Patent Publication Number: US-2016238669-A1

Title: Power System State Estimation Device and Power System State Estimation Method for Same

Description:
TECHNICAL FIELD 
     The present invention relates to a power system state estimation device and a power system state estimation method for the same to estimate a state of a power system. 
     BACKGROUND ART 
     In a power system, or a power distribution system, it is important to perceive a state of the entire power system for properly controlling and managing the system even when a power flow fluctuates due to variation in loads or the like. As a technique for perceiving the state in the entire power system, Patent Document 1, for example, discloses a technique, based on measured values such as a voltage and a current by sensors installed in the power distribution system and a power flow calculation with system configuration data, of calculating a correction amount for the system state with an estimated values of measurement errors and the power flow calculation to accurately estimate real values of the system state. 
     PRIOR ART DOCUMENT 
     Patent Document 
     Patent document 1: Japanese Patent Application Publication No. 2008-154418 
     SUMMARY OF THE INVENTION 
     Problem to be Solved by the Invention 
     The technique disclosed in Patent Document 1 described above allows, based on the measured values such as the voltage and the current by the sensors installed in the power distribution system and the power flow calculation with the system configuration data, for calculating the correction amount for the system state with the estimated values of the measurement errors and the power flow calculation to accurately estimate the real values of the system state. 
     However, the above technique is assumed to have an observable system in which sensors are redundant in number relative to amounts of system state, which causes a problem such that the technique cannot be applied to estimation of a state amount in an unobservable system in which sensors are insufficient in number relative to the amounts of system state. 
     Accordingly, the present invention is to solve such a problem, and an object of the present invention is to estimate and perceive a state amount with an estimated range of the state amount even in an unobservable subsystem where only a part of the state amount is measured, in addition to perceiving a state amount in an observable subsystem in an power system. 
     Means for Solving the Problem 
     To solve the problem described above, the present invention is configured as follows. 
     In short, the present invention is to provide a power system state estimation device for estimating a state amount of a power system having: a calculation unit which executes calculations on the power system; a system division unit which is inputted with system information and a measured value of the state amount of the power system to divide the power system into an observable subsystem and an unobservable subsystem with reference to a calculation result of he calculation unit; a state estimation unit which is inputted with the system information and the measured value of the state amount to calculate an estimated value of the state amount in the observable subsystem divided by the system division unit with reference to the calculation result of the calculation unit; and a state range estimation unit which is inputted with the system information, the measured value of the state amount and a constraint value of the state amount of the power system to calculate an estimated range of the state amount in the unobservable subsystem divided by the system division unit. 
     Other devices/units will be described in a detailed description of an embodiment. 
     Advantageous Effects Of The Invention 
     According to the present invention, a state amount can be estimated and perceived with an estimated range of a state amount even in an unobservable subsystem where only a part of the state amount is measured, in addition to perceiving a state amount in an observable subsystem in an power system. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram showing a configuration example of a power system state estimation device according to an embodiment of the present invention and its relation with a power system, information acquisition devices for perceiving a state of the power system, and peripheral devices of the subject device. 
         FIG. 2  is a diagram showing a configuration example of respective elements in the power system according to the embodiment of the present invention and an example of node numbers assigned to respective elements; 
         FIG. 3  is a diagram showing notations such as parameters of an SVR and an SVC in a power system control system according to the embodiment of the present invention; 
         FIG. 4  is a schematic diagram showing a calculation method for the maximum value of each element in a redundant solution from the maximum value of a solution norm in the redundant solution; 
         FIG. 5  is a schematic diagram showing a calculation method for a value range of the redundant solution; 
         FIG. 6  is a flowchart showing an algorithm of a third method with which the value range of the redundant solution according to the embodiment of the present invention is limited; 
         FIGS. 7A and 7B  show an example of a screen display on a display device, in which  FIG. 7A  shows representative values of the state amount and a range of the state amount in each node, and  FIG. 7B  shows a system diagram of the power system; 
         FIG. 8  is a table showing an example of a system log which a recording device outputs; and 
         FIG. 9  is an exemplary flowchart showing processing of the power system state estimation device according to the embodiment of the present invention. 
     
    
    
