Patent ID: 11953398
Assignee: DALIAN UNIVERSITY OF TECHNOLOGY
Field: Measurement (Instruments)
Classification: CPC G | IPC G

Claim 0:
1. A quasi-static calculation method for lateral unbalanced force of transmission lines, comprising the following steps:
step 1: determining mean wind-induced lateral unbalanced force
for a span conducting wire with an initial vertical height difference c0 at both ends, displacement components of one end in longitudinal direction (x-axis direction), vertical direction (y-axis direction) and lateral direction (z-axis direction) are respectively ΔX, ΔY and ΔZ, and a standard cubic equation with H1 as a variable is obtained:, H
       1
       3
      
      +
      
       
        (
        
         
          
           E
           ⁢
           A
           ⁢
           
            q
            2
           
           ⁢
           
            L
            2
           
          
          
           2
           ⁢
           4
           ⁢
           
            H
            0
            2
           
          
         
         -
         
          
           E
           ⁢
           
            A
            ⁡
            (
            
             
              Δ
              ⁢
              
               
                Y
                 
               
               2
              
             
             +
             
              Δ
              ⁢
              
               Z
               2
              
             
             +
             
              2
              ⁢
              L
              ⁢
              Δ
              ⁢
              X
             
             +
             
              2
              ⁢
              
               c
               0
              
              ⁢
              Δ
              ⁢
              Y
             
            
            )
           
          
          
           2
           ⁢
           
            L
            2
           
          
         
         -
         
          H
          0
         
        
        )
       
       ⁢
       
        H
        1
        2
       
      
      +
      
       
        (
        
         -
         
          
           E
           ⁢
           A
           ⁢
           Δ
           ⁢
           Z
           ⁢
           λ
          
          
           L
           3
          
         
        
        )
       
       ⁢
       
        H
        1
       
      
      +
      
       E
       ⁢
       
        A
        ⁡
        (
        
         
          -
          
           
            
             q
             2
            
            ⁢
            
             L
             2
            
           
           
            2
            ⁢
            4
           
          
         
         -
         
          δ
          
           2
           ⁢
           
            L
            3
           
          
         
        
        )
       
      
     
     =
     0
    
   
   
    
     (
     1
     )
    
   
  
 

 
  
   
    
     δ
     =
     
      
       ∫
       0
       L
      
      
       
        (
        
         
          
           (
           
            
             ∫
             0
             L
            
            
             
              ∫
              0
              
               x
               2
              
             
             
              
               (
               
                
                 
                  
                   f
                   ¯
                  
                  
                   R
                   ⁢
                   e
                   ⁢
                   f
                  
                 
                 (
                 
                  x
                  1
                 
                 )
                
                ⁢
                d
                ⁢
                
                 x
                 1
                
               
               )
              
              ⁢
              d
              ⁢
              
               x
               2
              
             
            
           
           )
          
          2
         
         +
         
          
           
            L
            2
           
           (
           
            
             ∫
             0
             x
            
            
             
              
               
                f
                ¯
               
               
                R
                ⁢
                e
                ⁢
                f
               
              
              (
              
               x
               1
              
              )
             
             ⁢
             d
             ⁢
             
              x
              1
             
            
           
           )
          
          2
         
         -
         
          2
          ⁢
          L
          ⁢
          
           
            ∫
            0
            L
           
           
            
             ∫
             0
             
              x
              2
             
            
            
             
              (
              
               
                
                 
                  f
                  ¯
                 
                 
                  R
                  ⁢
                  e
                  ⁢
                  f
                 
                
                (
                
                 x
                 1
                
                )
               
               ⁢
               d
               ⁢
               
                x
                1
               
              
              )
             
             ⁢
             d
             ⁢
             
              
               x
               2
              
              ·
              
               
                ∫
                0
                x
               
               
                
                 
                  
                   f
                   ¯
                  
                  
                   R
                   ⁢
                   e
                   ⁢
                   f
                  
                 
                 (
                 
                  x
                  1
                 
                 )
                
                ⁢
                d
                ⁢
                
                 x
                 1
                
               
              
             
            
           
          
         
        
        )
       
       ⁢
       dx
      
     
    
   
   
    
     (
     2
     )
    
   
  
 

 
  
   
    
     λ
     =
     
      
       ∫
       0
       L
      
      
       
        (
        
         
          
           ∫
           0
           L
          
          
           
            ∫
            0
            
             x
             2
            
           
           
            
             (
             
              
               
                
                 f
                 ¯
                
                
                 R
                 ⁢
                 e
                 ⁢
                 f
                
               
               (
               
                x
                1
               
               )
              
              ⁢
              d
              ⁢
              
               x
               1
              
             
             )
            
            ⁢
            d
            ⁢
            
             x
             2
            
           
          
