Patent Publication Number: US-11658478-B2

Title: Grid connected inverter, and method for reducing grid frequency variation

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is based upon and claims priority to Japanese Patent Application No. 2020-048626 filed on Mar. 19, 2020, the entire contents of which are incorporated herein by reference. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present disclosure relates to a grid connected inverter which reduces a grid frequency variation by the so-called Virtual Synchronous Generator (VSG), and a method for reducing the grid frequency variation. 
     2. Description of the Related Art 
     Inverters that convert DC power output from renewable energy power systems, such as photovoltaic power systems or the like, into AC power and supply the AC power to power grids, are normally operated in synchronism with the grid frequency under Phase Locked Loop (PLL) control. As is well known, unlike a synchronous generator having a rotor, the inverter is a stationary type electrical apparatus including semiconductor power switching elements. Hence, the inverter is not provided with a function to reduce the grid frequency variation by an inertial force of the rotor. 
     Particularly if the number of renewable energy power systems increases, and the number of synchronous generators relatively decreases, the grid frequency is liable to vary significantly due to sudden changes in the load and output fluctuations of the renewable energy power systems. 
     Accordingly, a technique for stabilizing the system by providing a pseudo inertial force in the inverter, to realize a frequency variation reducing function of the synchronous generator, is referred to as the Virtual Synchronous Generator (VSG) control. Such a frequency variation reducing function is sometimes also referred to as the VSG function. 
       FIG.  1    is a schematic diagram illustrating a configuration of a grid connected inverter having the VSG function described above. 
     In  FIG.  1   , a synchronous generator  30  is connected to a power grid  10 , to supply AC power to a load  60 . In addition, a PLL circuit  20  is connected to the power grid  10 , and a grid frequency f g  detected by the PLL circuit  20  and a derivative df g /dt thereof are input to an inverter  50  via a signal line  21 . The inverter  50  includes a main circuit including power semiconductor switching elements for performing DC-AC conversion, and a control circuit for controlling the main circuit. 
     A DC input side of the inverter  50  is connected to a renewable energy power system, such as a photovoltaic power system  40  or the like, and an AC output side of the inverter  50  is connected to the power grid  10 . 
     The number of each of the synchronous generator  30  and the inverter  50  (and the photovoltaic power system  40 ) illustrated in  FIG.  1    is not limited to one. In a case where a plurality of synchronous generators and a plurality of inverters are connected to the power grid  10 , the synchronous generator  30  illustrated in  FIG.  1    corresponds to a group or collection of the plurality of synchronous generators, and the inverter  50  illustrated in  FIG.  1    corresponds to a group or collection of the plurality of inverters. 
     According to the configuration described above, the inverter  50  reduces the variation of the grid frequency f g  caused by sudden changes in the load  60  or the like, by the VSG function described below. 
     In other words, an active power (command) output from the inverter  50  is computed from the following formula (1) which is indicated as a formula 18 in “Grid Tied Converter with Virtual Kinetic Storage”, M. P. N van Wesenbeeck et al., 2009 IEEE Bucharest Power Tech Conference, June 28th-July 2nd (hereinafter simply referred to as “Non-Patent Document 1”), for example. 
     
       
         
           
             
               
                 