     EMBODIMENTS FOR CARRYING OUT THE INVENTION 
     Hereinafter, a description will be given of an embodiment of the present invention with reference to the drawings. 
       FIG. 1  is a diagram showing a configuration example of a power system state estimation device  100  according to an embodiment of the present invention, and its relation with a power system  101  and information acquisition devices (sensor  102 , communication line  103 ) which perceive a state of the power system, and peripheral devices (display device  111 , recording device  112 ) of the power system state estimation device  100 . 
     In  FIG. 1 , a state amount such as a voltage and a current of the power system  101  is measured by the sensor  102  so as to be outputted via the communication line  103  to the power system state estimation device  100  as a measured value of the state amount. The power system state estimation device  100  estimates the state amount of the power system  101  to output the estimated result to the display device  111  and the recording device  112 . It is noted that respective signals from a state estimation unit.  106 , a system information database  108  and a state range estimation unit  107  cross with one another at some points on the way to the display device  111  or the recording device  112 . Those points are shown by black boxes, which indicate that respective signals described above cross with one another but are never connected. 
     Next, a configuration of the power system state estimation device  100 , the display device  111  and the recording device  112  will be described in order. 
     In addition, the power system  101  will be described in detail by way of an example, in a description of a mathematical model of the power system used in the power system state estimation device  100 . 
     It is noted that the sensor  102  and the communication line  103  are common ones, and detailed descriptions thereof are omitted. 
     &lt;&lt;Power System State Estimation Device&gt;&gt; 
     The power system state estimation apparatus  100  is configured to include a measured value database  104 , a system division unit  105 , a state estimation unit  106 , a state range estimation unit  107 , a system information database  108 , a constraint condition database  109 , and a calculation unit  110 . 
     The measured value database  104  records the measured value of the state amount in the power system obtained via the communication line  103 . 
     The system division unit  105  is inputted with system information and the measured value of the state amount of the power system  101  to divide the power system into an observable subsystem and an unobservable subsystem with reference to a calculation result of the calculation unit  110 . 
     The state estimation unit  106  is inputted with the system information and the measured value of the state amount of the power system  101  to calculate an estimated value of the state amount in the observable subsystem divided by the system division unit  105 . 
     The state range estimation unit  107  is inputted with the system information and the measured value of the state amount of the power system  101  and a constraint value of the state amount of the power system  101  to calculate an estimated range of the state amount in the unobservable subsystem divided by the system division unit  105 , with reference to the calculation result of the calculation unit  110 . 
     The system information database  108  records the system information about the configuration of the power system  101 , such as line impedance and system topology. 
     The constraint condition database  109  records the constraint value of the state amount cDf the power system  101 . 
     The calculation unit  110  executes calculations on the power system. Further, the calculation unit  110  calculates about the system division unit  105 , the state estimation unit  106  and the state range estimation unit  107 , to assist a function and an operation of each device ( 105 ,  106 ,  107 ) 
     Still further, the power system state estimation unit  100 , having the configuration described above, is inputted with the measured value of the state amount, estimates the state amount of the power system and outputs the estimated value of the state amount and the estimated range of the state amount. 
     It is noted that the power system state estimation device  100  and functions, operations, calculation methods and the like of respective devices constituting the device  100  will be described later in more detail in the description of the mathematical model to be described later. 
     &lt;&lt;Display Device and Recording Device&gt;&gt; 
     In  FIG. 1 , the power system state estimation device  100  is connected with the display device  111  and the recording device  112  as peripheral devices. 
     The display device  111  outputs the estimated value of the state amount and the estimated range of the state amount as output of the power system state estimation device  100  on a screen in numerical values or in a graph. Also, in conjunction with a system diagram showing system information, the display device  111  displays the estimated value of the state amount and the estimated range of the state amount. It is noted that a function of the display device  111  will be described later in detail. 
     The recording device  112  records, in the same manner, the estimated value of the state amount and the estimated range of the state amount as the system log. Further, the recording device  112  outputs the record to a recording medium it is noted that a function of the recording device  112  will be described later in detail. 
     Still further, in  FIG. 1 , the display device  111  and the recording device  112  have been shown as devices connected outside the power system state estimation device  100 , but they may be arranged as parts in the power system state estimation device  100 . 
     &lt;&lt;Mathematical Model of Power System&gt;&gt; 
     Next, the function of the power system state estimation device  100  ( FIG. 1 ) will be described by presenting the power system in a mathematical model. It is noted that the system information database  108  ( FIG. 1 ) records the system topology of the power system, using the mathematical model to be described below. 
     Referring to  FIG. 2  showing an example of the power system, the system topology of the power system will be described. First, a node number presenting the system topology of the power system, an adjacency matrix and a hierarchical matrix will be described in order. 
     &lt;&lt;Node Number&gt;&gt; 
       FIG. 2  is a diagram showing a configuration example of respective elements in the power system  101  according to the embodiment of the present invention and an example of node numbers assigned to respective elements. 
       FIG. 2  shows a state in which (AC) voltage is transmitted from a power transmission end  201  to a power distribution line  211  in the power system  101 . 
     The power distribution line  211 , first, includes a load end  202  connected with a load  212 , and then, a branch end  203 . The power distribution line  211  branches off at the branch end  203  to a first power distribution system  234  and a second power distribution system  237 . 
     The first power distribution system  234  includes an SVR  245  having an SVR end  204  at an input side and an SVR end  205  at an output side, respectively. Then, the SVR end  205  is connected with a load end  206  connected with a load  216 . 
     Further, the second power distribution system  237  branched off at the branch end  203 , first, includes a load end  207  connected with a load  217 , and then, an SVC end  208  connected with an SVC  218 . Furthermore, the SVC end  208  is connected with a load end  209  connected with a load  219 . 
     It is noted that the SVR stands for a Step Voltage Regulator and the SVC stands for a Static Var Compensator. In addition, the SVR and SVC are used to regulate the voltage of the power system, and then fail in the category of a voltage regulator. 
     In  FIG. 2 , the SVR and SVC are exemplified as voltage regulators in series with the power system and in parallel with the power system, respectively, but the voltage regulators on the power system are not limited thereto. 
     Further, if a distributed power source is connected to the power system, it is shown in the same manner as the load end  202 . Still further, in  FIG. 2 , the sensor  102  ( FIG. 1 ) is not shown. 
     At mathematical modeling of the above power system, node numbers are assigned to main points in the power system. 
     As shown in  FIG. 2 , node numbers 1 to 9 are exclusively assigned to the power transmission end  201 , the load end  202 , the branch end  203 , the SVR end  204 , the SVR end  205 , the load end.  206 , the load end  207 , the SVC end  208  and the load end  209 , respectively in order. 
     In addition, the connection relation between nodes is presented by the adjacency matrix and the hierarchy matrix to be described later. Firstly, the adjacency matrix will be explained, and secondly, the hierarchy matrix will be explained. 
     &lt;&lt;Adjacency Matrix&gt;&gt; 
     A description will be given of the adjacency matrix. The adjacency matrix defines an adjacency relation of an upstream and downstream (the upstream is closer to the power transmission end and the downstream is further from the power transmission end) of the nodes as a mathematical presentation. In addition, depending on the upstream and downstream, an upstream adjacency matrix U and a downstream adjacency matrix D are defined, respectively. Next, they will be described in order. 
     &lt;&lt;Upstream Adjacency Matrix U&gt;&gt; 
     An element u p  of the upstream adjacency matrix U is defined to be an upstream adjacent node (node number) of a node p. According to the definition, the example in  FIG. 2  is expressed with the upstream adjacent node of each node, starting from the the node 1, left to right, in order as in the following Equation 1. It is noted that 0 indicates no corresponding node. 
     In  FIG. 2 , P of the node p is defined as 0≦p≦8. 
     [Equation 1] 
       U=[0 1 2 3 4 5 3 7 8]  Equation 1
 
