Patent Publication Number: US-9887642-B2

Title: Method and apparatus for estimated inductance and current feedback control of a grid-connected inverter with nonlinear inductor

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of priority from pending U.S. Provisional Patent Application Ser. No. 62/330,862, filed May 3, 2016, and entitled “ENHANCED GRID-CONNECTED INVERTER WITH THE CAPABILITY OF NONLINEAR INDUCTOR IN THE OUTPUT FILTER HANDLING,” which is incorporated by reference herein in its entirety. 
    
    
     TECHNICAL FIELD 
     The present application generally relates to a control of grid connected inverters and, more specifically, a control using an adaptive observer providing concurrent estimation of grid current and an inverter-side inductor. 
     BACKGROUND 
     In grid-connected DC-AC converters (hereinafter“inverters”) an output filter can be used to improve power quality. The output filter can be an LCL filter. It has been known that hardware size and costs of the inductors on the LCL filter can possibly be reduced by operating the inductors in the nonlinear region of their B-H curve, as this enables smaller inductors. However, various technical problems have obstructed their use in a range of applications. One such technical problem can be stability in the LCL filters. Another technical problem, or set of technical problems, can result from inherent uncertainty of the non-linear inductor&#39;s instant inductance value. These may prevent or obstruct use of non-linear LCL filter inductors for some applications. One such application can be grid-connected inverters using, for feedback control, a measurement of current through one LCL inductor, and an estimate of current through another of the inductors. This can enable reduction in the number of current sensors and, hence, costs. However, the estimation of inductor current requires prior knowledge of the inductance values. Use of non-linear inductors, which have inherent variance in their inductance value, can remove the capability the estimation. This can necessitate additional current sensors, negating savings from reduced inductor size. 
     Accordingly, there is a need in the grid connected inverter field for a technical solution to reduce inductor costs without incurring other hardware costs or complexities. 
     SUMMARY 
     Disclosed methods can include transmitting an inverter output current from an inverter phase output, through a non-linear inverter side inductor, and through a grid side inductor to a grid phase, while concurrently estimating an inductance of the non-linear inverter side inductor and a current passing through the grid side inductor, in combination with measuring a current through the inverter side inductor, and generating a corresponding inverter side inductor current measurement; calculating a weighted average current, the weighted average current being a weighted average of the estimated current through the grid side inductor and the inverter side inductor current measurement, applying a weighting that is based, at least in part, on the estimated inductance of the inverter side inductor. Methods can further include generating a feedback signal, based at least in part on the weighted average current and, based at least in part on the feedback signal, controlling oscillation of an inverter feeding the inverter phase output. 
     Disclosed apparatuses can include a grid connected inverter with a nonlinear inductor in the output LCL filter, and a processor particularly configured to provide concurrent estimation of inductor value and grid current, and feedback using the estimation. 
     Benefits and advantages include technical solutions to the problem of employing non-linear inductors to reduce inductor size, without necessitating additional current sensors. Secondary benefits can include improvements in stability, and in efficiency by reducing core losses. 
     This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to limit the scope of the claimed subject matter. Furthermore, the claimed subject matter is not limited to implementations that solve or are directed to solving technical problems or disadvantages noted in any part of this disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawing figures depict one or more implementations in accord with the present disclosure, by way of example only, not by way of limitation. In the figures, like reference numerals refer to the same or similar elements. 
         FIG. 1  shows a block diagram of relevant portions of one example three phase grid-connected inverter topology. 
         FIG. 2  shows one block diagram of one implementation of one example adaptive observer inverter controlling system, according to one or more aspects of this disclosure. 
         FIG. 3  shows a B-H curve of one example Kool Mμ core. 
         FIG. 4  graphically shows one simulation model of one example nonlinear inductor. 
         FIG. 5A  shows a real value and estimated value of one example grid current;  FIG. 5B  illustrates an error of estimation. 
         FIG. 6A  shows a real value and estimated value of one example inverter inductor;  FIG. 6B  shows an example error of estimation. 
         FIG. 7  shows a block diagram of portions of one constructed and tested implementation of one example adaptive observer inverter controlling system according to one or more aspects of the disclosure. 
         FIGS. 8A-8C  show experimental results for one high loaded inverter, with  FIG. 8A  showing one example grid-side current feedback and nonlinear inductor;  FIG. 8B  showing one example WAC feedback, and nonlinear inductor; and  FIG. 8C  showing one example WAC feedback with adaptive observer and nonlinear inductor. 
         FIG. 9  shows experimental results of grid-current THD with different feedback methods in the presence of nonlinear inductor: (1) Grid current feedback, (2) combinational feedback, (3) combinational feedback and nonlinear inductor estimation 
     
    
    