                   
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     In the formula (1), P inv  denotes the output active power (command) of the inverter  50 , P 0  denotes a set value of the output active power of inverter  50 , k vd  denotes a pseudo attenuation coefficient, k vi  denotes a pseudo inertia coefficient (or coefficient of pseudo inertia), f g  denotes an actual grid frequency, and f 0  denotes a reference frequency (for example, 50 [Hz] or 60 [Hz]) of the power grid  10 . 
     According to the formula (1), the output active power P inv  of the inverter  50  is computed by subtracting an adjusting amount from the set value P 0  of the output active power, where the adjusting amount is a sum of a pseudo attenuation component (a second term on the right side of the equal sign in the formula (1)) according to a grid frequency variation, that is, a frequency deviation (f g −f 0 ), and a pseudo attenuation component (a third term on the right side of the equal sign in the formula (1)) according to a derivative (df g /dt) of the grid frequency f g . By operating the inverter  50  using the value of this output active power P inv  as the output active power command, it is possible to reduce the variation of the grid frequency f g . 
     Conventionally, suitably selected fixed values are used for the pseudo attenuation coefficient k vd  and the pseudo inertia coefficient k vi . 
     In addition, Japanese Laid-Open Patent Publication No. 2019-3454 (hereinafter simply referred to as “Patent Document 1”), at paragraphs 0026-0033, FIGS. 1 and 2, or the like, describes a VSG function similar to that of the Non-Patent Document 1. More particularly, a frequency variation reducing amount corresponding to an inertial force of the synchronous generator is computed by a generator inertial force generating unit provided in the control circuit of the inverter, based on a phase error with respect to a grid voltage that is obtained by delaying a response of a PLL circuit upon a sudden change of the load, and the computed frequency variation reducing amount is added to an active power target value of the inverter, to improve the decreasing grid frequency. 
     According to the technique described in the Non-Patent Document 1, after the frequency deviation (f g −f 0 ) reaches a maximum value, a control is carried out so that the grid frequency f g  converges to the reference frequency f 0 . However, because the pseudo inertia coefficient k vi  is set to the fixed value, there is a problem in that a convergence speed of the grid frequency f g  is slow. 
     On the other hand, the Patent Document 1 does not describe a specific method for recovering the grid frequency f g , which varies when the sudden changes in the load occur, within a short period of time. 
     SUMMARY OF THE INVENTION 
     Accordingly, it is an object in one aspect of the embodiments to provide a grid connected inverter which reduces the grid frequency variation, and a method for reducing the grid frequency variation, which can reduce the grid frequency variation caused by sudden changes in the load, or an output variation of the renewable energy power system within a short period of time. 
     According to one aspect of the embodiments, a grid connected inverter connectable to a power grid having a synchronous generator connected thereto, and operable according to an output active power command generated by a virtual synchronous generator control function, to thereby reduce grid frequency variation, includes a main circuit including power semiconductor switching elements that turn on and off according to the output active power command, to perform DC-AC conversion; and a control circuit configured to control the main circuit, wherein the output active power command is represented by a sum of a set value of the output active power of the grid connected inverter, a value obtained by multiplying a pseudo attenuation coefficient to a frequency deviation between a grid frequency and a reference frequency, and a value obtained by multiplying a pseudo inertia coefficient to a derivative value of the grid frequency, and wherein the control circuit adjusts the pseudo inertia coefficient after the grid frequency reaches a maximum point of frequency variation to a value smaller than the pseudo inertia coefficient before the grid frequency reaches the maximum point of frequency variation. 
     A renewable energy power system may be connected as a DC power supply of the grid connected inverter. 
     The control circuit may adjust the pseudo inertia coefficient to reduce the grid frequency variation caused by sudden changes in a load connected to the power grid, or an output variation of a renewable energy power system connected as a DC power supply of the grid connected inverter. 
     The control circuit may set the value of the pseudo inertia coefficient after the grid frequency reaches the maximum point of frequency variation, based on an inertia of the synchronous generator before the grid frequency reaches the maximum point of frequency variation, and an inertia coefficient of the entire power grid after the grid frequency reaches the maximum point of frequency variation. 
     The control circuit may compute the inertia of the synchronous generator before the grid frequency reaches the maximum point of frequency variation, based on a mechanical input variation of the synchronous generator, a load variation that is independent of the grid frequency, an attenuation coefficient of the synchronous generator, the pseudo attenuation coefficient, the pseudo inertia coefficient, the frequency deviation between the grid frequency and the reference frequency, the derivative value of the grid frequency, and the pseudo inertia coefficient, at a time before the grid frequency reaches the maximum point of frequency variation. 
     According to another aspect of the embodiments, a method for reducing grid frequency variation in a grid connected inverter, connectable to a power grid having a synchronous generator connected thereto, includes operating the grid connected inverter according to an output active power command, generated by a virtual synchronous generator control function, and represented by a sum of a set value of the output active power of the grid connected inverter, a value obtained by multiplying a pseudo attenuation coefficient to a frequency deviation between a grid frequency and a reference frequency, and a value obtained by multiplying a pseudo inertia coefficient to a derivative value of the grid frequency; and adjusting the pseudo inertia coefficient after the grid frequency reaches a maximum point of frequency variation to a value smaller than the pseudo inertia coefficient before the grid frequency reaches the maximum point of frequency variation, to generate the output active power command after the grid frequency reaches the maximum point of frequency variation. 