     &lt;&lt;Downstream Adjacency Matrix D&gt;&gt; 
     Next, a description will be given of the downstream adjacency matrix D. 
     Each element d np  of the downstream adjacency matrix D is defined to be a downstream adjacent node (node number) of the node p in a path to a node n. It is noted that 0 indicates no corresponding node. 
     Further, while the upstream adjacent node is unique, the downstream adjacent node may not be unique due to branching, and in  FIG. 2 , the path to the node 5 has a different downstream adjacent node of the node 3 from that to the node 8. 
     In addition, on lines 8 and 9 in the equation 2 to be described later, numbers 0,0,0 are present between the downstream adjacency matrix elements d 8,3  and d 6,7 , and between d 9,3  and d 9,7  corresponding to the node number 3 (presented as 7) and the node number 7 (presented as 8). The elements where numbers 0, 0, 0 are present originally correspond to the node numbers 4, 5 and 6. However, since the paths to the nodes 7 and 8 in  FIG. 2  do not include nodes (4, 5, 6), the numbers 0, 0, 0 are presented. 
     Such a presentation is for the convenience of the mathematical processing of the present system, and comes from the definitions described above. According to these definitions, all the elements are written for the example shown in  FIG. 2 , to obtain the following determinant in Equation. 2. 
     
       
         
           
             
               
                 
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     &lt;&lt;Hierarchy Matrix C D &gt;&gt; 
     Next, a description will be given of a hierarchy matrix C D . 
     The hierarchy matrix C D  is defined to be a mathematical presentation which presents a connection relation of the upstream and downstream regardless of whether they are adjacent or not. Respective elements C Dnp  take values described in the following. Equation 3A according to the connection relation. Further, the definition by the elements is applied to the example shown in  FIG. 2 , to obtain the hierarchical matrix C D  as a determinant shown in the following Equation 3B. 
     
       
         
           
             
               
                 
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     The downstream adjacency matrix D and the hierarchical matrix C D  are redundant presentations of the upstream adjacency matrix U, and are frequently used presentations for describing the mathematical model. Further, as long as the topology of the power distribution system remains unchanged, the downstream adjacency matrix D and the hierarchical matrix C D  have unchanged constants, and therefore may be generated for implementation in  30  advance based on the upstream adjacency matrix U. 
     To clarify the presentations of the mathematical model below, the above-mentioned U p , d np , C Dnp  will be presented as u (P) d(n, p), C D (n, P) as appropriate. 
     &lt;&lt;Parameter Presentation on SVR and SVC&gt;&gt; 
     Before a power equation at the node p is described, presentations of parameters and the like on the SVR (Step Voltage Regulator) and the SVC (Static Var Compensator) will be described. 
       FIG. 3  is a diagram showing presentations of the parameters and the like on an SVR  345  and an SVC  318  in the power system control system according to the embodiment of the present invention. 
     In  FIG. 3 , the SVR  345  is arranged between the node p and the upstream adjacent node u (p). A tap ratio for voltage regulation by the SVR  345  is presented as τ p . 
     In addition, assuming that there is no SVR  345 , a resistance component of impedance of the branch between the node u (p) and the node p is presented as r u(P) → P  and a reactance component as X u(p)→p . 
     Further, assuming that the node p is connected with a load  319  and the SVC  318 , a current flowing into or out of the load  319  or the SVC  318  at the node P is presented as I p . 
     It is noted that the SVC  318  is presented by a general symbol of a capacitor, but the SVC  318  has a function capable of supplying a lagging current, in addition to a leading current of the capacitor. 
     &lt;&lt;Power Equation for Branch&gt;&gt; 
     Next, a description will be given of power equations (Equations 4A and 4B) for a branch. These equations are established between adjacent nodes (u(p), p). It is noted that an element connecting adjacent nodes is referred to as a branch. 
     In the following Equation 4A, a voltage and a current of the node P are presented as V p  and I p . 
     Further, a passing current which passes through the node P is presented as I′ n (p) for a node current In at any downstream node n. 
     Still further, as a presentation of circuit impedance which is set in the system information database  108  ( FIG. 1 ), the branch from the adjacent node u(p) to the node p (corresponding to the power distribution line) is presented as u(p)→p as a subscript, as described above. 
     In other words, the resistance component and the reactance component of the impedance are presented as r u(p)→p , x u(p)→p , and the impedance is presented as (r u(p)→p +jx u(p)→p ). 
     As mentioned above, the τ p  is the tap ratio of the SVR. 
     It is noted that, in the Equations 4A and 4B, V p , V u(p) , I p , I′ n(p) , I n  as AC (complex numbers) are presented with dots of a modified symbol over the characters, but the dots are omitted in the description for the convenience of presentation. 
     
       
         
           
             
               
                 
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     It is noted that, in Equation 4A, a term including a coefficient τ p  is associated with the SVR (Step Voltage Regulator) and a pole transformer, and a term including I′ n(p)  is associated with the SVC (Static Var Compensator) and a load. 
     Further, in Equation 4B, the subscript d(n, p) of the τ is, as described above, the element d np  of the downstream adjacency matrix D, and furthermore, d(n, d(n, p)) indicates a relation between the n and the d(n, p) to follow downstream in order. 
     &lt;&lt;Presentation by Determinant of Power Equation&gt;&gt; 
     As to the power equation, a linear equation on the voltage and the current of the branch u(p)→p described in Equations 4A, 4B is established for combinations of all the adjacent nodes. In short, the number of equations is (N-1) for the number of nodes N, and they are collectively presented as the following matrix equation. 
     
       
         
           
             
               
                 
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     Here, A is a coefficient matrix formed with the impedance r u(p)→p  and x u(p)→p , the tap ration τ p  and the hierarchy matrix C D (n, p) Since there are 2N variables (N voltages and N currents), the size of A is (N-1)×2N. 
     Further, in Equation 4A, since all the terms include either V p  or I p , [0 N-1 ] on the right side in Equation 5 is a vector to make all the elements 0. 
     &lt;&lt;Extended Matrix Equation&gt;&gt; 
     Since Equation 5 does not include the information of the measured value of the state amount recorded in the measured value database  104 , terms which relate to matrices M V , M I  describing presence or absence of the measured value of the state amount on the voltage and current are added to extend Equation 5 to the following Equation 6. It is noted that the matrices M V , M I  will be described later. 
     
