Abstract:
The present invention provides for the integration of distributed renewable energy sources/storages utilizing a cascaded DC-AC inverter, thereby eliminating the need for a DC-DC converter. The ability to segment the energy sources and energy storages improves the maintenance capability and system reliability of the distributed generation system, as well as achieve wide range reactive power compensation. In the absence of a DC-DC converter, single stage energy conversion can be achieved to enhance energy conversion efficiency.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to currently co-pending U.S. Provisional Patent Application No. 61/495,458, entitled “System and Method for Single-Phase, Single-Stage Grid-Interactive Converter”, filed on Jun. 10, 2011, and incorporated herein by reference. 
    
    
     STATEMENT OF GOVERNMENT INTEREST 
     This invention was made with Government support under DE-FC26-07NT43221 awarded by the Department of Energy. The Government has certain rights in the invention. 
    
    
     BACKGROUND OF THE INVENTION 
     Distributed generation, also commonly referred to as on-site generation, dispersed generation, embedded generation, decentralized generation, decentralized energy or distributed energy, generates electricity from multiple small energy sources. The distributed power generation system comprising the multiple small energy sources is interconnected with the same transmission grid as the larger central power generation stations. Various technical and economic issues arise in the integration of these distributed generation resources into the grid, such as power quality, voltage stability, presence of harmonics and reliability. 
     Grid-interactive converters/inverters with energy storage system (ESS) for distributed power generation (DG) systems have been gaining popularity in response to the requirements for energy source/storage diversity, environmental concerns, desired cost reduction, improvements in efficiency, etc. One main challenge in DG systems is determining how to improve the efficiency, reduce the cost and improve the power quality of the current DG systems known in the art. 
     Traditionally, AC link systems and DC link systems have been employed in distributed generation systems known in the art. In the DC link system, as shown in  FIG. 1(   a ), multiple DC-DC converters are used to interface different energy sources/storages. In order to achieve the DC-AC energy conversion, the DC-DC converters are usually connected to one inverter with a common DC bus. As a result, the DC link system configuration still requires DC-DC and DC-AC conversion stages. Limited switching frequency on the common inverter results in the requirement for a large AC filter and large electrolyte capacitors, which seriously impacts the overall system efficiency, cost and lifetime of the inverter. Furthermore, in order to achieve desired wide range reactive power compensation, the DC capacitor and inverter must be oversized to generate the possible AC output voltage. 
     As shown with reference to  FIG. 1 , in the traditional AC link system, multiple DC-DC converters and DC-AC inverters are used to interface different energy sources/storages as shown in  FIG. 1(   b ). Though the real and reactive power can be distributed flexibly between different energy sources/storages utilizing the AC link system, the system requires multiple energy conversion stages. Moreover, due to the common AC bus used in the AC link system, the power flow control must to be carefully designed, particularly to address the needs of weak power systems. In an AC link system, the overall system cost will increase due to the need for multiple converters and inverters and large passive components. In addition, the transformer utilized in both the DC link and the AC link systems described above will increase the overall system size and reduce the power density. 
     Accordingly, what is needed in the art is a distributed generation (DG) system that exhibits high power density, high power efficiency, high power quality and high system reliability. 
     SUMMARY OF INVENTION 
     The present invention provides for the integration of distributed renewable energy sources/storages utilizing a cascaded DC-AC inverter, thereby eliminating the need for a DC-DC converter. The ability to segment the energy sources/storages will improve the maintenance capability and system reliability. In the absence of a DC-DC converter, single stage energy conversion can be achieved. 
     A system in accordance with the present invention integrates scalable cascaded architecture, an advanced Discrete Fourier Transform (DFT) Phase Locked Loop (PLL) method, a hybrid modulation technique and a sophisticated power allocation strategy to achieve wide range reactive power compensation, improve the system power density, and enhance energy conversion efficiency. 
     In a specific embodiment of the invention, an inverter for a distributed generation system is provided. The inverter may include a main DC to AC inverter to be coupled to the at least one distributed energy source, the main DC to AC inverter to provide a real power of the inverter and to provide a first portion of a reactive power of the inverter and at least one auxiliary DC to AC inverter in cascade with the main DC to AC inverter, each of the at least one auxiliary DC to AC inverters to be coupled to one of each of the at least one energy storages and each of the at least one auxiliary DC to AC inverters to provide a remaining portion of the reactive power of the inverter, wherein the sum of the first portion of the reactive power from the main inverter and each of the remaining portions of the reactive power from each of the auxiliary inverters equals a total reactive power of the inverter for the distributed generation system. 
