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Sequence component analysis of the balanced 3 phase system both for balanced & unbalanced loading conditions.
Ex: Series admittance of 3 phase transmission line ie mutually coupled lines
Admittance matrix is
Admittance matrix representation is
Symmetrical component transformation mutually coupled coils in to three uncoupled coils.
[Ts] is the transformation matrix= 1 1 1 1 a2 a 1 a a2
Therefore transformed admittance matrix becomes
If the system elements are balanced i.e.
[Y012] will be diagonal and mutually coupled 3 phase system is transformed
into 3 uncoupled symmetrical systems. similar to 1phase LF. If [Yabc] is in unbalanced state i.e. elements are unbalanced
[Y012] will be full and sequence networks will be mutually coupled.Symmetrical component frame of reference for problem
analysis fails.Phase coordinate method is adopted.
Compound admittancesConcept based on the use of matrix to represent
admittance of the network.Consider 6 mutually coupled single admittances whose
primitive network is
Primitive admittance matrix relates the nodal injected currents to branch voltages as
By partitioning eqn becomes
This partitioning is eqvt to grouping 6 coils into two compound coils, each composed of 3 individual admittances.
Primitive network using compound admittances
If the coupling between two groups of admittances are bilateral
i.e. if yik=yki
Consider a network represented by single admittances
compound admittances
Its primitive network and primitive admittance matrix is shown below
Primitive network for compound admittance
Actual network admittance matrix can be obtained by linear transformation.
Admittance matrix can be formed by inspectionwhen there is no coupling b/n compound admittance
Diagonal term is the sum of the individual branch admittances connected to the node corresponding to that term.
Off-diagonal term is the negated sum of the branch admittances connected b/n the two corresponding nodes.
Three phase Models of Transmission linesLine parameters are calculated from the line
geometrical characteristics.These are expressed as a series impedance and
shunt admittance per unit length of line. Lumped pi model of a 3 phase short line
Admittance matrix for the short line b/n i & k in terms of 3x3 matrix quantities is given by
Mutually coupled 3 phase transmission linesElectrostatic and Electromagnetic coupling.Two coupled lines form a subsystem of four
system bus bars. Eg.
Admittance matrix for sub system is given by
Mutually coupled parallel transmission lines (bus bars of coupled lines are not distinct)
Shunt ElementsShunt reactors and capacitors.Data given in terms of rated MVA & kV.Admittance in pu is calculated Eg: Shunt capacitor bank
Series ElementsSeries capacitor bank b/n node i&k
Three phase models of synchronous machinesModeled by their sequence impedances because
of the symmetry of phase windings.Phase component representation of generator in
terms of sequence impedance matrix is
Phase component impedance matrix is
Nodal admittance matrix is given by the inverse of impedance matrix which relates the injected currents at generator bus bar to their nodal voltages.
Phase Component model of the generator
Sequence component model of the machine
The voltage at the internal or excitation bus bar form a balanced 3 phase set. i.e
Three phase models of TransformerBeing a balanced 3 phase device it is represented
by its eqvt. sequence network.Also modeled in phase coordinate method based
on primitive admittance matrix.
Primitive Admittance model of 3 phase transformer
Consider a 2 winding 3 phase transformer
The primitive network and primitive admittance matrix is given by
Assuming flux path to be symmetrically distributed between windings
If inter phase coupling neglected, the coupling is modeled as for a single phase unit
Admittance matrix is
Admittance model of actual connected network is formed by linear transformation.
Consider a star-star connected transformer with neutral solidly grounded
The connection matrix is given by
Nodal admittance matrix is
Similarly for a star G-delta transformer
Two winding transformer represented as two compound coils
Sequence component modeling of 3 phase TransformerWith reference to star G-delta connection,
its y node can be partitioned into self and mutual elements and transformation can be applied to obtain the sequence admittance submatrces.
Three phase load flow solutionFor assessing the unbalanced operation of
an interconnected system and any significant load unbalance, three phase LF algorithm is necessary.
Formulation of unbalanced load flow problem requires the formation of nodal admittance matrix of the unbalanced network.
This is assembled by taking one element at a time and modifying the matrix of partial network to reflect the addition. The process is continued till all elements such as machines, lines, transformers, shunt parameters etc are considered.
Overall system admittance matrix is formed by combining the subsystem admittance matrix.
>The self admittance of any bus bar is the sum of all the individual self admittance matrices at that bus bar.
>The mutual admittance between any two bus bars is the sum of the individual mutual admittance matrices from all the subsystems containing those two nodes.
Three phase system behavior is given by the nodal equation,
[I] - [Y] [V]=0Where Y is the nodal admittance matrix or
system admittance matrix containing all sort of unbalances of the system.
