Ward Reduction
It is a method of elimination of nodes [1]. The Gaussian elimination technique is used to obtain the reduced equivalents for the external networks. One of the basic ideas in Ward-type equivalent circuit representation is to treat the generation and load in the network being reduced as constant-current quantities. Hence, a Ward equivalent will yield an exact representation for the linear problem but only an approximate equivalent for nonlinear applications, like the ac power flow problem. Let a power system be described by the following set of nodal equations,
Y_bus V=I, where〖 Y〗_bus is the n×n bus admittance matrix, V is the n×1 vector of complex voltages at all nodes, and I is the n×1 vector of complex currents injected at all nodes.
After elimination of the kth node, Y_bus is modified as,
▁Y_ij^((new) )= ▁Y_ij-(▁Y_ik ▁Y_kj)/▁Y_kk ;
∀ i,j=1,…,n;i,j ≠k
The current vector I is also modified as,
▁I_i^((new) )= ▁I_i-▁Y_ik/▁Y_kk I_k;
∀ i=1,…,n;i ≠k
The superscript (new) distinguishes the elements of the new (n-1)×(n-1) Y_bus from the original ×n Y_bus. Every step of the Gaussian elimination yields an equivalent circuit of the network at the base case. If the network is reduced to r nodes, a new bus admittance matrix of dimension r×r, and a new current injection vector of dimension r×1 are obtained. This reduced network carries full information of the original power system at the base case.
Kron Reduction
Gaussian elimination avoids the need of matrix inversion while solving the nodal equation of large power systems. Moreover it also leads to reduced order network equivalents. This is used to analyze power system with special focus on voltages at some selected buses. For this purpose selective numbering of s...
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... nodes is same as the sum of injections at the aggregated nodes,
▁S_a=∑_(i∈{A})▒▁S_i
The transformation of network is described as
[■(▁I_R@▁I_A )]=[■(▁Y_RR&▁Y_RA@▁Y_AR&▁Y_AA )][■(▁V_R@▁V_A )]
⟹[■(▁I_R@▁I_a )]=[■(▁Y_RR&▁Y_Ra@▁Y_aR&▁Y_aa )][■(▁V_R@▁V_a )],
For current and voltages at the retained nodes to remain unchanged
▁Y_RA ▁V_A= ▁Y_Ra ▁V_a or, ▁Y_Ra= ▁Y_RA ▁ϑ where, ▁ϑ= ▁V_a^(-1) ▁V_A
Equation (11) can be represented as
▁V_a ▁I_a^*=▁V_A^T ▁I_A^*
Putting values in (16) from (12) gives
▁V_a ▁Y_aR^* ▁V_R^*+▁V_a ▁Y_aa^* ▁V_a^*=▁V_A^T ▁Y_AR^* ▁V_R^*+▁V_A^T ▁Y_AA^* ▁V_A^*
This condition holds true if,
▁Y_aR=〖▁ϑ^(*T) ▁Y〗_AR
▁Y_aa=〖▁ϑ^(*T) ▁Y〗_AA ▁ϑ
Thus method can be electrically interpreted as all the nodes to be aggregated are connected together by ideal transformer with transformation ratios that give a common secondary voltage, ▁V_a.
Job's method is ordinarily applicable to system in which only one complex is present. Robert Gould and Vosburgh [12] have applied above method to system where second equilibrium exists and showed that it also obey's Beer'slaw.
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