     DETAILED DESCRIPTION 
     In the following detailed description, numerous specific details are set forth by way of examples in order to provide a thorough understanding of the relevant teachings. However, it should be apparent that the present teachings may be practiced without such details. In other instances, well known methods, procedures, components, and/or circuitry have been described at a relatively high-level, without detail, in order to avoid unnecessarily obscuring aspects of the present teachings. 
       FIG. 1  shows a block diagram of relevant portions of one example three phase grid-connected inverter topology  100 . The topology  100  can include a three-phase inverter block  103  connected to power grid  102  through an LCL-filter arrangement that includes, for each of the three phases, an inverter side inductor  106  connected in series (through a line that is visible in  FIG. 1  but not separately numbered) with a grid side inductor  109 , and includes a filter capacitor  109  (Cf) coupled between a ground reference (visible in  FIG. 1  but not separately numbered) and the above-described connection between inductors  106  and  109 . LCL-filter can be a resonant circuit and, for purposes of stability at its resonance frequency, can be damped by resistances represented collectively in  FIG. 1  by the damper block  113 . 
     As described above, known techniques may select and arrange the inverter side inductors  106  and grid side inductor  109  to operate in their linear regions. As also described, this can necessitate use of larger and higher cost inductors. Disclosed apparatuses and methods, in contrast, provide features that include, as will be described in greater detail, non-linear inductor implementation of inverter side inductors  106 , together with feedback control for the three-phase grid-connected inverter topology  100  providing stable control, and utilizing adaptive estimation of inductance and inductor current—with the non-linear inductors—providing reduced sensor costs, along with other benefits. 
       FIG. 2  shows a functional block diagram of one implementation of one example adaptive observer, non-linear LCL inductor feedback controlling system  200  according to one or more aspects of this disclosure. For brevity, description will alternatively recite “adaptive observer, non-linear LCL filter inductor, feedback controlling system”  200  in the abbreviated form “feedback control system”  200 . Various implementations according to the  FIG. 2  feedback control system  200  can be applied to various grid-connected inverter systems, including three phase grid-connected inverter systems having, for example, a topology as illustrated in  FIG. 1 . Accordingly, for purposes of illustration through examples, certain exemplary operations by the feedback control system  200  will be described in reference to FIG. 1 . It will be understood that these are only examples, and are not intended to limit implementations or practices of systems such as illustrated by  FIG. 2  to the  FIG. 1  grid-connected inverter topology. 
     Referring to  FIG. 2 , the feedback control system  200  can utilize a reference current  201  feeding a proportional resonant current control  202  that is coupled to an inverter block  203  implemented, for example, with insulated-gate bipolar transistors (IGBTs). One example implementation can be the six IGBTs (visible but not separately numbered) forming the  FIG. 1  inverter block  103 . Each of the three phase outputs of the inverter block  203  can be scaled by a weighted grid voltage V n , also labeled as “ 204 , ” resulting in an output inverter voltage V i  on the output line  205  ( FIG. 2  shows only one representative example of the phase output lines  205 ). In an aspect, V n  can be considered a disturbance and, accordingly, V n  can consist of the grid voltage V g  multiplied by a factor smaller than integer 1, for example, to increase a dynamic range of the controller system  200 . 
     Each phase output line  205  can connect, via a corresponding phase structure of a specially configured, non-linear inductor LCL filter  211 , to a grid voltage  212 . Each phase of the specially configured, non-linear inductor LCL filter  211  can include an inverter-side, non-linear inductor  206  connected by a line (visible but not separately numbered) in series with a grid-side inductor  209  to a grid voltage  212 . Further features and aspects of the inverter side non-linear inductors  206 , as well as methods and processes of combinational feedback that include estimating their currents, will be described in greater detail later in this disclosure. 
     Referring to  FIG. 2 , each phase of the LCL filter  211  can include a filter capacitor  210  of value C f  coupled between a ground (represented in  FIG. 2  by a ground symbol) and the line connecting the inverter-side inductor  206  to the grid-side inductor  209 . The LCL filter  211  can include damping resistors (not visible in  FIG. 2 ), such as the  FIG. 1  damper block  113 . 
     Combination feedback structures of the  FIG. 2  feedback control system  200 , and aspects thereof, and including adaptive observer features will now be described. Referring to  FIG. 2 , an inverter inductor current sensor  220  senses a current through a representative one of the non-linear inverter inductors  206 , and sends a corresponding I 1  measurement value to a “ 1 −β” sub-block (visible but not separately numbered) of an adaptive combinational feedback weighting block  214 . A non-linear adaptive observer block  213 , which will be alternatively referenced as “online parameter estimator” block  213 , can be configured to receive V i , V g  and V L  and output in response, applying estimation algorithms described in greater detail in later paragraphs, an estimated instant inductance, labeled “L 1 ,” of the non-linear inverter-side inductor  206  and an estimated instant current through the grid-side inductor  209 , labeled “I L2 .” 
     The non-linear adaptive observer block  213  can be configured to apply the following estimation Equations (1) to obtain the estimated instant inductance L 1  of the non-linear inverter-side inductor  206 , which is represented in Equation (1) as {circumflex over (L)} 1 , and the estimated grid-side inductor  209  current I L2 , which is represented in Equation (1) as Î g : 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                   Equations 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     1 
                     ) 
                   
                 
               
             
           
         
       
     
     in which R 1 , and R 2  are the resistance of L 1  (the non-linear inverter-side inductor  206 ) and L 2 , (the grid-side inductor  209 ), respectively; I i , is the measured current through the inverter side inductor  206 ; L 2  is the value of the grid end inductor  209 ; C f  is the filter capacitor value, and f i =K i Ĩ i , f g =K g Ĩ i  and f c =K c Ĩ i  represent observer correction functions. V i  is the inverter voltage and can be readily detected, for example, by the status of inverter switches (not separately visible in  FIG. 2 ) and dc-link voltage. V g  is the grid voltage, and can be measured by a voltage sensor. 
     Referring to  FIG. 2 , the estimated instant inductance {circumflex over (L)} 1  of the non-linear inverter-side inductor  206  can be input to, and can vary, the “ 1 −β” sub-block and the “β” sub-block of the adaptive combinational feedback weighting block  214 . 
     The estimated instant current through the grid-side inductor  209 , I L2 , can be input to the “β” sub-block of the adaptive combinational feedback weighting block  214 . Resulting outputs from the adaptive combinational feedback weighting block  214  are the “ 1 −β” weighted value of the measured inverter inductor current I L1  and the “β” weighted value of the estimated current grid side inductor current I L2 , and these are summed to provide the weighted average, I feedback , labeled  215 . The weighted average I feedback    215  can be input to a feedback gain block  216 , and the result subtracted from the reference current  201  for input to the proportional resonant current control  202 , described above. The feedback gain block  216  can be, for example, a constant gain. 
     Referring to  FIG. 2 , the proportional resonant current control  202  can be configured to apply as its transfer function Equation (2):
 
 G   c ( s )= G   αβ ( s )+ G   hc ( s )   Equation (2).
 