     In a particular embodiment, the main DC to AC inverter and the auxiliary DC to AC inverters may be H-bridge cells. 
     In an additional embodiment, the main DC to AC inverter may switch at a fundamental frequency and the auxiliary DC to AC inverters may switch at a pulse-width-modulated frequency. 
     In operation, a method for inverting DC to AC in a distributed generation system, is provided by the present invention. The method may include calculating a reactive power allocation coefficient, providing a real power and a first portion of a reactive power of the inverter from a main DC to AC inverter coupled to the at least one distributed energy source, wherein the first portion of the reactive power is dependent upon the reactive power allocation coefficient and providing a remaining portion of the reactive power of the inverter from each of at least one auxiliary DC to AC inverters coupled in cascade to the main DC to AC inverter and coupled to each of the at least one energy storages, wherein the remaining portion of the reactive power is dependent upon the reactive power allocation coefficient and wherein the sum of the first portion of the reactive power from the main inverter and the remaining portion of the reactive power from each of the auxiliary inverters equals a total reactive power of the inverter. 
     The present invention can achieve high power efficiency, high power density, high power quality and high system reliability as a result of the single stage energy conversion, transformer-free structure, wide range reactive power compensation, multilevel AC output with equivalent high frequency, possible low frequency for each switching device, and redundant capability. Additionally, the incorporation of the present invention into future smart grid implementations is promising. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a fuller understanding of the invention, reference should be made to the following detailed description, taken in connection with the accompanying drawings, in which: 
         FIG. 1  illustrates the reported grid-connected inverters with ESS: (a) DC link system; and (b) AC link system as is known in the prior art. 
         FIG. 2  illustrates a proposed single-stage grid-interactive inverter with ESS in accordance with an embodiment of the present invention. 
         FIG. 3  illustrates a proposed DG System with ES: (a) Topology; and (b) Real and reactive power allocation between DES and ES in accordance with an embodiment of the present invention. 
         FIG. 4  illustrates the power allocation vector diagram between DES and ES in accordance with an embodiment of the present invention. 
         FIG. 5  illustrates the proposed DFT PLL method in accordance with an embodiment of the present invention. 
         FIG. 6  illustrates the experimental results of DFT PLL, CH1 is input signal and CH2 is output signal: (a) input signal is square wave and output signal is sinusoidal wave, (b) input signal is triangular wave and output signal is sinusoidal wave, (c) the input signal frequency increases from 60 Hz to 70 Hz, (d) the input signal frequency decreases from 60 Hz to 50 Hz, in accordance with an embodiment of the present invention. 
         FIG. 7  illustrates the proposed dual-stage DFT PLL algorithm in accordance with an embodiment of the present invention. 
         FIG. 8  illustrates the proposed Reactive Power Allocation Strategy in accordance with an embodiment of the present invention: (a) voltage reference generation for main inverter and auxiliary inverters; (b) voltage synthesis of v main  and v auxi12    
         FIG. 9  illustrates the operation conditions of single-stage unit with different k when P and Q changes from 0 to 1.0 pu, in accordance with an embodiment of the present invention. 
         FIG. 10  illustrates the operation conditions of single-stage unit with different k when P=1.0 pu. and Q changes from 0 to 1.0 pu, in accordance with an embodiment of the present invention. 
         FIG. 11  illustrates the control system block for the DG system in grid-connected mode in accordance with an embodiment of the present invention. 
         FIG. 12  illustrates the Grid current and total harmonic distortion as P_grid=0.5 pu, Q_grid=1.0 pu with different ‘k’, in accordance with an embodiment of the present invention. 
         FIG. 13  illustrates the two capacitor voltages in self-startup and grid-connected mode in accordance with an embodiment of the present invention. 
         FIG. 14  illustrates the real and reactive power delivered to grid in grid-connected mode in accordance with an embodiment of the present invention. 
         FIG. 15  illustrates the real and reactive power from cascaded inverter, main inverter and auxiliary inverter with ‘k’=0.7 in accordance with an embodiment of the present invention. 
         FIG. 16  illustrates the hardware prototype of the proposed system in accordance with an embodiment of the present invention. 
         FIG. 17  illustrates the experimental results of real and reactive power delivered to grid P_grid, Q_grid, and their reference P*_grid, Q*_grid in accordance with an embodiment of the present invention. 
         FIG. 18  illustrates the experimental results of power allocation: (a) Real power allocation of cascaded inverter, main inverter and auxiliary inverter, P_inv, P_main, P_auxi, (b) Reactive power allocation of cascaded inverter, main inverter and auxiliary inverter, Q_inv, Q_main, Q_auxi, in accordance with an embodiment of the present invention. 