         
         -
         
          L
          ⁢
          
           
            ∫
            0
            x
           
           
            
             
              
               f
               ¯
              
              
               R
               ⁢
               e
               ⁢
               f
              
             
             (
             
              x
              1
             
             )
            
            ⁢
            d
            ⁢
            
             x
             1
            
           
          
         
        
        )
       
       ⁢
       dx
      
     
    
   
   
    
     (
     3
     ), where: H0 represents initial horizontal tension; EA represents Young's modulus; q represents weight per unit length of the span conducting wire; L represents horizontal span; both δ and λ represent constants related to mean wind pressure; fRef(x) is a distribution function of the mean wind pressure along line direction; and H1 represents horizontal tension of the span conducting wire under combined action of support displacement and mean wind, which can be obtained by solving formula (1) via a Cardan's formula;
further, the mean wind-induced lateral unbalanced force at both ends of the span conducting wire is obtained:, T
       _
      
      
       z
       ⁢
       0
      
     
     =
     
      
       
        1
        L
       
       ⁢
       
        
         ∫
         0
         L
        
        
         
          ∫
          0
          
           x
           2
          
         
         
          
           (
           
            
             
              
               f
               ¯
              
              
               R
               ⁢
               e
               ⁢
               f
              
             
             (
             
              x
              1
             
             )
            
            ⁢
            
             dx
             1
            
           
           )
          
          ⁢
          d
          ⁢
          
           x
           2
          
         
        
       
      
      +
      
       
        
         H
         1
        
        ⁢
        Δ
        ⁢
        Z
       
       L
      
     
    
   
   
    
     (
     4
     )
    
   
  
 

 
  
   
    
     
      
       T
       _
      
      
       z
       ⁢
       1
      
     
     =
     
      
       
        1
        L
       
       ⁢
       
        
         ∫
         0
         L
        
        
         
          ∫
          0
          
           x
           2
          
         
         
          
           (
           
            
             
              
               f
               ¯
              
              
               R
               ⁢
               e
               ⁢
               f
              
             
             (
             
              x
              1
             
             )
            
            ⁢
            d
            ⁢
            
             x
             1
            
           
           )
          
          ⁢
          d
          ⁢
          
           x
           2
          
         
        
       
      
      -
      
       
        ∫
        0
        x
       
       
        
         
          
           f
           ¯
          
          
           R
           ⁢
           e
           ⁢
           f
          
         
         (
         
          x
          1
         
         )
        
        ⁢
        d
        ⁢
        
         x
         1
        
       
      
      +
      
       
        
         H
         1
        
        ⁢
        Δ
        ⁢
        Z
       
       L
      
     
    
   
   
    
     (
     5
     ), where: subscripts 0 and 1 respectively represent one side of the span conducting wire without support displacement and one side of the span conducting wire with support displacement, the same below;
step 2: determining fluctuating wind-induced lateral unbalanced force
considering quasi-static background response only, the fluctuating wind-induced lateral unbalanced force at both ends of the span conducting wire is calculated by an influence line method:, T
       
        z
        ⁢
        0
       
      
      (
      t
      )
     
     =
     
      
       L
       n
      
      ⁢
      
       
        ∑
        
         i
         =
         1
        
        n
       
       
        
         (
         
          1
          -
          
           
            x
            i
           
           L
          
          +
          
           
            
             Δ
             ⁢
             Z
            
            L
           
           ⁢
           
            φ
            h
           
          
         
         )
        
        ⁢
        
         
          F
          Ref
         
         (
         
          
           x
           i
          
          ,
          t
         
         )
        
       
      
     
    
   
   
    
     (
     6
     )
    
   
  
 

 
  
   
    
     
      
       T
       
        z
        ⁢
        1
       
      
      (
      t
      )
     
     =
     
      
       L
       n
      
      ⁢
      
       
        ∑
        
         i
         =
         1
        
        n
       
       
        
         
          (
          
           
            
             x
             i
            
            L
           
           -
           
            
             
              Δ
              ⁢
              Z
             
             L
            
            ⁢
            
             φ
             h
            
           
          
          )
         
         
          z
          ⁢
          
           1
           ι
          
         
        
        ⁢
        
         
          F
          Ref
         
         (
         
          
           x
           i
          
          ,
          t
         
         )
        
       
      
     
    
   
   
    
     (
     7
     ), where: FRef(xi,t) represents fluctuating wind load acting on each point of the span conducting wire; n represents number of wind speed simulation points; xi represents x-coordinate of the fluctuating wind load; φh represents increment of horizontal tension caused by unit wind load, and the expression thereof is:, φ
      h
     
     =
     
      
       
        
         
          1
          -
          
           
            x
            1
           
           /
           L
          
         
         