     The operating may operate the grid connected inverter according to the output active power command represented by a sum of the set value of the output active power of the grid connected inverter, and the value obtained by multiplying the pseudo attenuation coefficient to the frequency deviation between the grid frequency and the reference frequency, when the frequency deviation between the grid frequency and the reference frequency is smaller than a first threshold value, and the output active power command at a time after the grid frequency reaches the maximum point of frequency variation, when the frequency deviation between the grid frequency and the reference frequency exceeds first threshold value. 
     The adjusting may adjust the pseudo inertia coefficient to reduce the grid frequency variation caused by sudden changes in a load connected to the power grid, or an output variation of a renewable energy power system connected as a DC power supply of the grid connected inverter. 
     Other objects and further advantages of the present invention will now be apparent from the description set forth below in conjunction with the drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    is a schematic diagram illustrating a configuration of a grid connected inverter having the VSG function. 
         FIG.  2    is a diagram illustrating a configuration of a grid connected inverter according to one embodiment of the present invention. 
         FIG.  3    is a diagram for explaining a maximum point of frequency variation in one embodiment of the present invention. 
         FIG.  4    is a schematic diagram illustrating a recovery state of a grid frequency for a case where a pseudo inertia coefficient is adjusted according to one embodiment of the present invention, and a case where the pseudo inertia coefficient is fixed as in a conventional case. 
         FIG.  5    is a flow chart illustrating a method of computing an output active power according to a grid frequency variation in one embodiment of the present invention. 
         FIG.  6    is a waveform diagram illustrating an example of the grid frequency variation in one embodiment of the present invention. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Embodiments of the present invention will be described below, by referring to the drawings. 
       FIG.  2    is a diagram illustrating a configuration of a grid connected inverter  150  according to one embodiment of the present invention, which is connectable to the power grid  10 . The grid connected inverter  150  includes an inverter main circuit  100  and a control circuit  500 . In  FIG.  2   , those parts that are the same as those corresponding parts in  FIG.  1    are designated by the same reference numerals, and a description thereof may be omitted. 
     The control circuit  500  generates an output active power command P inv , and controls the inverter main circuit  100  based on the output active power command P inv . The control circuit  500  includes subtractors  71  and  74 , a differentiator  72 , multipliers  75  and  76 , and an adder  73 . 
     In  FIG.  2   , the grid frequency fg detected by the PLL circuit  20  is input to the subtractor  71  which computes a deviation (f g −f 0 ) from the reference frequency f 0 . The multiplier  75  multiplies the deviation (f g −f 0 ) by the pseudo attenuation coefficient k vd , and inputs a result of this multiplication to the adder  73 . 
     In addition, the grid frequency fg detected by the PLL circuit  20  is also input to the multiplier  76  which multiplies the grid frequency f g  by the pseudo inertia coefficient k vi  which is a variable, and a result of this multiplication is input to the differentiator  72  which differentiates the result of this multiplication. An output of the differentiator  72  is input to the adder  73 . 
     The adder  73  adds the two inputs from the multiplier  75  and the differentiator  72 , respectively, and an output of the adder  72  is input to the subtractor  74 . The subtractor  74  subtracts the output of the adder  73  from a set value P 0  of an output active power of the grid connected inverter  150 , and obtains the output active power command P inv . This output active power command P inv  is input to a current reference value generator  80 . 
     In addition to the output active power command P inv , a reactive power command Q inv  is separately input to the current reference value generator  80 . The current reference value generator  80  generates d-axis and q-axis current command values i dref  and i qref , based on these inputs thereto, and outputs the d-axis and q-axis current command values i dref  and i qref  to a current controller  90 . 
     On the other hand, a DC power supply  101  and a DC intercondenser  102 , corresponding to a renewable energy power system, are connected in parallel on a DC input side of the inverter main circuit  100 . In this example, the inverter main circuit  100  is formed by a three-phase inverter main circuit having a known configuration including the power semiconductor switching elements. An AC output side of the inverter main circuit  100  is connected to the power grid  10 . 
     Output currents i a , i b , and i c  of each of the phases of the inverter main circuit  100  are detected by a current detector  103 , and converted into d-axis and q-axis currents i d  and i q  by a coordinate transformation circuit  91 . The d-axis and q-axis currents i d  and i q  are input to the current controller  90 . In addition, output voltages v a , v b , and v c  of each of the phases of the inverter main circuit  100  are converted into d-axis and q-axis voltages v d  and v q  by a coordinate transformation circuit  92 . The d-axis and q-axis voltages v d  and v q  are input to the current controller  90 . 
     A phase angle ρ, used for performing the coordinate transformation, is input to each of the coordinate transformation circuits  91  and  92 , and a coordinate transformation circuit  93  which will be described later. 
     The current controller  90  generates d-axis and q-axis modulation commands m d  and m q , based on the d-axis and q-axis current command values i dref  and i qref  from the current reference value generator  80 , the d-axis and q-axis currents i d  and i q  from the coordinate transformation circuit  91 , and the d-axis and q-axis voltages v d  and v q  from the coordinate transformation circuit  92 . The d-axis and q-axis modulation commands m d  and m q  are converted into modulation signals m a , m b , and m c  of each of the three phases, by the coordinate transformer  93  and a modulation signal generator  94 . These modulation signals m a , m b , and m c  are input to the inverter main circuit  100 . In the inverter main circuit  100 , the semiconductor switching elements are turned on and off by a Pulse Width Modulation (PWM) control or the like, using the modulation signals m a , m b , and m c , and DC power of the DC power supply  101  is converted into AC power and supplied to the power grid  10 . 
     An equation of motion (oscillation or swing equation) for a model of the synchronous generator  30  may be represented by the following formula (2). 
     