         FIG. 19  illustrates the experimental results of two capacitor voltages, Vcap1 and Vcap2: (a) Self-startup; (b) Grid-connected mode, in accordance with an embodiment of the present invention. 
         FIG. 20  illustrates Table I showing the system parameters in accordance with an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     With reference to  FIG. 2 , an embodiment of the present invention in a distributed generation (DG) system  10  having distributed energy sources (DES)  15  and energy storages (ES)  20  is illustrated. The output of the distributed energy sources (DES)  15  and the energy storages (ES)  20  are coupled to a cascaded DC-AC inverter  25 . The output of the cascaded DC-AC inverter  25  is coupled to the grid  35  through an appropriate filter  30 . 
     In an exemplary embodiment of the present invention, one distributed energy source (DES)  15  and two energy storages (ES)  20   a ,  20   b  are interfaced to the inverter  25  utilizing three cascaded H-bridge cells  40   a ,  40   b ,  40   c , as illustrated with reference to  FIG. 3(   a ). In the single-phase single-stage DG system  10  in accordance with the exemplary embodiment of the present invention, as shown in  FIG. 3(   a ), the “main” inverter cell  4   a  is coupled to the distributed energy source  15  and the “auxiliary” inverter cells  4   b ,  4   c  are coupled to the energy storages  20   a ,  20   b . In a particular embodiment, the energy storages  20   a ,  20   b  may be ultra-capacitors. In additional embodiments, the energy storages  20   a ,  20   b  may include battery storage, flow batteries, flywheel, superconducting magnetic energy storage (SMES), compressed air energy storage (CAES), or other various energy storages solutions known in the art. The different energy sources and energy storages integrating with the cascaded inverter can be selected based upon a specific application. In a particular embodiment, the voltage ratio between V dc , associated with the main inverter cell  4   a , and V cap1  and V cap2 , associated with the auxiliary inverter cells  4   b ,  4   c , may be 2:1:1, respectively. The “main” and “auxiliary” inverters may switch at the fundamental and a pulse-width modulated (PWM) frequency, respectively. 
     The real and reactive power allocation between the distributed energy sources  15  and the energy storages  20   a ,  20   b  is shown with reference to  FIG. 3(   b ). P_grid  120  and Q_grid  125  are representative of the real power P and the reactive power Q delivered to the grid  35 . P_main  70  and Q_main  80  represent the real power P and reactive power Q generated from the main inverter cell  40   a . Q_auxi 1   90  and Q_auxi 2   100  represent the reactive power Q from the auxiliary inverter cells  40   b ,  40   c . As such, in the present invention, the main inverter cell  40   a  provides all of the real power  120  and a portion of the reactive power Q  125  to the grid and the auxiliary inverters  40   b ,  40   c  provide the remaining portion of the reactive power  125 . P_auxi 1   50  and P_auxi 2   60  are delivered from the main inverter  4   a  to the auxiliary inverter cells  4   b ,  4   c  to charge the energy storages  20   a ,  20   b  during start-up, compensate the for power loss and maintain the energy storages  20   a ,  20   b  voltage during grid-connected mode. In addition, the low-order harmonic voltages  105  generated by the main inverter  4   a  resulting from the fundamental switching frequency are cancelled by the equivalent negative harmonic voltages  110 ,  115  generated from the auxiliary inverters  4   b ,  4   c . The distributed power generation system  10  is able to operate in both stand-alone mode and grid-connected mode through a static transfer switch (STS)  130 . 
     One advantage of the cascaded inverter structure of the present invention is that the DC voltage of each inverter cell can be reduced. However, the reduced DC voltage will affect the reactive power generation capability. In order to achieve wide range reactive power compensation, it is desired that the reactive power is provided by all the inverter cells instead of a single inverter cell. The desired distribution of the reactive power among the inverter cells  4   a ,  4   b ,  4   c  is based upon a proposed reactive power allocation (RPA) strategy. 
     A vector diagram is shown in  FIG. 4  illustrating the reactive power distribution between the distributed energy source  15  and the energy storages  20   a ,  20   b . The rotation frequency of the pq frame is the system frequency.            ,           and           and are the vectors of i L1 , v inv , and v g  respectively where the latter can be identified with reference to  FIG. 3(   a ).           is the fundamental component of the main inverter  4   a  output voltage.           and           is the q-axis and p-axis component of           respectively in the vector diagram. Since the real power is entirely provided by the main inverter  4   a , the p-axis component of           is the same as that of          . k is defined as reactive power allocation coefficient (RPAC). α is the phase shift angle between           and          . β is the angle between           and          . δ is the phase shift between           and          . θ is the switching angle of main inverter.
     In grid-connected mode, the real and reactive power delivered to the grid  35 , as well as grid voltage, are known. The average real and reactive power delivered to the grid  35  can be given by: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           P_grid 
                           = 
                           