          H
          1
         
        
        ⁢
        
         
          ∫
          0
          
           x
           1
          
         
         
          
           
            d
            ⁢
            
             w
             ¯
            
           
           
            d
            ⁢
            x
           
          
          ⁢
          d
          ⁢
          x
         
        
       
       -
       
        
         
          
           x
           1
          
          /
          L
         
         
          H
          1
         
        
        ⁢
        
         
          ∫
          
           x
           1
          
          L
         
         
          
           
            d
            ⁢
            
             w
             ¯
            
           
           
            d
            ⁢
            x
           
          
          ⁢
          dx
         
        
       
      
      
       
        L
        EA
       
       -
       
        
         q
         ⁢
         
          
           ∫
           0
           L
          
          
           
            
             d
             ⁡
             (
             
              
               y
               0
              
              +
              
               v
               ¯
              
             
             )
            
            
             d
             ⁢
             x
            
           
           ⁢
           
            (
            
             
              2
              ⁢
              x
             
             -
             L
            
            )
           
           ⁢
           d
           ⁢
           x
          
         
        
        
         2
         ⁢
         
          H
          1
          2
         
        
       
       +
       
        
         
          
           ∫
           0
           
            x
            1
           
          
          
           
            
             (
             
              
               d
               ⁢
               
                w
                ¯
               
              
              
               d
               ⁢
               x
              
             
             )
            
            2
           
           ⁢
           d
           ⁢
           x
          
         
         +
         
          
           ∫
           
            x
            1
           
           L
          
          
           
            
             (
             
              
               d
               ⁢
               
                w
                ¯
               
              
              
               d
               ⁢
               x
              
             
             )
            
            2
           
           ⁢
           d
           ⁢
           x
          
         
        
        
         H
         1
        
       
       -
        
       
        
         Δ
         ⁢
         
          Z
          ⁡
          (
          
           
            
             ∫
             0
             
              x
              1
             
            
            
             
              
               d
               ⁢
               
                w
                ¯
               
              
              
               d
               ⁢
               x
              
             
             ⁢
             d
             ⁢
             x
            
           
           +
           
            
             ∫
             
              x
              1
             
             L
            
            
             
              
               d
               ⁢
               
                w
                ¯
               
              
              
               d
               ⁢
               x
              
             
             ⁢
             d
             ⁢
             x
            
           
          
          )
         
        
        
         
          H
          1
         
         ⁢
         L
        
       
      
     
    
   
   
    
     (
     8
     )
    
   
  
 

 
  
   
    
     
      
       d
       ⁡
       (
       
        
         γ
         0
        
        +
        
         v
         ¯
        
       
       )
      
      
       d
       ⁢
       x
      
     
     =
     
      
       
        q
        
         2
         ⁢
         
          H
          1
         
        
       
       ⁢
       
        (
        
         L
         -
         
          2
          ⁢
          x
         
        
        )
       
      
      +
      
       
        Δ
        ⁢
        y
       
       L
      
     
    
   
   
    
     (
     9
     )
    
   
  
 

 
  
   
    
     
      
       d
       ⁢
       
        w
        ¯
       
      
      
       d
       ⁢
       x
      
     
     =
     
      
       1
       
        
         H
         1
        
        ⁢
        L
       
      
      ⁢
      
       
        ∫
        0
        L
       
       
        
         ∫
         0
         
          x
          2
         
        
        
         (
         
          
           
            
             
              f
              ¯
             
             
              R
              ⁢
              e
              ⁢
              f
             
            
            (
            
             
              x
              1
             
             ⁢
             λ
             ⁢
             1
             ⁢
             
              x
              1
             
            
            )
           
           ⁢
           d
           ⁢
           
            x
            2
           
          
          -
          
           
            1
            
             H
             1
            
           
           ⁢
           
            
             ∫
             0
             X
            
            
             
              
               
                f
                ¯
               
               
                R
                ⁢
                e
                ⁢
                f
               
              
              (
              
               x
               1
              
              )
             
             ⁢
             λ1
             ⁢
             
              x
              1
             
            
           
          
          +
          
           
            Δ
            ⁢
            Z
           
           L
          
         
        
       
      
     
    
   
   
    
     (
     10
     ), where: y0, v and w respectively represent initial vertical displacement, vertical displacement under combined action of mean wind and support displacement, and lateral displacement caused by mean wind of the span conducting wire;
finally, total lateral reaction force is a sum of the mean wind-induced lateral unbalanced force and the fluctuating wind-induced lateral unbalanced force, the total lateral unbalanced force is obtained:

{circumflex over (T)}z0(t)=Tz0+Tz0(t)  (11)

{circumflex over (T)}z1(t)=Tz1+Tz1(t)  (12).