     In an aspect the proportional resonant current control  202  can be configured to include, in its transfer function, a model of the reference signal and, assuming a sinusoidal reference signal, the transfer function should contain two poles on jω. The proportional resonant current control  202  can be configured to eliminate a steady state error at a desired frequency by applying an infinite gain at that frequency. 
     Equation (2) can be modelled as a sum of Equations (3) and (4), which will now be described. In an implementation, Equation (3) can model certain aspects of the proportional resonant current control  202 , as the inverse transformation of the synchronous PI controller back to the stationary frame: 
     
       
         
           
             
               
                 
                   
                     
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                   ⁢ 
                   
                       
                   
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                     ( 
                     3 
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     where ω c , is the cut-off frequency and can provide a more robust implementation. 
     Being a three phase proportional resonant controller, the non-diagonal terms of G αβ (s) can be zero, implying that the α and β axes are independent of each other and, can be treated as two single-phase systems. 
     By paralleling several resonant blocks tuned to resonate at different frequencies, the proportional resonant controller  202  can be configured as harmonic compensators. The transfer function of a harmonic compensator is given by Equation (4). 
     
       
         
           
             
               
                 
                   
                     
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                   ⁢ 
                   
                       
                   
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                     ( 
                     4 
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     The inventors provide, for purposes of assisting readers in understanding and practicing according to disclosed aspects, using example Equations (5) through (23), and without subscribing to or limiting any aspect of this disclosure to any particular scientific theory, what the inventors believe to be illustrative models forming Equation (1). 
     This can start with an LCL model with inductor resistances, as represented by Equation (5): 
     
       
         
           
             
               
                 
                   
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                   Equation 
                   ⁢ 
                   
                       
                   
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                     ( 
                     5 
                     ) 
                   
                 
               
             
           
         
       
     
     in which R 1  and R 2  are the resistance of L 1  and L 2 , respectively. Note that V i  is the inverter voltage and it is known according to the status of inverter switches and dc-link voltage. V g  is the grid voltage and measured by voltage sensors. 
     Next, the system estimated equation can be formed according to the system model of Equation (6), as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
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                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
           
         
       
     
     The estimated states or parameters are shown with the {tilde over (□)} sign. 
     Estimation errors of 
                   I   ~     i     =       I   i     -       I   ^     i         ,         I   ~     g     =       I   g     -       I   ^     g         ,         V   ~     c     =         V   c     -         V   ^     c     ⁢           ⁢   and   ⁢           ⁢     1       L   ~     1           =       1     L   1       -     1       L   ^     1             ,         
and subtracting Equation (6) from Equation (5), estimation error equations can be written as Equation (7):
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
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                                 I 
                                 ~ 
                               
                               i 
                             
                           
                           - 
                           
                             
                               1 
                               
                                 L 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 V 
                                 c 
                               
                               ^ 
                             
                           
                           - 
                           
                             
                               1 
                               
                                 
                                   L 
                                   ~ 
                                 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 V 
                                 ^ 
                               
                               c 
                             
                           
                           + 
                           
                             
                               1 
                               
                                 
                                   L 
                                   ~ 
                                 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 V 
                                 ^ 
                               
                               i 
                             
                           
                           + 
                           
                             f 
                             i 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               I 
                               ~ 
                             
                             . 
                           
                           g 
                         
                         = 
                         
                           
                             
                               - 
                               
                                 
                                   R 
                                   2 
                                 
                                 
                                   L 
                                   2 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 I 
                                 ~ 
                               
                               g 
                             
                           
                           + 
                           
                             
                               1 
                               
                                 L 
                                 2 
                               
                             
                             ⁢ 
                             
                               
                                 V 
                                 ~ 
                               
                               c 
                             
                           
                           + 
                           
                             f 
                             g 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               V 
                               ~ 
                             
                             . 
                           
                           c 
                         
                         = 
                         
                           
                             
                               1 
                               
                                 C 
                                 f 
                               
                             
                             ⁢ 
                             
                               
                                 I 
                                 i 
                               
                               ^ 
                             
                           
                           - 
                           
                             
                               1 
                               
                                 C 
                                 f 
                               
                             
                             ⁢ 
                             
                               
                                 I 
                                 ~ 
                               
                               g 
                             
                           
                           + 
                           
                             f 
                             c 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
           
         
       
     
     where f i =K i Ĩ i , f g =K g Ĩ i  and f c =K c Ĩ i  represent the observer correction functions. 
     With a strategy of reducing estimation error to zero, Equation (7) can be rewritten as Equation (8): 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 I 
                                 ~ 
                               
                               . 
                             
                             i 
                           
                         
                       
                       
                         
                           
                             
                               
                                 I 
                                 ~ 
                               
                               . 
                             
                             g 
                           
                         
                       
                       
                         
                           
                             
                               
                                 V 
                                 ~ 
                               
                               . 
                             