                             
                               
                                 
                                   V 
                                   g 
                                 
                                 ⁢ 
                                 
                                   V 
                                   inv 
                                 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 L 
                               
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             δ 
                           
                         
                       
                     
                     
                       
                         
                           Q_grid 
                           = 
                           
                             
                               
                                 V 
                                 g 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 L 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     V 
                                     inv 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   δ 
                                 
                                 - 
                                 
                                   V 
                                   g 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where V g  is the amplitude of the grid voltage; V inv  is the amplitude of the cascaded inverter output voltage; ω is the fundamental frequency; L=L 1 +L 2  is the total filter inductor. The filter capacitor is neglected. 
     According to (1), the phase shift between grid voltage and cascaded inverter output voltage can be calculated as: 
     
       
         
           
             
               
                 
                   δ 
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           P_grid 
                           × 
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                         
                         
                           
                             Q_grid 
                             × 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ω 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                           
                           + 
                           
                             V 
                             g 
                             2 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     The average real and reactive power delivered to grid can also be represented in (3): 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           P_grid 
                           = 
                           
                             
                               
                                 
                                   V 
                                   g 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                               
                               2 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             β 
                           
                         
                       
                     
                     
                       
                         
                           Q_grid 
                           = 
                           
                             
                               
                                 
                                   V 
                                   g 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                               
                               2 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             β 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where I L1  is the amplitude of the inductor current. 
     So, the angle between grid voltage and inductor current can be obtained in (4): 
     
       
         
           
             
               
                 
                   β 
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         Q_grid 
                         P_grid 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     The phase shift between cascaded inverter output voltage and inductor current can be obtained by adding (2) to (4) as:
 
α=β+δ  (5)
 
     Based on (1), the V inv  can be derived in (6): 
     
       
         
           
             
               
                 
                   
                     V 
                     inv 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 Q_grid 
                                 × 
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 L 
                               
                               
                                 V 
                                 g 
                               
                             
                             + 
                             
                               V 
                               g 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               P_grid 
                               × 
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               L 
                             
                             
                               V 
                               g 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     The Fourier series expansion of the quasi-square-wave of main inverter output voltage is given by: 
     
       
         
           
             
               
                 
                   
                     v 
                     main 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           3 
                           , 
                           5 
                           , 
                           … 
                         
                         ∞ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             4 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               V 
                               dc 
                             
                           
                           
                             n 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               n 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               n 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   - 
                                   φ 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         v 
                         
                           main 
                           ⁢ 
                           _ 
                           ⁢ 
                           F 
                         
                       
                       + 
                       
                         v 
                         
                           main 
                           ⁢ 
                           _ 
                           ⁢ 
                           h 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Where φ is phase-shift angle between             and          , V dc  is the main inverter dc input voltage,
               v     main   ⁢   _   ⁢   F       =         4   ⁢           ⁢     V   dc       π     ⁢     cos   ⁡     (   θ   )       ⁢     sin   ⁡     (       ω   ⁢           ⁢   t     -   φ     )               
is the fundamental component of v main ,
 
               v     main   ⁢   _   ⁢   h       ⁢       ∑       n   =   1     ,   3   ,   5   ,   …     ∞     ⁢           ⁢         4   ⁢           ⁢     V   dc         n   ⁢           ⁢   π       ⁢     cos   ⁡     (     n   ⁢           ⁢   θ     )       ⁢     sin   ⁡     (     n   ⁡     (       ω   ⁢           ⁢   t     -   φ     )       )                 
is the harmonic component of v main .
 
     According to  FIG. 4 , the transient fundamental component of the main inverter output voltage can be given as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           v 
                           
                             main 
                             ⁢ 
                             _ 
                             ⁢ 
                             F 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               v 
                               invp 
                             
                             + 
                             
                               kv 
                               invq 
                             
                           
                           = 
                           
                             
                               
                                 V 
                                 inv 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 •sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       t 
                                     
                                     - 
                                     β 
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 V 
                                 inv 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 α•cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       t 
                                     
                                     - 
                                     β 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             V 
                             inv 
                           
                           ⁢ 
                           
                             
                               
                                 
                                   ( 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   t 
                                 
                                 - 
                                 β 
                                 + 
                                 γ 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Where, 
     
       
         
           
             
               
                 γ 
                 = 
                 
                   arcsin 
                   ⁡ 
                   
                     ( 
                     
                       
                         k 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                       
                         
                           
                             
                               ( 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     ) 
                   
                 
               
               , 
               
                 
                   v 
                   invp 
                 
                 = 
                 
                   
                     V 
                     inv 
                   
                   ⁢ 
                   cos 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     α•sin 
                     ⁡ 
                     
                       ( 
                       
                         
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                         - 
                         β 
                       
                       ) 
                     
                   
                 
               
               , 
               
                 
                   v 
                   invq 
                 
                 = 
                 
                   
                     V 
                     inv 
                   
                   ⁢ 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     α•cos 
                     ⁡ 
                     
                       ( 
                       
                         
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                         - 
                         β 
                       
                       ) 
                     
                   
                 
               
             
             ⁢ 
             
               
 
             
           
         
       
     
       FIG. 4  shows that the selection of θ with respect to the reactive power allocation and θ is derived in (9): 
     
       
         
           
             
               
                 
                   θ 
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                     [ 
                     
                       
                         
                           V 
                           inv 
                         
                         
                           V 
                           dc 
                         
                       
                       × 
                       
                         π 
                         4 
                       
                       × 
                       
                         
                           