       
         
           
             
               
                 
                   
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     In the formula (2), M denotes an inertia of the synchronous generator  30  (of reduced model for a case where the inertias of a plurality of synchronous generators  30  connected to the power grid  10  are integrated into one inertia), f m  denotes a rotation frequency (of the rotor) of the synchronous generator  30 , ΔP m  denotes a mechanical input variation of synchronous generator  30 , and ΔP e  denotes an electrical output variation of the synchronous generator  30 . 
     From the formula (1) described above, a variation ΔP inv  of the active power output from the grid connected inverter  150  can be expressed by the following formula (3). 
     
       
         
           
             
               
                 
                   
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     In addition, as represented by the following formula (4), a sum of the active power variation ΔP inv  and the electrical output variation ΔP e  of the synchronous generator  30 , is equal to an active power variation ΔP L  of the load  60 . Further, the active power variation ΔP L  is a sum of a load variation ΔP L0  that is independent of the grid frequency f g , and a product of an attenuation coefficient D and the frequency deviation (f g −f 0 ) of the synchronous generator  30 .
 
Δ P   inv   +ΔP   e   =ΔP   L   =ΔP   L0   +D ( f   g   −f   0 )  [formula (4)]
 
     If f m ≈f g  in the formula (2) described above, an extension equation of motion of the synchronous generator  30  can be represented by the following formula (5), based on the formulas (2) through (4). 
     
       
         
           
             
               
                 
                   
                     
                       
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     In the formula (5) described above, (M+k vi ) denotes an inertia coefficient of the entire power grid  10  (including the inertia output by the grid connected inverter  150 ), and (D+k vd ) denotes an attenuation coefficient of the entire power grid  10 . 
     In this embodiment, in a case where the grid frequency f g  decreases from the reference frequency f 0  due to the load variation, for example, the pseudo inertia coefficient k vi  is increased to increase the inertia coefficient (M+k vi ) of the entire power grid  10  of the formula (5) to a large value, during a period of time in which the grid frequency f g  separates more from an estimated convergence value f ∞ , in a region where the grid frequency f g  is smaller than the convergence value f ∞ . On the other hand, the pseudo inertia coefficient k vi  is decreased to decrease the inertia coefficient (M+k vi ) of the entire power grid  10  of the formula (5) to a small value, during a period of time in which the grid frequency f g  approaches more toward the convergence value f ∞ , in the region where the grid frequency f g  is smaller than the convergence value f ∞ . As a result, it is possible to increase a recovery speed of the grid frequency f g . The inertia coefficient (M+k vi ) of the entire power grid  10 , and the inertia M of the synchronous generator  30 , have positive values. 
     In other words, in the case where the grid frequency f g  varies in the region where the grid frequency f g  is smaller than the convergence value f ∞ , the pseudo inertia coefficient k vi  is increased until the grid frequency f g  reaches a maximum point of frequency variation, and the pseudo inertia coefficient k vi  is decreased during the period of time in which the grid frequency f g  approaches the convergence value f ∞  after reaching the maximum point of frequency variation. 
     By adjusting the pseudo inertia coefficient k vi  of the grid connected inverter  150  in this manner, it is possible to recover the grid frequency f g  in a short period of time compared to the conventional case where the pseudo inertia coefficient k vi  is set to a fixed value. 
     Because the derivative value (df g /dt) of the grid frequency f g  is zero at the maximum point of frequency variation described above, the formula (5) can be expressed by the following formula (6). In the formula (6), f min  denotes a minimum frequency of the maximum point of frequency variation, and t 2  denotes a time when the maximum point of frequency variation occurs.
 
