of the MM filter against the MR filter.  FIG.  21    demonstrates the battery charging/discharging power, and as expected the MM (W.S=48) (signal  2101 ) has higher charging and discharging power than the MR filter (W.S=45) (signal  2102 ) due to its time delay characteristic. 
       FIG.  22 A  represents a comparison of power smoothing performance of the SG filter against the MR filter. Further,  FIG.  22 B  is an enlarged view of the representation of  FIG.  22 A . As shown in  FIG.  22 A  and  FIG.  22 B , signal  2202  shows the power smoothing performance of the MR filter (W.S=45). Further,  FIG.  23    represents a battery SoC comparison of the SG filter against the MR filter, according to aspects of the present disclosure. Signal  2201  indicates the PV output generated by the PV module. The SG filter, used for solar power smoothing, surpasses the MA, DMA and MM filters in terms of both power leveling and tracking capacity. The SG filters smoothing performance is exhibited in  FIGS.  22 A and  22 B , and although both the SG filters (signals  2203 , and  2204 ) have good quality tracking, they still have poor battery SoC, and this is clearly visible from  FIG.  23    (indicated by signals  2301  and  2303 ). In  FIG.  23   , signal  2302  shows maintaining the battery state of charge by the MR filter. Even though the SG filter (W.S=53) (signal  2301 ) has a slight lag from SG (W.S=27) (signal  2303 ), it is still large enough to have a significant impact on the battery&#39;s SoC performance (shown in  FIG.  23   ). 
       FIG.  24    represents a battery charging/discharging power performance comparison of the SG filter against the MR filter. Since the MR filter (W.S=45) (signal  2402 ) has unique power trailing, it results in a relatively less battery charging and discharging power than the SG (W.S=53) (signal  2401 ) which can be observed from  FIG.  24   . 
       FIG.  25 A  represents a comparison of power smoothing performance of the GF against the MR filter.  FIG.  25 B  is an enlarged view of the representation of  FIG.  25 A . As shown in  FIG.  25 A  and  FIG.  25 B , signal  2501  indicates the PV output generated by the PV module. The Gaussian Filter has good power tracking (displayed in  FIG.  24   ), but the GF tends to over smooth the solar power data which results in the loss of the PV power signal characteristics, as shown by signal  2503 , and  2504  in  FIG.  25   .A and  FIG.  25 B  The MR filter of window size 45 (signal  2502 ) has improved power smoothing in comparison with the GF. 
       FIG.  26    represents a battery SoC comparison of the GF against the MR filter, and  FIG.  27    represents a battery charging/discharging power performance comparison of the GF against the MR filter according to aspects of the present disclosure. From  FIGS.  25 A and  25 B , it can be concluded that over smoothing of the solar power negatively effects the battery as it results in extremely poor SoC management. Both the GF filters (as displayed by signals  2601  and  2603 ) have poor SoC management performance. On the other hand, the MR (W.S=45) (signal  2602 ) has proper SoC management while having a larger window size. 
     Additionally, over smoothing also greatly increases the battery charging and discharging power which can be seen in  FIG.  27   . Since the use of MR filter does not result in the loss of the original PV power characteristic and due to its exceptional power tracking (shown in  FIG.  25   ), the MR filter results in a reduced battery charging/discharging power ( FIG.  27   ) shown as signal  2702 . while having appropriate SoC management (shown in  FIG.  26   ). Signal  2701  indicates the battery charging and discharging power associated with the GF (W.S.=48). 
     The fluctuation rate of the PV power to be dispatched into the grid is analyzed. The ramp rate of the unsmoothed solar power is compared to the proposed MR filter and various other previously mentioned filters combined with the SoC feedback control  FIG.  28 A  shows the ramp rate comparison of the PV power against the LPF and the MR filter. Further  FIG.  28 B  is an enlarged view of the representation of  FIG.  28 A . As shown in  FIG.  28 A  and  FIG.  28 B , signal  2801  indicates unfiltered PV power. Although the LPF filter manages to reduce the ramp rate (as shown by signal  2803 ), the MR filter (signal  2802 ) achieves significantly better ramp rate reduction along with considerably improved smoothing, tracking and battery SoC performance. 
       FIG.  29 A  represents a solar power ramp rate comparison of the MA filter against the MR filter.  FIG.  29 B  is an enlarged view of the representation of  FIG.  29 A . As shown in  FIG.  29 A  and  FIG.  29 B , signal  2901  indicates unfiltered PV power. The MR filter (signal  2902 ) achieves significantly better ramp rate reduction as compared with MA filter (as shown by signal  2903 ). 
       FIG.  30 A  represents a solar power ramp rate comparison of the MM filter against the MR filter, according to aspects of the present disclosure. Further,  FIG.  30 B  is an enlarged view of the representation of  FIG.  30 A . As shown in  FIG.  30 A  and  FIG.  30 B , signal  3001  indicates unfiltered PV power. The MM filter manages to reduce the ramp rate (as shown by signal  3003 ). However, the MR filter (signal  3002 ) achieves significantly better ramp rate reduction along with considerably improved smoothing, tracking and battery SoC performance. 
       FIG.  31 A  represents a solar power ramp rate comparison of SG against MR filter, according to aspects of the present disclosure. Further,  FIG.  31 B  is an enlarged view of the representation of  FIG.  31 A . As shown in  FIG.  31 A  and  FIG.  31 B , signal  3101  indicates unfiltered PV power. The SG filter reduces the ramp rate (as shown by signal  3103 ). However, the MR filter (signal  3102 ) achieves a significantly ramp rate reduction. 
     As evident from  FIGS.  29 A,  29 B,  30 A,  30 B,  31 A, and  31 B , even though ramp rate reduction can be achieved by all the filters described and simulated, the MR filter achieves the most suitable ramp rate control. 
     Moreover, the present disclosure is not limited to the specific circuit elements described herein, nor is the present disclosure limited to the specific sizing and classification of these elements. For example, the skilled artisan will appreciate that the circuitry described herein may be adapted based on changes on battery sizing and chemistry, or based on the requirements of the intended back-up load to be powered. 
     The functions and features described herein may also be executed by various distributed components of a system. For example, one or more processors may execute these system functions, wherein the processors are distributed across multiple components communicating in a network. The distributed components may include one or more client and server machines, which may share processing, in addition to various human interface and communication devices (e.g., display monitors, smart phones, tablets, personal digital assistants (PDAs)). The network may be a private network, such as a LAN or WAN, or may be a public network, such as the Internet. Input to the system may be received via direct user input and received remotely either in real-time or as a batch process. Additionally, some aspects of the present disclosures may be performed on modules or hardware not identical to those described. Accordingly, other aspects of the present disclosures are within the scope that may be claimed. 
     The above-described hardware description is a non-limiting example of corresponding structure for performing the functionality described herein. 
     Obviously, 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 disclosure may be practiced otherwise than as specifically described herein.