       
         
           
             
               
                 
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                   6 
                 
               
             
           
         
       
     
     In Equation 6, V 1  . . . V N , I 1  . . . I N  on the right side are measured values of the state amount of the voltage and current at the node p (1≦p≦N). It is noted that, on the left side in Equation 6, (V 1  . . . V N , I 1  . . . I N ) having dots of a modified symbol of a vector value of AC (complex number) over the characters present internal states of the voltage and current at each node. 
     Further, on the right side in Equation 6, (V 1  . . . V N , I 1  . . . I N ) having a modified symbol “˜” over the character present the measured values (including not only actual measured values but also estimated values). However, in the description, the characters are shown without the modified symbol “dot” or “˜” for the convenience of the presentations. 
     Further, M V , M I  in Equation 6 are matrices which describe the presence or absence of the measured values of the state amount on the voltage and current as described above, respectively, and M V  (voltage) has elements shown in the following Equation 7. 
     
       
         
           
             
               
                 
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                    
                   
                       
                   
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                   7 
                 
               
             
           
         
       
     
     In the example shown in Equation 7, elements of “1” in columns 2, 3, 5 indicate that the voltage has been measured by the sensor  102  ( FIG. 1 ) at the nodes  2 , 3 ,  5 , and associate the state amount V p  (with a “dot”) with the measured value of the state amount V p  (with “˜”) 
     Further, M I  (current) associates the state amount I p  (with a “dot”) with the measured value of the state amount I p  (with “˜”) to be presented in the same manner. However, in a case where the measurement points for the voltage are different from those for the current, the composition of 1, 0 in the determinant will be different. 
     Equation 6 is a fundamental equation for state estimation and will be presented hereinbelow in a simplified form as the following equation. 
     &lt;&lt;Presentation of Simplified Equation&gt;&gt; 
     [Equation 8] 
       Sy=b   Equation 8
 
     In Equation 8, S on the left side is a coefficient matrix composed of A, M V , M I  shown on the left side in Equation 6, and y is a variable vector constituted by the state amounts V p , I p  (1≦p≦N) shown on the left side in the equation 6. 
     Further, in Equation 8, b on the right side corresponds to the entire right side in Equation 6, and is a constant vector composed of 0 N-1 , M V , M I  and the measured values of the state amounts V p , I p  (1≦p≦N). 
     With the mathematical model above, a description will be given of a function of the system division unit  105  ( FIG. 1 ). 
     &lt;&lt;General Solution to Underdetermined Problem&gt;&gt; 
     First, Equation 8 will be solved for the variable vector y. In a case where Equation 8 is an underdetermined problem due to lack of the measured values of the state amounts V p , I p , that is, in a case where the coefficient matrix S has a rank deficiency, Equation 8 is solved by using a pseudo-inverse matrix S +  of S. 
     In short, a general solution of the variable vector y for minimizing an error norm in Equation 8 is given by the following Equation 9, using the pseudo-inverse matrix S +  of S. 
     It is noted that the norm (norm, vector norm) corresponds to a “length” of a vector, or a “distance” in a vector space. 
     [Equation 9] 
         y=S   +   b +Nul( S ) z    Equation 9
 
     The left side of Equation 9 is, as described above, the general solution of a variable vector for minimizing the error norm. 
     The first terra on the right side in Equation 9 presents a particular solution y 0  which minimizes the solution norm, of the general solution y in a row space of the coefficient matrix S. 
     In addition, the second term on the right side presents a redundant solution w in a null space Nul(S) of the coefficient matrix S. In Equation 9, “z” is an arbitrary vector. It is noted that, in accordance with a customary practice, the null space is presented with Nul. 
     &lt;&lt;Presentation of Equation Separated into Observable Subsystem and Unobservable Subsystem&gt;&gt; 
     Here, focusing on the null space Nul(S), assuming that i-th elements of the base are all 0s, i-th elements of the corresponding redundant solution w are always 0s, resulting in that the general solution y does not have redundancy with respect to the i-th elements. 
     Based on this reference, the elements of the general solution y are rearranged to Y U  without redundancy and Y R  with redundancy, and elements of S +  Nul(S) are rearranged accordingly, so that. Equation 9 is rewritten as shown in the following Equation 10. 
     
       
         
           
             
               
                 
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                   10 
                 
               
             
           
         
       
     
     On the left side in Equation 10, a node for which Y U  includes the state amount is observable and a node for which y R  includes the state amount is unobservable. Thus, the system division unit  105  ( FIG. 1 ) divides the unobservable power system in which the measured values of the state amounts lacks with respect to the state amounts into the observable subsystem, and the unobservable subsystem. 
     It is noted that one of the voltage and current at the same node may not have redundancy while the other may have it. The observable subsystem and unobservable subsystem obtained in this case may have the voltage and current separately. 
     The above process is performed by the system division unit  105  ( FIG. 1 ), for dividing the power system into the observable subsystem where the state amount defined based on the above measured values of the state amount does not have redundancy and the unobservable subsystem where the state amount defined based on the measured values of the state amount has redundancy. 
     Further, when dividing the power system into the observable subsystem and unobservable subsystem, the system division unit  105  performs it based on redundancy of the solution of the state amount obtained by solving simultaneous equations regarding the state amount, the system information and the measured values of the state amount with the calculation unit  110  ( FIG. 1 ). 
     &lt;&lt;Presentation of Equations Separated for Observable Subsystem and Unobservable Subsystem&gt;&gt; 
     Next, a description will be given of a function of the state estimation unit  106  ( FIG. 1 ). The upper row of Equation is an equation for the state estimation on the observable subsystem, and the following Equation 11 is obtained from the upper row of Equation 10. 
     [Equation 11] 
         y   U   =P   U   b+K   U   z    Equation 11
 