                             
                               ( 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     where V dc  is the main inverter DC input voltage. 
     Based on the vector analysis of  FIG. 4 , extracting the real and reactive components of v inv  is essential to establish the desired power allocation. In order to extract the real and reactive components of v inv , a Discrete Fourier Transform (DFT) Phase-Locked Loop (PLL) method is proposed, as shown in  FIG. 5 . 
     The DFT PLL method of the present invention can extract the phase, frequency and amplitude information from any signal. The DFT PLL method of the present invention is capable of extracting the fundamental component from any input signal without damping. This is in contrast to the prior art in which the harmonic components of the input signal degrades the PLL performance. As such, the proposed DFT PLL method is immune to the degradation effect resulting from the harmonic components. Additionally, in the DFT PLL method of the present invention, the controlled DC component is only related to the angle of the input signal which allows the control system for the PLL to be applicable within a wide range, and moreover the amplitude change of the input signal will not have an impact on the DFT PLL performance. This is in contract to the prior art, in which one DC component is related to amplitude and angle of the input signal and is controlled to be zero and used to achieve the PLL. As such, the proposed DFT PLL method is immune to harmonics and signal amplitude which may affect the PLL performance. As illustrated in  FIG. 5 , any signal including harmonic components can be expressed as follows: 
     
       
         
           
             
               
                 
                   
                     u 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         u 
                         o 
                       
                       + 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           ∞ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           [ 
                           
                             
                               u 
                               k 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   + 
                                   
                                     θ 
                                     k 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         u 
                         o 
                       
                       + 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           ∞ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 u 
                                 ky 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 u 
                                 kx 
                               
                               ⁢ 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Where u 0  is the dc component; ω is the fundamental frequency, 
     
       
         
           
             
               
                 u 
                 kx 
               
               = 
               
                 
                   2 
                   T 
                 
                 ⁢ 
                 
                   
                     ∫ 
                     0 
                     T 
                   
                   ⁢ 
                   
                     
                       u 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           kω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ⅆ 
                       t 
                     
                   
                 
               
             
             ; 
             
                 
             
             ⁢ 
             
               
                 u 
                 ky 
               
               = 
               
                 
                   2 
                   T 
                 
                 ⁢ 
                 
                   
                     ∫ 
                     0 
                     T 
                   
                   ⁢ 
                   
                     
                       u 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         k 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                       ) 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ⅆ 
                       t 
                     
                   
                 
               
             
             ; 
             
               
                 θ 
                 k 
               
               = 
               
                 
                   tg 
                   
                     - 
                     1 
                   
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       u 
                       ky 
                     
                     / 
                     
                       u 
                       kx 
                     
                   
                   ) 
                 
               
             
             ; 
             
               T 
               = 
               
                 2 
                 ⁢ 
                 
                   π 
                   / 
                   
                     ω 
                     . 
                   
                 
               
             
           
         
       
     
     For PLL, only the fundamental component is of concern. The fundamental component is converted to a DC component by (9): 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               u 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             · 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   t 
                                 
                                 ) 
                               
                             
                           
                           = 
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         u 
                                         o 
                                       
                                       + 
                                       
                                         
                                           ∑ 
                                           
                                             k 
                                             = 
                                             1 
                                           
                                           ∞ 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           [ 
                                           
                                             
                                               u 
                                               k 
                                             
                                             ⁢ 
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   
                                                     k 
                                                     ⁢ 
                                                     
                                                         
                                                     
                                                     ⁢ 
                                                     ω 
                                                     ⁢ 
                                                     
                                                         
                                                     
                                                     ⁢ 
                                                     t 
                                                   
                                                   + 
                                                   
                                                     θ 
                                                     k 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                     ] 
                                   
                                   · 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         t 
                                       
                                       ) 
                                     
                                   
                                 
                                 = 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       u 
                                       1 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         θ 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       = 
                                       1 
                                     
                                     ∞ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     [ 
                                     
                                       
                                         u 
                                         kt 
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               k 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               ω 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               t 
                                             
                                             + 
                                             
                                               φ 
                                               k 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               u 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             · 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   t 
                                 
                                 ) 
                               
                             
                           
                           = 
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         u 
                                         o 
                                       
                                       + 
                                       
                                         
                                           ∑ 
                                           
                                             k 
                                             = 
                                             1 
                                           
                                           ∞ 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           [ 
                                           
                                             
                                               u 
                                               k 
                                             
                                             ⁢ 
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   
                                                     k 
                                                     ⁢ 
                                                     
                                                         
                                                     
                                                     ⁢ 
                                                     ωt 
                                                   
                                                   + 
                                                   
                                                     θ 
                                                     k 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                     ] 
                                   
                                   · 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         t 
                                       
                                       ) 
                                     
                                   
                                 
                                 = 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       u 
                                       1 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         θ 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       = 
                                       1 
                                     
                                     ∞ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     [ 
                                     
                                       
                                         u 
                                         kt 
                                         ′ 
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               k 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               ω 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               t 
                                             