Δ P   L0 =−( D+k   vd )( f   min   −f   0 )+Δ P   m ( t   2 )  [formula (6)]
 
       FIG.  3    is a diagram for explaining the maximum point of frequency variation in one embodiment of the present invention.  FIG.  3    illustrates the minimum frequency f min , the time t 2 , or the like of the maximum point of frequency variation, as well as a time t 1  before the time t 2 , a corresponding frequency f 1  at the time t 1 , and a rate of frequency change, (df 1 /dt), which is a derivative of the frequency f 1  and is indicated by a one-dot chain line. 
     Next, the convergence value f ∞  of the grid frequency f g , required to adjust the pseudo inertia coefficient k vi  as described above, may be estimated in the following manner. 
     First, a relationship represented by the following formula (7) exists between an amount of change, ΔF, of the rotation frequency of the synchronous generator  30 , and the load variation ΔP L0  described above. In the formula (7), s denotes the Laplace operator, and G(s) denotes the transfer function.
 
Δ F ( s )= G ( s )Δ P   L0 ( s )  [formula (7)]
 
     From the formula (7) and the final value theorem, the convergence value f ∞  of the grid frequency f g  can be estimated by the following formula (8). 
     
       
         
           
             
               
                 
                   
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       FIG.  4    is a schematic diagram illustrating a recovery state of the grid frequency f g  for a case where the pseudo inertia coefficient k vi  is adjusted according to this embodiment, and a case where the pseudo inertia coefficient k vi  is fixed as in the conventional case. As may be seen from  FIG.  4   , the time required to restore the grid frequency f g  to fall within a predetermined range centered on the convergence value f ∞  in this embodiment, as indicated by a solid line, is shorter than that of the conventional case indicated by a dotted line. This shorter time required to restore the grid frequency f g  in this embodiment contributes to a quicker stabilization of the grid frequency f g  when compared to the conventional case. 
     Next, a method of computing the output active power of the grid connected inverter  150  according to the variation state of the grid frequency f g , will be described with reference to  FIG.  5    and  FIG.  6   . 
     A process of a flow chart illustrated in  FIG.  5    is performed by the control circuit  500  of the grid connected inverter  150  illustrated in  FIG.  2   , at a predetermined control period. The control circuit  500  controls the inverter main circuit  100  of the grid connected inverter  150 . In this example, k denotes a current time, and (k−1) denotes a time one control period before the current time k. 
     Further,  FIG.  6    is a waveform diagram illustrating an example in which the grid frequency f g  varies, that is, the grid frequency variation occurs, in one embodiment of the present invention. 
     In the following, a case where the grid frequency fg decreases will be described in this embodiment. However, in a modification of this embodiment, for example, the present disclosure is similarly applicable to a case where the grid frequency f g  increases. 
     First, in  FIG.  5   , a determination is made to determine whether or not an absolute value of the deviation between the grid frequency f g  at the current time k and the reference frequency f 0  (for example, 50 [Hz]) is smaller than the first preset threshold value c 1 , in step S 1 . Step S 1  is performed to determine whether or not the grid frequency f g  decreased (or increased) significantly. 
     If the grid frequency f g  did not decrease significantly and the decision result in step S 1  is YES, the active power command of a normal mode is output to the grid connected inverter  150  in step S 8 . On the other hand, if the grid frequency f g  did decrease significantly and the decision result in step S 1  is NO, the process advances to step S 2 . 
     The pseudo inertia coefficient k vi  of the grid connected inverter  150  is set to a first pseudo inertia coefficient k vi1  and the active power command P inv (k) at the current time k is computed from the formula (1) and input to the grid connected inverter  150 , in step S 2 . 
     Next, a determination is made to determine whether or not an absolute value of a difference between the grid frequency f k  at the current time k and the grid frequency f k-1  at the time (k−1) is smaller than a second threshold value c 2 , in step S 3 . Step S 3  is used to determine whether or not the grid frequency f g  reached a maximum point of frequency variation illustrated in  FIG.  6   . 
     A relationship of magnitudes of the first threshold value c 1  and the second threshold value c 2  is c 1 &gt;&gt;c 2 , as illustrated in  FIG.  6   . For example, c 1 =0.3, and c 2 =0.001. 