     Solving Equation 11 gives the estimated value of the state amount. 
     Since K U  on the right side in Equation 11 is a zero matrix (term having no redundancy), the estimated value of the state amount in the observable subsystem is uniquely determined as a particular solution y U0  based on P U  derived from the pseudo-inverse matrix S +  and b. 
     In the observable subsystem, the solution y U0  is the estimated value of the state amount which minimizes the error norm derived from Equation 8, that is, which satisfies the power equation and the measured value of the state amount with the least square error. 
     The above calculation for calculating the estimated value of the state amount in the observable subsystem is executed by the state estimation unit  106  ( FIG. 1 ) with the calculation unit  110  ( FIG. 1 ). 
     Further, solving the equation 11 to set the solution as an estimated value of a state amount is equivalent to “the state estimation unit sets the solution of the state amount obtained by solving the simultaneous equations on the state amount, the system information and the measured value of the state amount in the observable subsystem by the calculation unit as the estimated value of the state amount”. 
     &lt;&lt;State Estimation Equation on Unobservable Subsystem&gt;&gt; 
     Next, a description will be given of a function of the state range estimation unit  107  ( FIG. 1 ). The lower row in above Equation is an equation for the state estimation on the unobservable subsystem, and the following equation 12 is obtained from the lower row in the above equation 10. 
     [Equation 12] 
         y   R   =P   R   b+K   R   z    Equation 12
 
     Solving Equation 12 gives the estimated value of the state amount. 
     The first term on the right side in Equation 12 is a particular solution y R0  and is uniquely determined based on P R  derived from the pseudo-inverse matrix S +  and b. 
     While, the second term on the right side derived from the null space Nul (S) is a redundant solution W R . Since K U  is a zero matrix to the base of the Nul (S) which is linearly independent, each column of K R  is also a linearly independent base. The redundant solution W R  takes an arbitrary value on a span (K R ). It is noted that the span (K R ) indicates a subspace which spans from the linearly independent base of K R  and is presented according to a usual presentation. 
     Here, a value range of the state amount is limited by adding a unique constraint condition to the power system in Equation 12. In other words, the estimated range of the state amount is calculated. 
     The above calculation for calculating estimated range of the state amount in the unobservable subsystem above is executed by the state range estimation unit  107  ( FIG. 1 ) with the calculation unit  110  ( FIG. 1 ). 
     Further, the above-mentioned “solving Equation 12 and adding the unique constraint condition to the power system in Equation 12 to limit the value range of the state amount for calculating the estimated range of the state amount” can also be described as follows. That is, the description above is equivalent to a description of “the state range estimation unit sets the value range of the particular solution of the state amount and the general solution which is a sum of the redundant solution as an estimated range of the state amount, the particular solution and the general solution being obtained by solving the simultaneous equations regarding the state amount, the system information and the measured value of the state amount in the unobservable subsystem with the calculation unit”. 
     By calculating the estimated range of the state amount for the state amount in the unobservable subsystem as described above, the estimation for the state amount in the unobservable subsystem where sensors are insufficient in number to the system state amount can be obtained. 
     Various methods can be conceivable for limiting the value range of the state amount to calculate the estimated range of the state amount, and three of them are shown below. 
     &lt;&lt;First Method for Limiting Value Range of Redundant Solution&gt;&gt; 
     A first method for limiting a value range of a redundant solution is to calculate, based on the nature of a row space being orthogonal to a null space in a coefficient matrix S, a value range of a redundant solution. 
     The orthogonality is established for the particular solution Y 0  and the redundant solution w in Equation 9. Further, since the redundant solution w R  in Equation 12 is a vector in which elements to be always 0 are removed from the redundant solution w in Equation 9 and rearranged, the orthogonality is also established for the particular solution Y R0  and the redundant solution W. 
     Therefore, the following Equation 13 is established for solution norms of the general solution y R , the particular solution y R0  and the redundant solution w R  in Equation 12. 
     [Equation 13] 
       ∥ y   R ∥ 2   =∥y   R0 ∥ 2   +∥w   R ∥ 2    Equation 13
 
     Here, since the particular solution y R0  is unique, the solution norm can also be uniquely calculated. Further, assuming that each state amount cannot take a value greater than the constraint value set in the constraint condition database  109  for the general solution y R , the solution norm at that time is the maximum value ∥w R ∥ max . 
     The constraint value of the state amount is a rated current, for example, for the currents in the load node and the SVC node, and may be a threshold voltage of an overvoltage protection relay arranged in the system for the voltage. Thus, by setting the maximum value ∥w R ∥ max  of the solution norm of the general solution y R , the range of the solution norm ∥w R ∥ of the redundant solution w R  is defined by the following Equation 14. 
     [Equation 14] 
       ∥ w   R ∥ 2   −∥y   R ∥ 2   −∥y   R0 ∥ 2   ≦∥y   R ∥ max   2   −∥y   R0 ∥ 2   =∥w   R ∥ max   2    Equation 14
 
     &lt;&lt;Calculation of Maximum Value of each Element in Redundant Solution&gt;&gt; 
     Next, the maximum value of each element in the redundant solution w R  is calculated from the maximum value ∥w R ∥ max  of the solution norm of the redundant solution w R . 
       FIG. 4  is a schematic diagram showing a method for calculating the maximum value of each element of the redundant solution w R  from the maximum value ∥w R ∥ max  of the solution norm of the redundant solution w R . 
     In  FIG. 4 , a reference numeral  401  indicates a state space having state amounts of respective elements in a general solution y R  as axes. The state space includes a first axis, a second axis, . . . , and an i-th axis. 
     A reference numeral  402  indicates a vector (particular solution vector) of a particular solution y R0 =P R b. 
     A reference numeral  403  indicates a subspace span (K R ) where the redundant solution w R  is present. 
     A synthetic vector of the particular solution vector  402  and an arbitrary vector on the subspace  403  is a general solution vector. 
     A reference numeral  404  is a cross section of a hypersphere to be described later. 
     A reference numeral  405  is a unit vector to be described later. 
     Here, the fact that the range of the solution norm of the redundant solution w R is limited indicates that, in the subspace  403 , the vector of the redundant solution w R  is present inside the hypersphere  404  having the maximum value ∥w R ∥ max  as a radius. It is noted that the reason for calling the hypersphere  404  as a “hypersphere” is that the hypersphere is a spherical surface defined by the first axis, the second axis, . . . , and the i-th axis. 
     In this case, when the length of the unit vector  405  of which gradient in the i-th axis direction is the maximum on the subspace  403  is multiplied by ∥w R ∥ max , the i-th element in the redundant solution w R  takes the maximum value. Such a unit vector f 1  is calculated by the following Equation 15. 
     [Equation 15] 
         f   i   =p   i   /∥p   i   ∥, p   i   =K   R ( K   R   T   K   R ) −1   K   R   T   e   i    Equation 15
 