                             c 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                       
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     
                                       R 
                                       1 
                                     
                                     
                                       L 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   K 
                                   i 
                                 
                               
                             
                             
                               0 
                             
                             
                               
                                 - 
                                 
                                   1 
                                   
                                     L 
                                     1 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 K 
                                 g 
                               
                             
                             
                               
                                 - 
                                 
                                   
                                     R 
                                     2 
                                   
                                   
                                     L 
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 1 
                                 
                                   L 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   1 
                                   
                                     C 
                                     f 
                                   
                                 
                                 + 
                                 
                                   K 
                                   c 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   1 
                                   
                                     C 
                                     f 
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                           
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       I 
                                       ~ 
                                     
                                     i 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       I 
                                       ~ 
                                     
                                     g 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       V 
                                       ~ 
                                     
                                     c 
                                   
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             R 
                                             1 
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             I 
                                             ^ 
                                           
                                           i 
                                         
                                       
                                       - 
                                       
                                         
                                           V 
                                           ^ 
                                         
                                         c 
                                       
                                       + 
                                       
                                         V 
                                         i 
                                       
                                     
                                   
                                 
                                 
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                 
                               
                               ] 
                             
                             ⁡ 
                             
                               [ 
                               
                                 1 
                                 
                                   
                                     L 
                                     ~ 
                                   
                                   1 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     8 
                     ) 
                   
                 
               
             
           
         
       
     
     These equations therefore have the form of {tilde over ({dot over (X)})}=A{tilde over (X)}+ω T {tilde over (θ)}, in which {tilde over (X)}and {tilde over (θ)} include estimation errors, and A and Ω can be obtained. The following Equation (9) Lyapunov function can be used to obtain an adaption:
 
 V={tilde over (X)}   T   {tilde over (X)}+{tilde over (θ)}=Ĩ   i   2   +Ĩ   g   2   +{tilde over (V)}   c   2   +α{tilde over (L)}   1   2 ≧0   Equation (9)
 
     The time derivative of the Lyapunov function can be given by Equation (10):
 
{dot over ( V )}=2{tilde over ( I )} i {tilde over ({dot over ( I )})} i +2{tilde over ( I )} g {tilde over ({dot over ( I )})} g +2{tilde over ( V )} c {tilde over ({dot over ( V )})} c +2α{tilde over ( L )} 1 {tilde over ({dot over ( L )})} 1   Equation (10)
 
     Equation (10) can be negative definite to support Equation (9) as a Lyapunov function. This can be used to derive the adaption law. By substituting Equation (7) into Equation (10), {dot over (V)} can be broken down in three parts as {dot over (V)} 1 , {dot over (V)} 2  and {dot over (V)} 3 , as represented by the following Equation (11): 
     
       
         
           
             
               V 
               . 
             
             = 
             
               
                 [ 
                 
                   
                     
                       - 
                       
                         
                           I 
                           ~ 
                         
                         i 
                       
                     
                     ⁢ 
                     
                       1 
                       
                         L 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         V 
                         ~ 
                       
                       c 
                     
                   
                   + 
                   
                     
                       ( 
                       
                         
                           I 
                           ~ 
                         
                         i 
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             - 
                             
                               R 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               I 
                               ^ 
                             
                             i 
                           
                         
                         - 
                         
                           
                             V 
                             ^ 
                           
                           c 
                         
                         + 
                         
                           V 
                           i 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         
                           
                             L 
                             ~ 
                           
                           1 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       
                         I 
                         ~ 
                       
                       g 
                     
                     ⁢ 
                     
                       K 
                       g 
                     
                     ⁢ 
                     
                       
                         I 
                         ~ 
                       
                       i 
                     
                     ⁢ 
                     
                       
                         
                           + 
                           I 
                         
                         ~ 
                       
                       g 
                     
                     ⁢ 
                     
                       
                         
                           V 
                           ~ 
                         
                         c 
                       
                       
                         L 
                         2 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           V 
                           ~ 
                         
                         c 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             1 
                             
                               C 
                               f 
                             
                           
                           + 
                           
                             K 
                             c 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         I 
                         ~ 
                       
                       i 
                     
                   
                   - 
                   
                     
                       
                         V 
                         ~ 
                       
                       c 
                     
                     ⁢ 
                     
                       
                         
                           I 
                           ~ 
                         
                         g 
                       
                       
                         C 
                         f 
                       
                     
                   
                   - 
                   
                     
                       
                         R 
                         2 
                       
                       
                         L 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         I 
                         ~ 
                       
                       g 
                       2 
                     
                   
                 
                 ] 
               
               + 
               
                   
                 
                   
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               - 
                               
                                 
                                   R 
                                   1 
                                 
                                 
                                   L 
                                   1 
                                 
                               
                             
                             + 
                             
                               K 
                               i 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             I 
                             ~ 
                           
                           i 
                           2 
                         
                       
                       ] 
                     
                     + 
                     
                       [ 
                       
                         α 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             L 
                             ~ 
                           
                           1 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               L 
                               ~ 
                             
                             . 
                           
                           1 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         V 
                         . 
                       
                       1 
                     
                     + 
                     
                       
                         V 
                         . 
                       
                       2 
                     
                     + 
                     
                       
                         V 
                         . 
                       
                       3 
                     
                   
                 
               
             
           
         
       
     
     The first part of Equation (11) can be written in quadratic form as Equation (12). 
     
       
         
           
             
               
                 
                   
                     
                       
                         X 
                         ~ 
                       
                       T 
                     
                     ⁢ 
                     Ω 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       X 
                       ~ 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             I 
                             ~ 
                           
                           i 
                         
                         ⁢ 
                         
                           
                             I 
                             ~ 
                           
                           g 
                         
                         ⁢ 
                         
                           
                             V 
                             ~ 
                           
                           c 
                         
                         ⁢ 
                         
                           1 
                           
                             
                               L 
                               ~ 
                             
                             1 
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     
                         
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   
                                     - 
                                     1 
                                   
                                   
                                     L 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       
                                         R 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         I 
                                         ^ 
                                       
                                       i 
                                     
                                   
                                   - 
                                   
                                     
                                       V 
                                       ^ 
                                     
                                     c 
                                   
                                   + 
                                   
                                     V 
                                     i 
                                   
                                 
                               
                             
                             
                               
                                 
                                   K 
                                   g 
                                 
                               
                               
                                 
                                   - 
                                   
                                     
                                       R 
                                       2 
                                     
                                     
                                       L 
                                       2 
                                     
                                   
                                 
                               
                               
                                 
                                   1 
                                   
                                     L 
                                     2 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   
                                     1 
                                     