                                             + 
                                             
                                               φ 
                                               k 
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Where u 1  is the desired fundamental component; θ 1  is the phase angle of fundamental component; u k  is the kth order harmonic component; θ k  is the phase angle of the kth order harmonic component; 
     And then the DC component is extracted by (12) 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             u 
                             
                               1 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               p 
                             
                           
                           = 
                           
                             
                               
                                 2 
                                 T 
                               
                               ⁢ 
                               
                                 
                                   ∫ 
                                   0 
                                   T 
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       u 
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     · 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           ω 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           t 
                                         
                                         ) 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                             
                             = 
                             
                               
                                 u 
                                 1 
                               
                               · 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     θ 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             u 
                             
                               1 
                               ⁢ 
                               q 
                             
                           
                           = 
                           
                             
                               
                                 2 
                                 T 
                               
                               ⁢ 
                               
                                 
                                   ∫ 
                                   0 
                                   T 
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       u 
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     · 
                                     
                                       cos 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           ω 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           t 
                                         
                                         ) 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                             
                             = 
                             
                               
                                 u 
                                 1 
                               
                               · 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     θ 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Based on (10)-(12), the θ 1  can be derived in (13):
 
θ 1 =tan −1 ( u   1q   /u   1p )  (13)
 
     Once the θ 1  is extracted from u(t), it can be treated as the regulated error signal as shown in  FIG. 5 . The proportional integral (PI) controller can control θ 1  to be zero in real time to achieve the PLL of u(t). 
     Experimental results are shown in  FIG. 6  to verify the performance of the proposed DFT PLL method under different conditions.  FIG. 6(   a ) shows the performance of the DFT PLL when the input signal is a square wave, which includes all-order harmonic components. With a square wave input, the output signal is the sinusoidal signal extracted from the square wave. It can be seen that the sinusoidal output signal can track the input signal within three cycles.  FIG. 6(   b ) illustrates the performance of the DFT PLL when the input signal is a triangular wave, which includes abundant low-order harmonic components. When the input signal is a triangular wave, the output signal is the sinusoidal signal extracted from the triangular wave. In the beginning, the output signal is opposite in phase with the input signal, but the sinusoidal signal can still track the input signal within five cycles.  FIGS. 6(   c ) and ( d ) show the dynamic response when the frequency of the input sinusoidal signal increases from 60 Hz to 70 Hz and decreases from 60 Hz to 50 Hz, respectively. As illustrated with reference to  FIG. 6 , it is obvious that the output sinusoidal signal can track the input signal within only one cycle. 
     As mentioned above, the output voltage of the cascaded inverter  25  dominates the power allocation between DES  15  and ES  20   a ,  20   b  because the same inductor current i L1  flows through the cascaded inverter  25 . Accordingly, a preferred dual-stage DFT PLL approach is developed to synchronize the inductor current i L1  and extract the active and reactive components of v inv  relative to i L1 , as shown with reference to  FIG. 7 . In the first stage, v g , which is u(t) in  FIG. 5 , is used for the PLL signal. The phase angle ωt corresponding to v g  is obtained and used for the synchronization of i L1  and v inv . In the second stage, i L1  and v inv  are synchronized with v g  by exploiting the same approach as shown in (11)-(13). The phase angles β and δ of i L1  and v inv  relative to v g  can be derived and then synthesize the α, which is the phase shift angle between i L1  and v inv . After obtaining the voltage amplitude V inv , α and ωt, the v invp  and v invq , which are the components of the v s  in phase with i L1  and leading 90° to i L1  respectively, can be obtained and matched with (8). The derived v invq  combing with the selected k may then be used to conduct the reactive power from distributed energy sources  15  and energy storages  20   a ,  20   b.    
     The reactive power allocation strategy with hybrid fundamental and PWM control, in accordance with the present invention, is illustrated with reference to  FIG. 8 . In order to effectively achieve the desired power allocation, it is vital to accurately calculate the v main , v auxi1  and v auxi2 . In  FIG. 8(   a ), the i L1  v g , and v inv  are firstly measured and sent to the “Dual-stage DFT PLL” module as shown in  FIG. 7 , which generates the α, V inv , v invp  and v invq . The detected PV voltage V dc , α, V inv , and k dedicate together to generate the switching angle of main inverter θ presented in (9). After obtaining θ, the main inverter  40   a  outputs the quasi-square-wave voltage v main , including real power component v invp , reactive power component kv invq  and harmonic component v invh  as described in (7)-(8). The two auxiliary inverters  40   b ,  40   c  generate the voltage v auxi12 , which is separated into v auxi1  and v auxi2 . Therefore, v main  and v auxi12  synthesize the desired inverter voltage v inv .  FIG. 8(   b ) shows the voltage synthesis of v main  and V auxi12 . The θ indicates the relationship between v main  and its fundamental component v main     —     F =v invp +kv invq . The v auxi12  includes two components: (1−k)v invq  and −v main     —     h . The v main     —     h  from main inverter  40   a  can be canceled by −v main     —     h  from auxiliary inverters  40   b ,  40   c  to improve the cascaded inverter output voltage quality. The auxiliary inverters  40   b ,  40   c  also supply the rest of the reactive power component (1−k)v invq  of v inv . In this way, the reactive power can be distributed between main  40   a  and auxiliary inverter cells  40   b ,  40   c.    
     In the present invention, k decides the weighted q-axis voltage component in the main inverter cell  40   a  and auxiliary inverter cells  40   b ,  40   c , and therefore determines the reactive power in each cell. k can be varied from 0˜1 to generate a wide range Q_grid under a required P_grid. However, inappropriate k will cause duty cycle saturation or over modulation resulting in degraded power quality. In addition, the appropriate range of k varies with P_grid. The design guidelines for selecting k are described as follows when P_grid and Q_grid vary from 0˜1 pu, respectively. 
     Firstly, in order to avoid the duty cycle saturation, the assigned q component of v main , i.e., kv invq , should be no bigger than the available maximum magnitude of the q component of v main , which is calculated as √{square root over ((4V dc /π) 2 −(V inv  cos α) 2 )}{square root over ((4V dc /π) 2 −(V inv  cos α) 2 )}. That is the fundamental switching angle θ shown in (9) should be controlled within [0, π/2]. The inequality condition in (14) is therefore derived and needs to be satisfied as the first limitation condition defined as Cond.1: 
     