     In Equation 15, e i  is a unit vector having the i-th element of 1, and p i  is a projection of the e i  to the span (K R ). 
     The i-th element of a vector ∥w R ∥ max f i  formed by the unit vector f i  multiplied by ∥w R ∥ max  is the maximum value w Rmaxi  on the i-th element in the redundant solution w R . 
     Further, the value range of the i-th element in the redundant solution w R  is [−w Rmaxi , w Rmaxi ]. The value range of the i-th element y Ri  in the general solution y R  for the value range of the redundant solution w R  and the i-th element y R0i  in the particular solution is given by the following Equation 16. 
     [Equation 16] 
         y   R i   ∈[y   R0 i   −w   Rmaxi   , y   R0 i   +w   Rmaxi ]  Equation 16
 
     &lt;&lt;(Case of Value Range of General Solution being out of Constraint Value&gt;&gt; 
     In a case where the value range of the general solution y R  is out of the constraint value, the value range is rounded off to the constraint value. By calculating each element by Equation 16, the error norm derived from Equation 8 is minimized for the unobservable subsystem. That is, the estimated range of the state amount can be obtained which satisfies the power equation and the measured value of the state amount with the least square error. 
     A first method to limit the value range of the redundant solution as described above is a method in which “the state range estimation unit limits the value range of the redundant solution of the state amount based on the constraint value of the state amount”. 
     Further, the first method is also referred to as a method in which “the state range estimation unit sets a sum of the particular solution vector of the state amount and the redundant solution vector obtained by solving the simultaneous equations with the calculation unit as the general solution vector, and subtracts the vector norm of the particular solution vector from the maximum value of the vector norm of the general solution vector defined by the constraint value, to calculate the maximum value of the vector norm of the redundant solution vector for limiting the value range of the redundant solution vector”. 
     Further, the first method for limiting the value range of the redundant solution has a feature in which the accuracy to limit the value range is low, but the calculation amount is small, as compared with a second and a third methods to be described later. 
     &lt;&lt;Second Method for Limiting Value Range of Redundant Solution&gt;&gt; 
     A second method for limiting the value range of the redundant solution is to calculate the value range of the redundant solution w having the constraint value of each state amount defined in the constraint, condition database  109  ( FIG. 1 ) as a boundary condition. 
       FIG. 5  is a schematic diagram showing a calculating method for the value range of the redundant solution. 
     In  FIG. 5 , a reference numeral  501  indicates a state space having the state amounts of respective elements in the general solution y R  as axes. The state space includes a first axis, a second axis, . . . , an i-th axis. 
     A reference numeral  502  indicates a vector (particular solution vector) of the particular solution y R0 =P R b. 
     A reference numeral  503  indicates a subspace span (K R ) where the redundant solution w R  is present. 
     A synthetic vector of the particular solution vector  502  and an arbitrary vector on the subspace  503  is the general solution vector which forms a subset.  504 . 
     A hyperplane  505  indicates the constraint values of the state amount which are present on the first: axis, the second axis, . . . , and the i-th axis, respectively, and is a (hyper) plane defined by the constraint values. 
     Defining the value range of the redundant solution w R  having the constraint values of the state amount as boundary conditions indicates that the subset  504  is cut off by the hyperplane  505 . 
     The value range of the redundant solution w R  is calculated by solving simultaneous inequalities of the following Equations 17A and 17B for w Ri . It is noted that W R  is a vector and W Ri  is an element contained therein. 
     [Equations 17A and 17B] 
       w R =K R z   Equation 17A
 
         y   Rlim1i   ≦y   R0 i   +w   ri   ≦y   Rlim2i    Equation 17B
 
     In Equation 17B, y Rlim1i , y Rlim2i  are upper and lower limit values of the state amount defined in the constraint condition database  109 . Since there are so many solutions for such simultaneous inequalities, a description thereof will be omitted in the present embodiment. 
     The second method for limiting the value range of the redundant solution as described above is a method in which “the state range estimation unit limits the value range of the redundant solution of the state amount based on the constraint. values of the state amount”. 
     Further, the second method is also referred to as a method in which “the state range estimation unit sets a sum of the particular solution vector of the state amount: and the redundant: solution vector obtained by solving the simultaneous equations with the calculation unit as the general solution vector, and subtracts the particular solution vector from the sum by setting the constraint values as boundary conditions of the general solution vector to limit the value range of the redundant solution vector.” 
     Still further, the second method for limiting the value range of the redundant solution has a feature in which the calculation amount is large and implementation is complex, but an accurate solution of the value range can be obtained —    
     &lt;&lt;Third Method for Limiting Value Range of Redundant Solution&gt;&gt; 
     A third method for limiting the value range of the redundant solution uses a nature that a particular solution obtained by the pseudo-inverse matrix is the minimum solution of the solution norm in Equation 9. Then, the value range of the voltage in the state amount is limited based on the rated value of the current defined in the constraint condition database  109  ( FIG. 1 ) 
     Firstly, a variable vector y is applied with weighting based on a type (voltage, current) of the state amount and Equation 8 is rewritten as the following Equation 18. 
     [Equation 18] 
         SH   −1 ( Hy )= b    Equation 18
 
     A weight matrix H shown on the left side in Equation 18 is a diagonal matrix having a weighting coefficient associated with each element of the variable vector as a diagonal component, and is described as follows by using “diag” indicating a diagonal matrix. 
         H =diag ([ h _ V 1,  h _ V 2, . . . ,  h _ VN, h _ I 1,  h _ I 2, . . . ,  h _ IN ]) 
     It is noted that h_Vp is a weighting coefficient corresponding to the voltage state amount of the node p, and h_Ip is a weighting coefficient corresponding to the current state amount I p  of the node P where 1≦p≦N. 
     By solving Equation 18 for Hy, a particular solution Hy H0  can be newly obtained which minimize a solution norm for Hy. The particular solution Hy H0  is obtained by the following Equation 19, by using pseudo-inverse matrix (SH −1 ) + . 
     [Equation 19] 
         Hy   H0 =( SH   −1 ) +   b    Equation 19
 