     The process returns to step S 2  if the grid frequency f g  did not reach the maximum point of frequency variation and the decision result in step S 3  is NO. On the other hand, if the grid frequency f g  did reach the maximum point of frequency variation and the decision result in step S 3  is YES, the convergence value f ∞  is estimated by the method described above, in step S 4 . 
     Next, a determination is made to determine whether or not the sign (or polarity) of f k −f k-1  changed, in step S 5 . Step S 5  is used to determine a point where restoring of the grid frequency f g  begins. 
     If the sign of f k −f k-1  did not change and the decision result in step S 5  is NO, the active power command P inv (k) at the current time k is computed from the formula (1) and input to the grid connected inverter  150 , in step S 9 , similar to step S 2 . 
     On the other hand if the sign of f k −f k-1  changed and the decision result in step S 5  is YES, a determination is made to determine whether or not the grid frequency f g , within a range smaller than the convergence value f ∞ , changes in a direction approaching the convergence value f ∞ , in step S 6 . 
     If the grid frequency f g  changes in the direction approaching the convergence value f ∞  and the decision result in step S 6  is YES, the pseudo inertia coefficient k vi  of the grid connected inverter  150  is set to a second pseudo inertia coefficient k vi2  smaller than the first pseudo inertia coefficient k vi1  (k vi2 &lt;k vi1  and 0&lt;k vi2 ), and the active power command P inv (k) at the current time k is computed from the formula (1) and input to the grid connected inverter  150 , in step S 10 . 
     On the other hand, if the grid frequency f g  does not change in the direction approaching the convergence value f ∞  and the decision result in step S 6  is NO, the active power command P inv (k) at the current time k is computed from the formula (1) and input to the grid connected inverter  150 , in step S 7 , similar to steps S 2  and S 9 . 
     As long as the grid frequency f g  is not restored to the reference frequency f 0 , either step S 7  or step S 10  of  FIG.  5    is performed. In addition, when the grid frequency f g  is restored to the reference frequency f 0 , the processes of step S 1  and subsequent steps are successively repeated. 
     As described above, by performing the processes described above according to the variation state of the grid frequency f g , the recovery speed can be increased for the case where the grid frequency f g  changes in the direction approaching the convergence value f ∞  after reaching the maximum point of frequency variation. 
     The operation to restore the grid frequency f g  to the reference frequency f 0  is performed by the inertial force of the synchronous generator  30 . However, in many cases, the adjusting function utilizing this inertial force does not activate immediately even when the grid frequency f g  varies. Accordingly, the VSG function according to this embodiment is effective in reducing the variation of the grid frequency f g  during a time until the adjusting function of the synchronous generator  30  becomes effective. 
     Next, a method for determining a numerical range for resetting the pseudo inertia coefficient k vi  when setting the second pseudo inertia coefficient k vi2  (the pseudo inertia coefficients k vi =k vi2 , used on or after the time t 2  corresponding to the maximum point of frequency variation), to compute the active power command P inv (k) in step S 10  of  FIG.  5   , will be described. 
     If a desired inertia coefficient of the entire power grid  10  after resetting the pseudo inertia coefficient k vi  has a desired value M′ (M′=M+k vi2  and M′&gt;0), this desired value M′ needs to be smaller than the inertia coefficient (M+k vi1 ) of the entire power grid  10  before resetting the pseudo inertia coefficient k vi  in order to improve the recovery speed of the grid frequency f g . For this reason, the following inequality (9) stands.
 
 k   vi2   &lt;k   vi1  (when 0&lt; k   vi1 ,0&lt; k   vi2 )
 
− M&lt;k   vi2  (when  k   vi2 &lt;0)  [formula (9)]
 
     However, according to the inequality (9) described above, the pseudo inertia coefficient k vi  cannot be reset if the inertia M of the synchronous generator  30  is unknown. 
     Accordingly, the numerical range for resetting the pseudo inertia coefficient k vi  may be obtained by the following method. 
     First, by substituting the formula (6) into the formula (5), the following formula (10) can be obtained. In the formula (10), t 1  denotes the time before the maximum point of frequency variation (time t 2 ), and f 1  denotes the grid frequency at this time t 1 , as illustrated in  FIG.  2   , and (df 1 /dt) denotes the frequency variation rate. 
     