       
         
           
             
               
                 
                   0 
                   ≤ 
                   
                     
                       
                         kV 
                         inv 
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                     
                     
                       
                         
                           
                             ( 
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   V 
                                   dc 
                                 
                                 / 
                                 π 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 V 
                                 inv 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   ≤ 
                   1 
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     Secondly, in order to avoid over modulation, the peak value of v auxi1 +v auxi2  should be no bigger than V cap1 +V cap2 , as the second limitation condition defined as Cond.2, which is shown in (15). 
     
       
         
           
             
               
                 
                   
                     
                       max 
                       ⁢ 
                       
                          
                         
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 k 
                               
                               ) 
                             
                             ⁢ 
                             
                               V 
                               invq 
                             
                           
                           - 
                           
                             V 
                             
                               main 
                               ⁢ 
                               _ 
                               ⁢ 
                               h 
                             
                           
                         
                          
                       
                     
                     
                       ( 
                       
                         
                           V 
                           
                             cap 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         + 
                         
                           V 
                           
                             cap 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                       
                       ) 
                     
                   
                   ≤ 
                   1 
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     An appropriate k can be obtained based on (14-15). Four cases with different k values are shown in  FIGS. 9(   a - 1 ,  b - 1 ,  c - 1 , and  d - 1 ) which illustrates Cond.1 variation as P_grid and Q_grid varies from 0 to 1.0 pu.  FIGS. 9(   a - 2 ,  b - 2 ,  c - 2 , and  d - 2 ) illustrates Cond.2 variation with different k under a wide P_grid and Q_grid variation range. As shown in the 3D plot, when ‘k’=0, 0.4 and 1, (14) and (15) can not be satisfied simultaneously. As such, the ‘k’=0.7 is a proper reactive power allocation coefficient under wide real and reactive power ranges. 
     In order to verify the above analysis further, the grid real power P_grid is fixed to 1.0 pu with varied grid reactive power Q_grid and ‘k’ as depicted in  FIG. 10 . The ‘k’ changes from 0 to 1 by 0.1. As shown in  FIG. 10(   a ), the allowable range of Cond.1 is in the interval [0, 1]. In this case, the reactive power allocation with ‘k=0.9’ and ‘k=1.0’ can not meet duty cycle saturation limitation condition in the whole grid reactive power.  FIG. 10(   b ) shows the allowable range of Cond.2 is in the interval [ 0 ,  1 ]. If the ‘k≦0.6, the over modulation limitation condition can not be satisfied in the whole grid reactive power. 
       FIG. 11  shows the control system design of proposed distributed power generation system  10  operating in grid-connected mode. The grid current reference i g     —     ref  is generated by controlling P_grid and Q_grid to track their reference. The grid current reference i g     —     ref  is generated by the ‘current reference generator’ module. An inner proportional (P) controller is cascaded with an outer loop proportional plus resonant (PR) controller to control the grid current i g  to track i g     —     ref  with zero steady-state error. The “reactive power allocation” module as shown in  FIG. 8  receives v inv     —     ref  from current controller and generates the references of v main , v auxi1  and v auxi2 . The “capacitor voltage balance control” module is developed to achieve two capacitor voltages, V cap1  and V cap2 , to track the reference V cap     —     ref . The capacitor voltage balance controller receives the inductor current i L1  and generates the inverter voltage compensation components v cmp1  and v cmp2 . The sum v cmp =v cmp1 +v cmp2  is added to v main , and sent to the hybrid fundamental and PWM control module to generate the main inverter voltage. Similarly, the v auxi1 −v cmp1  and v auxi2 −v cmp2  are sent to the control module respectively to generate the corresponding auxiliary inverter voltages. 