                                       C 
                                       j 
                                     
                                   
                                   + 
                                   
                                     K 
                                     c 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     1 
                                     
                                       C 
                                       f 
                                     
                                   
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     I 
                                     ~ 
                                   
                                   i 
                                 
                               
                             
                             
                               
                                 
                                   
                                     I 
                                     ~ 
                                   
                                   g 
                                 
                               
                             
                             
                               
                                 
                                   
                                     V 
                                     ~ 
                                   
                                   c 
                                 
                               
                             
                             
                               
                                 
                                   1 
                                   
                                     
                                       L 
                                       ~ 
                                     
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Calculating the Eigenvalues of Ω, it can be seen that there exists K c  and K g  that causes Ω to be negative semi-definite and subsequently {dot over (V)} 1 ≦0. 
     The second part of {dot over (V)} can be written as Equation (13): 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       . 
                     
                     2 
                   
                   = 
                   
                     [ 
                     
                       
                         ( 
                         
                           
                             - 
                             
                               
                                 R 
                                 1 
                               
                               
                                 L 
                                 1 
                               
                             
                           
                           + 
                           
                             K 
                             i 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           I 
                           ~ 
                         
                         i 
                         2 
                       
                     
                     ] 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     13 
                     ) 
                   
                 
               
             
           
         
       
     
     It can be assumed that an estimation error of a parameter cannot, at least in ordinary applications, be zero. It will therefore, be seen that {dot over (V)} 2  can have a desirably large negative value by selecting K i  negative and as large as desired. 
     According to the results obtained, {dot over (V)}is negative if {dot over (V)} 3  is positively bounded since a proper choosing of K i , will result in an overall negative summation. Also, as will be understood, {dot over (V)} 3  can produce the inductor estimation portion of Equations (1). 
     More specifically, referring to Equation (11), the third part of {dot over (V)} can be extracted and written as Equation (14):
 
 {dot over (V)}   3 =[α {tilde over (L)}   1   {tilde over ({dot over (L)})}   1 ]  Equation (14)
 
     where α has a positive value. The system and estimator model can be represented as Equations (15) and (16), respectively.
 
Z=θu   Equation (15)
 
{circumflex over (Z)}={circumflex over (θ)}u   Equation (16)
 
     wherein θ is the unknown parameter, Z is the output and u is the input. Now, the error signal can be, written as Equation (17): 
     
       
         
           
             
               
                 
                   ɛ 
                   = 
                   
                     
                       Z 
                       - 
                       
                         Z 
                         ^ 
                       
                     
                     
                       m 
                       s 
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     17 
                     ) 
                   
                 
               
             
           
         
       
     
     where m s   2  is the normalizing factor and can be written as Equation (18):
 
 m   s   2 =1 +Γu   2    Equation (18)
 
     and Γ is a positive number. 
     By introducing a cost function which minimizes the error, {circumflex over (θ)}can be computed. One example cost function can be Equation (19). 
     
       
         
           
             
               
                 
                   
                     J 
                     ⁡ 
                     
                       ( 
                       
                         θ 
                         ^ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           ɛ 
                           2 
                         
                         ⁢ 
                         
                           m 
                           s 
                           2 
                         
                       
                       2 
                     
                     = 
                     
                       
                         
                           ( 
                           
                             Z 
                             - 
                             
                               
                                 θ 
                                 ^ 
                               
                               ⁢ 
                               u 
                             
                           
                           ) 
                         
                         2 
                       
                       
                         2 
                         ⁢ 
                         
                           m 
                           s 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     19 
                     ) 
                   
                 
               
             
           
         
       
     
     Using the gradient base method, {circumflex over ({dot over (θ)})} can be given as Equation (20):
 
{circumflex over ({dot over (θ)})}=γεu   Equation (20)
 
     wherein γ is the adaption gain and has a positive value. By defining the estimation error as err=θ−{circumflex over (θ)}, the estimation error and the time derivative of the parameter estimation error may converge to zero in a short time. 
     One equation for L 1  can be Equation (21):
 
 LsI   L     1     =V   i   −R   L     1     I   L     1     −V   c    Equation (21)
 
     Therefore, as can be set forth using Equation (22) 
     
       
         
           
             
               
                 
                   
                     θ 
                     = 
                     
                       L 
                       1 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     Z 
                     = 
                     
                       
                         
                           V 
                           i 
                         
                         - 
                         
                           
                             R 
                             
                               L 
                               1 
                             
                           
                           ⁢ 
                           
                             I 
                             
                               L 
                               1 
                             
                           
                         
                         - 
                         
                           V 
                           c 
                         
                       
                       
                         s 
                         + 
                         β 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     u 
                     = 
                     
                       
                         sI 
                         
                           L 
                           1 
                         
                       
                       
                         s 
                         + 
                         β 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     22 
                     ) 
                   
                 
               
             
           
         
       
     
     in which the input and output transfer functions are divided by s+β to place the transfer function into a clearer form. It will be understood that this division can provide a degree of freedom for use in the designing procedure. Consequently, the estimated value of L 1  can be expressed as the following Equation (23): 
     
       
         
           
             
               
                 
                   
                     
                       
                         L 
                         ∼ 
                       
                       . 
                     
                     1 
                   
                   = 
                   
                     
                       
                         
                           γ 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   V 
                                   i 
                                 
                                 - 
                                 
                                   
                                     V 
                                     ^ 
                                   
                                   c 
                                 
                                 - 
                                 
                                   
                                     R 
                                     1 
                                   
                                   ⁢ 
                                   
                                     
                                       I 
                                       ^ 
                                     
                                     i 
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       L 
                                       ^ 
                                     
                                     1 
                                   
                                   ⁢ 
                                   
                                     
                                       I 
                                       ^ 
                                     
                                     i 
                                   
                                   ⁢ 
                                   s 
                                 
                               
                               
                                 
                                   m 
                                   s 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     s 
                                     + 
                                     β 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               s 
                               ⁢ 
                               
                                 
                                   I 
                                   ^ 
                                 
                                 i 
                               
                             
                             
                               s 
                               + 
                               β 
                             
                           
                           ) 
                         
                       
                       → 
                       
                         
                           
                             L 
                             ^ 
                           
                           . 
                         