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           M 
                           + 
                           
                             k 
                             vi 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           d 
                           ⁢ 
                           
                             f 
                             1 
                           
                         
                         
                           d 
                           ⁢ 
                           t 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           D 
                           + 
                           
                             k 
                             vd 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             f 
                             1 
                           
                           - 
                           
                             f 
                             0 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       Δ 
                       ⁢ 
                       
                         
                           P 
                           m 
                         
                         ( 
                         
                           t 
                           1 
                         
                         ) 
                       
                     
                     - 
                     
                       Δ 
                       ⁢ 
                       
                         P 
                         
                           L 
                           ⁢ 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     formula 
                     ⁢ 
                         
                     
                       ( 
                       10 
                       ) 
                     
                   
                   ] 
                 
               
             
           
         
       
     
     The following formula (11) can be obtained by rearranging the formula (10). In the formula (11), k vi  of the second term on the right side of the equal sign denotes the pseudo inertia coefficient (corresponding to k vi1  in steps S 2 , S 7 , and S 9  of  FIG.  5   ) before the maximum point of frequency variation. 
     The inertia M of the synchronous generator  30  is obtained from the formula (11), and the pseudo inertia coefficient k vi  is reset within the numerical range of the second pseudo inertia coefficient k vi2  defined in the inequality (9). Hence, even if the second pseudo inertia coefficient k vi2  assumes a negative value, it is possible to reset the second pseudo inertia coefficient k vi2  which satisfies k vi2 +M&gt;0. 
     
       
         
           
             
               
                 
                   M 
                   = 
                   
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                             
                               P 
                               m 
                             
                             ( 
                             
                               t 
                               1 
                             
                             ) 
                           
                         
                         - 
                         
                           Δ 
                           ⁢ 
                           
                             P 
                             
                               L 
                               ⁢ 
                               0 
                             
                           
                         
                         - 
                         
                           
                             ( 
                             
                               D 
                               + 
                               
                                 k 
                                 
                                   v 
                                   ⁢ 
                                   d 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 f 
                                 1 
                               
                               - 
                               
                                 f 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           d 
                           ⁢ 
                           
                             f 
                             1 
                           
                         
                         
                           d 
                           ⁢ 
                           t 
                         
                       
                     
                     - 
                     
                       k 
                       vi 
                     
                   
                 
               
               
                 
                   [ 
                   
                     formula 
                     ⁢ 
                         
                     
                       ( 
                       11 
                       ) 
                     
                   
                   ] 
                 
               
             
           
         
       
     
     Further, in the case where the inertia M of the synchronous generator  30  is unknown, the pseudo inertia coefficient k vi  may be reset by computing the inertia M of the synchronous generator  30  at the electric power company operating the power grid, and transmitting the value of the formula (9) described above, including the computed value of the inertia M, to the grid connected inverter  150 , for example. 
     According to the embodiments and modifications described above, it is possible to restore the grid frequency, which varied, within a short period of time, and contribute to the quick stabilization of the grid frequency, by varying the pseudo inertia coefficient after the time when the varied grid frequency reaches the maximum point of frequency variation to a predetermined value, to generate the output active power command of the grid connected inverter. Accordingly, it is possible to provide a grid connected inverter which reduces the grid frequency variation, and a method for reducing the grid frequency variation, which can reduce the grid frequency variation caused by sudden changes in the load, or an output variation of the renewable energy power system within a short period of time. 
     The present invention is not limited to the embodiments and modifications specifically disclosed above, and various variations, modifications, substitutions, combinations with other techniques, or the like, may be made without departing from the scope of the present invention. 
     All examples and conditional language recited herein are intended for pedagogical purposes to aid the reader in understanding the invention and the concepts contributed by the inventor to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions, nor does the organization of such examples in the specification relate to a showing of the superiority and inferiority of the invention. Although the embodiments of the present invention have been described in detail, it should be understood that the various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention.