Power Flow Studies
    ◮ Power flow studies are of great importance in planning and
     operation.
    ◮ A power flow study gives the magnitude and angle of the
     voltage at each bus.
    ◮ Once the bus voltage magnitudes and angles are known, the
     real and reactive power flow through each line can be
     computed and hence losses in a system.
    ◮ Power flow studies are a steady state analysis of a power
     system. They are called as load flow studies.
    ◮ Since the loads are specified in terms of power, the resulting
     equations are non-linear algebraic which need to be solved
     iteratively.
    ◮ We use numerical methods such as Gauss-Seidal and
     Newton-Raphson Methods for solving them.
          Power Flow Problem:
          Let Vi be the voltage at ith bus.
                                                   Vi = |Vi| δ
                                                                     i
          Let Yii and Yij be
                                      Y =|Y | θ                Y =|Y | θ
                                        ii        ii    ii       ij        ij     ij
                                                                                      i
          The net current injected into the network at bus ❖ is
                                                                                   N
                         I  =Y V1+Y V2+···+Y V =XY Vn
                          i       i1           i2                    iN    N              in
                                                                                  n=1
          where N be the total number of buses in the network. Let P and
                                                                                                     i
          Qi be the net real and reactive power entering the network at the
                  i
          bus ❖.
                                                P +Q =VI∗
                                                   i        i       i  i
                                                P −Q =V∗I
                                                   i        i       i   i
                                N
                    P −Q =V∗XY Vn
                     i    i   i    in
                               n=1
     On substitution,
                             N
              P −Q =|V| −δ X|Y ||Vn| θ +δ
               i    i   i   i    in    in  n
                             n=1
                       N
              P −Q =X|Y ||V||V | θ +δ −δ
               i    i      in i  n  in  n  i
                       n=1
     Equating real and and imaginary parts,
                   N
              P =X|Y ||V||V |cos(θ +δ −δ )
               i      in  i n    in  n   i
                  n=1
                   N
             Q =−X|Y ||V||V |sin(θ +δ −δ)
              i        in i n    in  n   i
                  n=1
         The above equations are power flow equations in the polar form.
         They are non linear functions of |V| and δ.
                                            P =f (|V|,δ)
                                                     1
                                            Q=f (|V|,δ)
                                                     2
                                                                     i
                                                                   ❖
                                      →P              →P
                                            gi               i,sch              }P,Q
                                      →Q              →Q                            i    i
                                             gi              i,sch
                                                     P ,Q
                                                       di    di