                         1 
                       
                     
                     = 
                     
                       
                         γ 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 V 
                                 i 
                               
                               - 
                               
                                 
                                   V 
                                   ^ 
                                 
                                 c 
                               
                               - 
                               
                                 
                                   R 
                                   1 
                                 
                                 ⁢ 
                                 
                                   
                                     I 
                                     ^ 
                                   
                                   i 
                                 
                               
                               - 
                               
                                 
                                   
                                     L 
                                     ^ 
                                   
                                   1 
                                 
                                 ⁢ 
                                 
                                   
                                     I 
                                     ^ 
                                   
                                   i 
                                 
                                 ⁢ 
                                 s 
                               
                             
                             
                               
                                 m 
                                 s 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   s 
                                   + 
                                   β 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             s 
                             ⁢ 
                             
                               
                                 I 
                                 ^ 
                               
                               i 
                             
                           
                           
                             s 
                             + 
                             β 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     By initiating {circumflex over (L)} 1  with a limited value, to provide a bounded {dot over (V)} 3 , and proper selection of adaptive gains, the negative definite condition of {dot over (V)} will be satisfied. 
     Lastly, the adaptive observer equations implemented in the grid-connected inverter controller can be defined as set forth in Equation (1) above. 
     According to the inverter characteristics, the calculated LCL-filter parameters are given in TABLE I. It should be noted that the maximum attenuation can be obtained when inverter-side and grid-side inverters have equal values. However, since filter volume minimization is an important criterion, the desired attenuation was considered 20%, which results in a smaller grid-side inductor. 
     
       
         
           
               
             
               
                 TABLE I 
               
             
            
               
                   
               
               
                 LCL filter parameters  
               
            
           
           
               
               
               
            
               
                 Symbol  
                 Parameter  
                 Value 
               
               
                   
               
            
           
           
               
               
               
               
            
               
                 P n   
                 Nominal active power  
                 7  
                 kW 
               
               
                 E n    
                 Line to line RMS  
                 380  
                 V 
               
               
                   
                 voltage  
                   
                   
               
               
                 f g    
                 Grid frequency  
                 50  
                 Hz  
               
               
                 f sw   
                 Switching frequency  
                 15  
                 kHz  
               
               
                 L 1   
                 Inverter side inductor  
                 8.64  
                 mH  
               
               
                 L 2    
                 Grid side inductor  
                 0.123  
                 mH  
               
               
                 C f    
                 Filter capacitor  
                 5.5  
                 μF 
               
               
                 R f    
                 Damping resistor  
                 1.56  
                 Ω 
               
               
                 ω res   
                 Resonant frequency  
                 38720 
                   rad / sec   
               
               
                   
               
            
           
         
       
     
     Powder cores are distributed air gap cores that can be suitable for high frequency applications, and compared to ferrite cores can exhibit higher saturation flux density, lower sensitivity to temperature changes and a softer saturation curve.  FIG. 3  shows the characteristic curve of a powder core. 
     For purposes of illustration, inductor parameters were calculated assuming operation limited to the linear region. Software used for the calculation was Magnetics Co. The linearity assumption was then relaxed and the calculations were repeated. The calculated inductor size, and corresponding flux density for the linear and relaxed assumption are shown in TABLE II. For example, the maximum flux density of the linear inductor L 1  is 0.44 Tesla, compared to 0.86 Tesla in the nonlinear implementation. The nonlinear inductor size, in terms of inductance, is about one tenth the size of the linear implementation. The simulation also calculates, for this example, a 40% reduction in core loss being provided by the non-linear implementation. The calculation applied the following Equation (24) which is a core loss density equation of the KoolM μ material:
 
 P   loss   =V× 120 B   2.09   F   1.46    Equation (24)
 
     where V is the core volume, B is the core maximum flux density, and F is the operating frequency of the inductor core. As indicated by these simulation results, in implementations according to this disclosure inductor capacity can be used more efficiently. Secondary benefits can include reduction in inductor size, as well as reduction in core losses compared to linear region inductors in equivalent scale inverter output filters. 
     
       
         
           
               
             
               
                 TABLE II 
               
             
            
               
                   
               
               
                 Linear and Non-Linear Inductor Parameters and Flux Density  
               
            
           
           
               
               
               
               
            
               
                   
                 L 1   
                   
                 L 1   
               
               
                 Parameters  
                 (Linear)  
                 L 2   
                 (Nonlinear) 
               
               
                   
               
               
                 Inductors  
                 8.64 mH 
                 0.123 mH 
                 8.64 mH 
               
               
                 Part number  
                 77102  
                 77440  
                 77908  
               
               
                 Material  
                 KoolM μ  
                 KoolM μ  
                 KoolM μ  
               
               
                 Winding number of turns  
                 226  
                 49  
                 646  
               
               
                 Core cross section (cm 2 )  
                 3.58  
                 1.977  
                 1.96  
               
               
                 Core magnetic path length (cm)  
                 24.3  
                 11.13  
                 22.1 
               
               
                 Number of stacks  
                 5  
                 1  
                 1  
               
               
                 Maximum flux density  
                 0.44  
                 0.2  
                 0.86  
               
               
                 (Tesla) 
               
               
                   
               
            
           
         
       
     
     In this example only L 1  was implemented as a non-linear inductor, because it is significantly larger than L 2  due to the attenuation factor of the particular LCL filter used for this example. The maximum flux density of nonlinear designation is chosen on the basis of 1.3 times of the inverter nominal current to guarantee reliable operation under unknown grid conditions which may cause inductor saturation. 
     A time variant current-dependent model of the inductor using the real B-H curve of the selected core is used in the simulation tests. This implementation is realized under Simulink software by using the block diagram depicted in  FIG. 4 . The model consists of a dependent current source, appropriate gain blocks, an integrator to derive the flux from voltage and a lookup table function which defines the relationship between flux density and magnetizing force which is depicted in  FIG. 3 . 
     To verify control structure with concurrent adaptive estimation of grid current and inverter-side inductor, a three phase 7 kW grid-connected inverter with output LCL filter is modeled. As discussed in the previous section, nonlinear inductors are model by the described block diagram of  FIG. 4 . The observer gains are K i =170, K c =7.7*10 −8 , K g =4.7×10 6 , α=1, γ=50000 and Γ=1.  FIG. 5A  shows the real and estimated value of current through the grid side inductor, and  FIG. 5B  shows the corresponding error.  FIG. 6A  shows the real and estimated value of the inductance of the inverter side inductor, and  FIG. 6B  shows the corresponding error. The results appear to verify convergence of the estimated current and inductor value. 
     TABLE III shows control parameters of conventional feedback methods, including a conventional weighted average method, and an implementation including an adaptive observer providing concurrent estimation of inductance of the inverter side inductor and current through the grid side inductor in accordance with one or more aspects of the present disclosure. The method in accordance with one or more aspects of the present disclosure includes an adaptive observer such as the  FIG. 2  non-linear adaptive observer block  213 , and a nonlinear inductor in the LCL-filter, such as the  FIG. 2  non-linear inverter-side inductor  206 . It can be seen that the THD improved larger control gains were used in implementations according to disclosed aspects. The cutoff-frequency in this case is 15 rad/sec. Using smaller harmonic compensator gains is another advantage of adaptive observer estimation based inverter control according to disclosed aspects. Use of harmonic compensators in accordance with disclosed aspects can provide lower phase margin and, therefore, smaller harmonic compensator gains may provide better stability margin. 
     It will be appreciated by persons of ordinary skill, upon reading this disclosure, that contrary to conventional controlling structures, grid-voltage feed forward is not needed in methods according to this disclosure. This can provide, among other benefits, robustness against grid voltage distortions. 
     
       
         
           
               
             
               
                 TABLE III 
               
             
            
               
                   
               
               
                 Controller parameters - nonlinear inductor and combinational feedback  
               
            
           
           
               
               
               
               
            
               
                   
                 Values-  
                 Values-  
                 Values-Nonlinear  
               
               
                   
                 Nonlinear  
                 Nonlinear  
                 inductor with  
               
               
                   
                 inductor with  
                 inductor with  
                 WAC feedback  
               
               
                   
                 conventional  
                 WAC feedback  
                 and adaptive  
               
               
                 Parameters  
                 feedback  
                 structure  
                 observer 
               
               
                   
               
            
           
           
               
               
               
               
            
               
                 PI proportional  
                 10  
                 20  
                 20  
               
               
                 gain  
                   
                   
                   
               
               
                 PI integral gain  
                 10 
                 20  
                 20  
               
               
                 Fundamental  
                 15  
                 145  
                 165  
               
               
                 harmonic  
                   
                   
                   
               
               
                 proportional gain  
                   
                   
                   
               
               
                 Fundamental  
                 588  
                 120  
                 170  
               
               
                 harmonic resonant  
                   
                   
                   
               
               
                 gain  
                   
                   
                   
               
               
                 5th harmonic  
                 260  
                 35  
                 14  
               
               
                 resonant gain  
                   
                   
                   
               
               
                 7th harmonic  
                 78  
                 40  
                 20  
               
               
                 resonant gain  
                   
                   
                   
               
               
                 Cutoff frequency  
                 5 ( rad / sec )  
                 15 ( rad / sec )  
                 15 ( rad / sec )  
               
               
                 THD  
                 12.61%  
                 5.41%  
                 3.42% 
               
               
                   
               
            
           
         
       
     
     TABLE IV depicts the individual harmonic current distortion of different structures according to IEEE-1547. 
     
       
         
           
               
             
               
                 TABLE IV 
               
             
            
               
                   
               
               
                 Individual harmonic current distortion  
               
            
           
           
               
               
               
               
            
               
                   
                   
                   
                 WAC Feedback  
               
               
                   
                 Conventional  
                 WAC Feedback  
                 with Adaptive  
               
               
                   
                 Method &amp;  
                 Method &amp;  
                 Observer &amp;  
               
               
                   
                 Nonlinear Ind.  
                 Nonlinear Ind.  
                 Nonlinear Ind. 
               
               
                   
               
               
                 h &lt; 11  
                 9.92%  
                 2.26%  
                 1.31%  
               
               
                 11 ≦ h &lt; 17  
                 0.75%  
                 0.55%  
                 0.60%  
               
               
                 17 ≦ h &lt; 23  
                 0.87%  
                 0.72%  
                 0.66%  
               
               
                 23 ≦ h &lt; 35  
                 0.89%  
                 1.21%  
                 0.50%  
               
               
                 35 ≦ h  
                 0.18% 
                 0.67%  
                 0.35%  
               
               
                 THD 
                 12.6%  
                 5.41%  
                 3.42% 
               
               
                   
               
            
           
         
       
     
     TABLE V shows the total harmonic distortion of different methods when the system sampling time is reduced to 5 μs. As can be seen, improvement can be provided by reducing sampling time. 
     