     Further, Equation 19 is solved for y H0  to obtain the following Equation 20. 
     [Equation 20] 
         y   H0   =H   −1 ( SH   −1 ) +   b    Equation 20
 
     Since Equation 18 and Equation 8 are equivalent, the nature that the solution y H0  is a solution to minimize the error norms on both sides in Equation 8 is equivalent. On the other hand, the solution y H0  in the equation 20 represents one of the general solutions of Equation 9, and the difference between the solution y H0  in Equation 20 and the particular solution y 0  in Equation 9 represents the redundant solution w in Equation 9. 
     &lt;&lt;Method for Calculating Estimated Voltage Range&gt;&gt; 
     Next, a method for calculating the estimated voltage range (lower limit voltage and upper limit voltage) will be described by using the weighted solution y H0  in Equation 20. The method mainly includes three steps. The steps will be described below. 
     [Step 1] 
     As a step 1, the weighting coefficient h_Ip associated with the current state amount among the diagonal components constituting the weight matrix H is set to a larger value than the weighting coefficient h_Vp associated with the voltage state amount. 
     [Step 2] 
     As a step 2, Equation 20 is solved to obtain a solution in which a sum of squares of the current state amounts I p  an respective nodes is small and a sum of squares of the voltage state amounts V p  in respective nodes is large. If the weighting coefficient h_Ip is set to be sufficiently large in the step 1, this is the solution to minimize the sum of squares of the current state amount among solutions which satisfy the power equation and a measurement equation with the minimum error norm. 
     [Step 3] 
     As a step 3, an absolute value of the current state amount at this time is evaluated, and if any of the state amounts does not exceed the rated current, h_Ip which has been set in the step 1 is reset to a relatively smaller value (for example, to a value having 95% of the original coefficient) , to reduce the weighting for the current. 
     Here, returning to the step 2 to solve Equation 20, a solution can be obtained, in which the sum of squares of the current state amount relatively turns to be large. 
     By repeating the above steps, the sum of squares of the current state amount turns to be gradually larger and the sum of squares of the voltage state amount turns to be gradually smaller in the solution y H0 . 
     In short, in the step 3, when an absolute value of any of the current state amounts exceeds the rated current, the solution at that time is, in a range to satisfy Equation 8 and the rated current defined in the constraint condition database  109 , the minimum solution of the voltage norm which minimizes the sum of squares of the voltage state amount. 
     In this case, if the reference of the voltage state amount is set, for example, at 0 volt, the minimum solution of the voltage norm represents a solution which gives the lower limit voltage. On the other hand, if the reference of the voltage state amount is set to a value sufficiently larger than a specified voltage, the minimum solution of the voltage norm represents the solution which gives the upper limit voltage. It is noted, as an example of being set to the sufficiently large value described above, the voltage state amount may be indicated by a difference from 10000 volts in a system having the specified voltage of 6600 volts. 
     Further, in a process of execution of steps 2 and 3 repetitively, since the weighting coefficient h_p is discretely reduced, the minimum solution of the voltage norm obtained in the process is not an accurate solution but an approximate solution. 
     Then, until the absolute value of the current state amount exceeds the rated current, the weighting coefficient h_Ip is adjusted so that the absolute value of the current state amount converges to the rated current, instead of simply reducing the weighting coefficient h_Ip. The adjustment allows the approximate accuracy of the minimum solution of the voltage norm to be more accurate. 
     Further, if the rated current is different depending on the node, weighting is made such that a product of the associated weighting coefficient h_Ip and the rated current is set to have the same value for each node. With this weighting, the solution obtained in the step 2 turns to be a current state amount which is normalized to the rated current. Alternatively, the same solution can be obtained by writing Equation 8 with the current state amount which is normalized, by the rated current in advance and solving Equation 18. 
     &lt;&lt;Flow of Calculating Step in Third Method&gt;&gt; 
     The calculation step in the third method described above will be shown by a flowchart below. 
       FIG. 6  is a flowchart showing the calculation step in the third method for limiting the value range of a redundant solution according to the embodiment of the present invention. 
     In  FIG. 6 , step S 601  is the step 1 described above, in which the weighting coefficient h_Ip is set to the sufficiently large coefficient as an initial value. Then, step S 602  is executed. 
     Step S 602  is the step 2 described above, and the weighted particular solution is calculated so that the sum of squares of the current state amount decreases, to obtain the solution. Then, step S 603  is executed. 
     Step S 603  is the step 3 described above, and determines if the absolute value of the current state amount exceeds the rated current. If the current state amount at any node does not exceed the rated current (N), step S 604  is executed. Alternatively, if the current state amount at any node exceeds the rated current (Y), step S 605  is executed. 
     In step S 605 , the solution at that time (particular solution weighted so that the sum of squares of the current state amount decreases and the current state amount at some node exceeds the rated current) is set to be the minimum solution of the voltage norm. 
     It is noted that, in step S 604  branched from step S 603  above, the weighting coefficient h_Ip is decreased to reduce the weighting on the current, and step S 602  is executed again. 
     The third method for limiting the value range of the redundant solution as described above is equivalent to a method in which “the state range estimation unit uses the calculation unit to weight the state amount and solve the simultaneous equations for obtaining a new solution representing one of the general solutions, and sets a voltage component of the solution at a stage where a current component of the solution reaches the rated current defined by the constraint value as the estimated range of the state amount, while weighting for the current component in the state amount is gradually reduced”. 
     Further, the third method for limiting the value range of the redundant solution has a feature in which the calculation amount is large, but implementation is simple and an approximate solution of the value range can be obtained. 
     The three methods for calculating the estimated range of the state amount using the state range estimation unit  107  ( FIG. 1 ) are described above, but the calculation method for the estimated range of the state amount using the constraint, value defined in the constraint condition database  109  ( FIG. 1 ) is not limited thereto. 
     &lt;&lt;Function of Display Device  111 &gt;&gt; 
     Next, a function of the display device  111  ( FIG. 1 ) will be described. 
       FIGS. 7A and 7B  are diagrams showing an example of a screen display on the display device  111 , in which  FIG. 7A  indicates representative values of the state amounts  702  and ranges of the state amounts  703  at respective modes, and  FIG. 7B  indicates a system diagram  701  of the power system. 