     The performance of the proposed reactive power allocation (RPA) strategy is firstly tested in simulation. The system parameters are shown in Table I. In order to investigate the effect of ‘k’ on power quality, the grid current and its FFT results with different ‘k’ are compared as real and reactive power delivered to the grid are fixed to 0.5 pu and 1.0 pu, respectively in  FIG. 12 . As shown in  FIG. 12(   a ), the grid current with ‘k’=0.4 has a total harmonic distortion (THD) of 16.37%. However, THD can be decreased to 0.47% as ‘k’=0.7 in  FIG. 12(   b ). It is clear that ‘k’=0.7 is a proper reactive power allocation coefficient. 
       FIG. 13-FIG .  15  show the simulation results with ‘k’=0.7.  FIG. 13  shows the two capacitor voltages, V cmp1  and V cmp2 , in self-startup and grid-connected mode. Before the normal operation of the DG system, the two capacitors are charged to the 0.5 pu by self-startup. After 3 s, the DG system operates in grid-connected mode. It can be seen from  FIG. 13  that the capacitors are kept approximately constant at 0.5 pu while the real and reactive power delivered to the grid varies. 
       FIG. 14  illustrates the dynamic response to step changes in the real power to grid reference P*_grid from 0 to 1.0 pu at 4 s and back to 0.5 pu at 6 s, reactive power to grid reference Q*_grid from 0 to 1.0 pu at 5 s and back to 0.5 pu at 7 s, and then to −0.5 pu at 8 s. It is obvious that the real and reactive power, P_grid and Q_grid, can quickly and smoothly track their references. 
       FIG. 15  depicts the real and reactive power from the cascaded inverter  25 , main inverter  40   a  and auxiliary inverter  40   b ,  40   c , while the P_grid and Q_grid vary. At steady-state, the real power provided by the main inverter  40   a  P_main is equal to the real power from the cascaded inverter  25  P_inv as shown in  FIG. 15(   a ). The real power from auxiliary inverter  40   b ,  40   c  P_auxi changes only during power transition to keep the capacitor voltage constant. The P_inv is a bit more than P_grid due to the system loss. As shown in  FIG. 15(   b ), the reactive power from the cascaded inverter  25  Q_inv is also more than Q_grid due to the system loss. The ratio between the reactive power from main inverter  40   a  Q_main and auxiliary inverters  40   b ,  40   c  Q_auxi is always 0.7 as P_grid and Q_grid vary. 
     A 3.5 kW hardware prototype has been built in the laboratory as shown in  FIG. 16 . In this exemplary embodiment of the present invention, the IGBT FMG2G100US60 has been chosen as the main inverter switch operating at the fundamental frequency and the MOSFET SUP85N15-21 has been used in the auxiliary inverters operating at 2.5 kHz. The control algorithms are implemented in dSPACE DS1104 controller. The experimental results at 350 W with ‘k’=0.7 are presented with reference to  FIG. 17  through  FIG. 19 . 
       FIG. 17  shows the response of real and reactive power to the grid at step changes in grid-connected mode. P*_grid increases from 20 W to 200 W and back to 100 W. Q*_grid increases from 0 to 200 VAR and then drops to 100 VAR, finally reaches to −100 VAR. The real and reactive power distribution between the main inverter  40   a  and auxiliary inverters  40   b ,  40   c  is shown in  FIG. 18 . It can be seen from  FIG. 18(   a ) that the P_inv is very close to P_main at steady state. P_auxi changes a small amount only to keep capacitor voltage constant. As anticipated, the ratio between Q_main and Q_auxi is maintained to be 0.7 in (b). Q_inv is more than Q_grid due to the system reactive power loss.  FIG. 19(   a ) shows that the two capacitor voltages charge to 25V quickly during the startup process. Capacitor voltages are stabilized at 25V in grid-connected mode as the power varies as shown in  FIG. 19(   b ). 
     In accordance with the present invention, a novel DFT PLL technique is developed to separate the real and reactive power between different energy sources/storages. The DFT PLL technique is immune to harmonics, signal amplitude, frequency and phase which may affect the PLL performance. The hybrid modulation technique and sophisticated power allocation strategy integrating the proposed DFT PLL method are developed to achieve wide range reactive power compensation and optimized real power distribution. 
     It will be seen that the advantages set forth above, and those made apparent from the foregoing description, are efficiently attained and since certain changes may be made in the above construction without departing from the scope of the invention, it is intended that all matters contained in the foregoing description or shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense. 
     It is also to be understood that the following claims are intended to cover all of the generic and specific features of the invention herein described, and all statements of the scope of the invention which, as a matter of language, might be said to fall therebetween. Now that the invention has been described,