       
         
           
               
             
               
                 TABLE V 
               
             
            
               
                   
               
               
                 Total harmonic distortion of different methods under 5 μs sampling time  
               
            
           
           
               
               
               
            
               
                   
                 Sampling  
                 THD  
               
               
                 Method  
                 Time  
                 (%) 
               
               
                   
               
               
                 Conventional Method &amp; Nonlinear  
                 5 μs 
                  3.1%  
               
               
                 Ind.  
                   
                   
               
               
                 WAC Feedback Method &amp;  
                 5 μs 
                 1.46%  
               
               
                 Nonlinear Ind.  
                   
                   
               
               
                 WAC Feedback with Adaptive  
                 5 μs 
                 1.01%  
               
               
                 Observer &amp; Nonlinear Ind. 
               
               
                   
               
            
           
         
       
     
       FIG. 7  shows a block diagram of portions of one constructed and tested implementation of one example adaptive observer inverter controlling system according to one or more aspects of the present disclosure. Some of the  FIG. 7  blocks are labeled with two item numbers separated by a forward slash. In such instances the rightmost number is an item number of  FIG. 1  or  FIG. 2 , to assist the reader associating  FIG. 7  blocks of the above-described  FIGS. 1 and 2 . 
     Referring to  FIG. 7 , in the experimental setup, a digital signal processor (DSP) based digital controller  701  has been implemented using TMS230F28335 to control a three phase 7 kW inverter  703  (corresponding to the  FIG. 2  inverter block  203 ) tied through an LCL filter  711  (corresponding to the  FIG. 2  LCL filter  211 ) and transformer  706  with the grid  102 . The DSP controller  701  implements an adaptive observer  713 , configured according to disclosed aspects of the  FIG. 2  non-linear adaptive observer block  213 , and a PR controller  702 , configured according to disclosed aspects of the  FIG. 2  proportional resonant current control  202 . The DSP controller  701  also implements a space vector pulse width modulation (SVPWM) unit  712  for controlling the inverter block  703 . Control interface is provided by a driver  704 . Voltage and current signals are measured using a sensor board  706  and the 12-bit resolution of internal analogue-to-digital converter in the DSP controller  702 . 
     Experimental tests were carried out to verify performance at different operating conditions. The sampling time is set to 25 μs and the switching frequency is set to 15 kHz with 3 μs the dead-time. The inverter is connected to the grid by a three phase isolated Y/Y-connected transformer. The transformer leakage inductance is 1 mH per phase, which can represent a weak grid condition. 
       FIGS. 8A-8C  show the experimental results under different controlling strategies.  FIG. 8A  depicts the waveform of injected current to the grid under proportional resonant control using measured grid current feedback. As can be seen, the THD is about 7.64%, which may not be acceptable to be injected into the grid according to some standards.  FIG. 8B  shows WAC feedback method results under the nonlinear behavior of LCL filter inductors. As can be seen from the simulation results, the injected current THD has reduced to 6.24% compared with the grid current feedback strategy.  FIG. 8C  depicts the experimental results of the tested implementation of WAC feedback with concurrent adaptive estimation of grid current and nonlinear inductors according to aspects of this disclosure. Adding the online adaptive observer for precise β assessment has improved THD to 3.64%. 
       FIG. 9  shows the results of the grid current THD under different control strategies. Curve 1 is the system output under conventional control strategy with grid current feedback. Curve 2 depicts the results under the WAC feedback of inverter current and grid current and curve 3 is the results of using the WAC feedback strategy with an online adaptive observer in accordance with one or more aspects of this disclosure. From the current THD comparison results, it can be seen and appreciated that methods using online adaptive observer according to one or more disclosed aspects can provide practical and beneficial solutions to current control of the grid connected inverters, including utilizing nonlinear inductors in the output LCL filter. 
     While the foregoing has described what are considered to be the best mode and/or other examples, it is understood that various modifications may be made therein and that the subject matter disclosed herein may be implemented in various forms and examples, and that the teachings may be applied in numerous applications, only some of which have been described herein. It is intended by the following claims to claim any and all applications, modifications and variations that fall within the true scope of the present teachings. 
     Unless otherwise stated, all measurements, values, ratings, positions, magnitudes, sizes, and other specifications that are set forth in this specification, including in the claims that follow, are approximate, not exact. They are intended to have a reasonable range that is consistent with the functions to which they relate and with what is customary in the art to which they pertain. 
     The scope of protection is limited solely by the claims that now follow. That scope is intended to be as broad as is consistent with the ordinary meaning of the language that is used in the claims when interpreted in light of this specification and the prosecution history that follows, and to encompass all structural and functional equivalents. Notwithstanding, none of the claims are intended to embrace subject matter that fails to satisfy the requirement of Sections 101, 102, or 103 of the Patent Act, nor should they be interpreted in such a way. Any unintended embracement of such subject matter is hereby disclaimed. 
     Except as stated immediately above, nothing that has been stated or illustrated is intended or should be interpreted to cause a dedication of any component, step, feature, object, benefit, advantage, or equivalent to the public, regardless of whether it is or is not recited in the claims. 
     It will be understood that the terms and expressions used herein have the ordinary meaning as is accorded to such terms and expressions with respect to their corresponding respective areas of inquiry and study except where specific meanings have otherwise been set forth herein. 
     Relational terms such as first and second and the like may be used solely to distinguish one entity or action from another without necessarily requiring or implying any actual such relationship or order between such entities or actions. The terms “comprises,” “comprising,” and any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. An element preceded by “a” or “an” does not, without further constraints, preclude the existence of additional identical elements in the process, method, article, or apparatus that comprises the element. 
     The Abstract of the Disclosure is provided to allow the reader to quickly identify the nature of the technical disclosure. It is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the claims. In addition, in the foregoing Detailed Description, it can be seen that various features are grouped together in various examples for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that any claim requires more features than the claim expressly recites. Rather, as the following claims reflect, inventive subject matter lies in less than all features of a single disclosed example. Thus the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as a separately claimed subject matter.