     In  FIG. 7B , the system diagram  701  of the power system is shown with an example of nodes and branches based on the system information recorded in the system information database  108  ( FIG. 1 ). The nodes in the system diagram  701  are each connected with the load or the SVC (Static Var Compensator). Further, the branch shown at approximately the center is connected with the SVR (Step Voltage Regulator). 
     In  FIG. 7A , the horizontal axis corresponds to an arrangement of respective nodes in the system diagram  701  in  FIG. 7B , and the vertical axis indicates the voltage. 
     In  FIG. 7A , the representative values of the state amounts  702  are displayed on a graph, in which the unique solution y U =y U0  calculated by the state estimation unit  106  ( FIG. 1 ) for the observable subsystem and the particular solution y R0  calculated by the state range estimation unit  107  ( FIG. 1 ) for the unobservable subsystem are associated with the nodes in the system diagram  701 . 
     Further, ranges of the state amount  703  are displayed on the graph, in which the value ranges of the general solution y R  calculated by the state range estimation unit  107  for the unobservable subsystem are associated with the nodes in the system diagram. 
     It is noted that the numerals  702  for the black circles indicate the representative values of the respective state amounts. Further, the numerals  703  for the two lines indicate the ranges of the respective state amounts. 
     Still further, since the solution y U  (=y U0 ) in the observable subsystem is unique, the width of the range of the state amount  703  for the observable subsystem is zero. 
     Since the general solution y R  in the unobservable subsystem has the value range of the redundant solution, the width (between the upper and lower limits) of the range of the state amount  703  has a given value. 
     Yet further, a numerical frame  704  describes the same information as the representative value of the state amount.  702  and the range of the state amount  703  with numeric values on the system diagram  701 . 
     &lt;&lt;Function of Recording Device  112 &gt;&gt; 
     Next, a function of the recording device  112  will be described. 
       FIG. 8  is a table showing an example of a system log outputted from the recording device  112 . 
     In  FIG. 8 , the system log exemplifies items of a timestamp, a node, a flag, a representative value of the state amount, a range (upper limit) of the state amount, and a range (lower limit) of the state amount. 
     The system log added with a timestamp showing a recorded date and time is outputted everytime the power system state estimation device  100  ( FIG. 1 ) is executed. 
     Further, the system log records the same information as the flag indicating which of the observable/unobservable subsystem the node belongs to, the representative value of the state amount  702  ( FIG. 7 ) and the range of the state amount  703  ( FIG. 7 ) with numerical values to be outputted. 
     Further, the items in the system log are not limited to the above. Depending on the calculation method, the maximum value ∥w R ∥ max  of the solution norm of the redundant solution in Equation 14, for example, is outputted. 
     &lt;&lt;Process Flow of Power System State Estimation Device&gt;&gt; 
     Next, a description will be given of a process flow of the power system state estimation device according to the embodiment. 
       FIG. 9  is a flowchart showing an exemplary process flow of the power system state estimation device according to the embodiment of the present invention. 
     In  FIG. 9 , step S 901  is for obtaining a measured value of the state amount. In short, the measured value of the state amount obtained via the communication line  103  ( FIG. 1 ) is recorded in the measured value database  104  ( FIG. 1 ) in step S 901 . 
     Step S 902  is for dividing the power system by the system. 
     division unit  105  ( FIG. 1 ). In step S 902 , the system division unit  105  is inputted with the system information and the measured value of the state amount to divide the power system into the observable subsystem and the unobservable subsystem (system division step). 
     Step S 903  is for estimating the state amount by the state estimation unit  106  ( FIG. 1 ). In step S 903 , the state estimation unit  106  is inputted with the system information and the measured value of the state amount to calculate the estimated value of the state amount in the observable subsystem (estimated value of the state amount calculation step). 
     Step S 904  is for estimating the state range by the state range estimation unit  107  ( FIG. 1 ). In step S 904 , the state range estimation unit  107  is inputted with the system information, the measured value of the state amount and the constraint value of the state amount to calculate the estimated value of the state range in the unobservable subsystem (estimated value of the state range calculation step). 
     Step S 905  is for displaying the information on a screen. In step S 905 , the state estimated value and the state estimated range are displayed on the display device  111  ( FIG. 1 ). 
     Step S 906  is for recording the system log. In step S 906 , the system log is recorded by the recording device  112  ( FIG. 1 ) Then, step  901  is repeated again. 
     The power system state estimation device  100  ( FIG. 1 ) executes the processing of each step above at regular intervals or in synchronous with obtaining the measured value of the state amount. 
     It is noted that, in the third method, the minimum solution of the voltage norm also includes the state amount relating to the observable subsystem, and the value thereof is equal to the estimated value of the state amount obtained by the calculation processing instep S 903 . Therefore, the calculation processing in step S 903  and step S 904  can be executed at the same time. 
     DESCRIPTION OF REFERENCE NUMERALS 
     
         
         
           
               100  power system state estimation device 
               101  power system 
               102  sensor (information acquisition device) 
               103  communication line (information acquisition device) 
               104  measured value database 
               105  system division unit 
               106  state estimation unit 
               107  state range estimation unit 
               108  system information database 
               109  constraint condition database 
               110  calculation unit 
               111  display device (peripheral device) 
               112  recording device (peripheral device) 
               201  power transmission end 
               202 ,  206 ,  207 ,  209  load end 
               203  branch end 
               204 ,  205  SVR end 
               208  SVC end 
               211  power distribution line 
               212 ,  216 ,  217 ,  219 ,  319  load 
               218 ,  318  SVC 
               234 ,  237  power distribution system 
               245 ,  345  SVR 
               401 ,  501  state space 
               402 ,  502  particular solution vector 
               403 ,  503  subspace 
               404  hypersphere 
               405  unit vector 
               504  subset 
               505  hyperplane 
               701  system diagram 
               702  representative value of state amount 
               702  range of state amount 
               704  numerical frame