Each bus bar is represented by three nodes, each representing a phase.
Each neutral is a node if not solidly grounded.
Each load is assumed decoupled in to three parts and each is connected to a node.
Notations used
AC system is considered to have a total of ‘n’ bus bars,
n=nb + ng nb=no. of actual system bus bars ng=no. of synch machines i, j - system bus barsi=1, nb all load bus bars+ all generator terminal
bus bari=nb + ng internal bus bar at slack machine
i=nb+1, nb+ng-1 all generator internal bus bar except the slack machine
reg- voltage regulatorInt- internal bus bar at a generatorgen- generatorp, m- three phases at a particular bus
The minimum and sufficient set of variables to define the three-phase system under steady-state operation.
The slack generator internal bus bar voltage magnitude V int j where j = nb + ng.
(The angle θint j is taken as a reference.)
The internal busbar voltage magnitude V int j and angles θint j at all other generators,
i.e. j = nb + 1, nb + ng - 1. (Only two variables are associated with each generator internal bus bar as the
three-phase voltages are balanced)
The three voltage magnitudes (Vip) and angles (θ
ip ) at every generator terminal bus bar and every load bus bar in the system, i.e. i = 1, nb and p = 1,3
The equations necessary to solve for the above set of variables are derived from the specified operating conditions,
The individual phase real and reactive power loading at every system bus bar.
The voltage regulator specification for every synchronous machine.
The total real power generation of each synchronous machine, with the exception of slack machine.
At slack machine, fixed voltage in phase and
magnitude, is applicable to the three-phase load flow.
The mathematical statement of the specified conditions in terms of Y matrix is as follows
1. For each of the three phases ( p ) at every load and generator terminal bus bar (i),
2. For every generator j,
where k is the bus number of the jth
generator’s terminal bus bar.
3. For every generator j , with the exception of the slack machine, i.e. j ≠ nb + ng,
the mutual terms Gjk and Bjk are nonzero only when k is the terminal bus bar of the jth generator.
The mathematical formulation of 3 phase LF problem is given by the above three set of independent algebraic eqn in terms of system variables.
Load Flow solution is the set of variables which makes up on substitution the mismatches in the eqn equal to zero.
The solution is obtained in an iterative manner using the Fast Decoupled algorithm.
The problem is defined as
The effects of Δθ on reactive power flows and ΔV on real power flows are ignored, therefore
[ I ] = [M] = [J] = [ N ] = 0 and [C] = [GI = 0.
In addition, the voltage regulator specification is assumed to be in terms of the terminal voltage magnitudes only and therefore
[D] = [H] = 0.
The equation in decoupled form is
for i, k = 1, nb and j, I = 1, ng - 1 (i.e. excluding the slack generator), ‘l ‘ refers to generator internal bus bar
and
for i, k = 1, nb and j , I = 1, ng (i.e. including the slack generator).
Jacobian elements are given as follows Consider for a generator internal bus
bar ‘l’
the sub matrix A &B is given by
Similarly E & F is given by
Where [Fjl]=0 for all j≠l because the jth generator has no connection with the lth generators internal bus bar
Sub matrix K,P,L&R is given by
[Lm
jk ]=0, when k is not the terminal bus bar of jth generator
[R jl]=0 for all j, l as the voltage regulator specification does not include the variables V int
Jacobian approximations1. At all nodes (all phases of all bus bars)
2. Between connected nodes of same phase,
3. The phase angle unbalance at any busbar will be small
4. The angle between different phases of connected bus bars will be 120o i.e.
Applying approximation to the jacobians we get the eqns as
where
All the terms in matrix [M] are constant and is same as –[B] matrix except for the off diagonal terms which connects nodes of different phases.
The reliability and speed of convergence can be improved with some modification in the defining function.
1. The left-hand side defining functions are redefined as [ΔPp
i / Vpi] , [ΔPgen j /V int j] and
[Qpi / Vp
i]2. In equation (1), the remaining right-hand-
side V terms are set to 1 p.u.3. In equation(2), the remaining right-hand-
side V terms are cancelled by the corresponding terms in the right-hand-side vector.
There fore the eqn becomes,
As Vreg is normally a simple linear function of
the terminal voltages, [L’] will be a constant matrix
Therefore, the Jacobian matrices [B’] and [ B’’ ] in equations have been approximated to constants.
The final algorithmic eqn may be written as
The eqns are now solved iteratively using the algorithms
Starting values for the iterations are assigned as
1. The non voltage-controlled bus bars are assigned 1 p.u. on all phases.
2. At generator terminal bus bars all voltages are assigned values according to the voltage regulator specifications.
3. All system bus bar angles are assigned 0, - 1200, + 1200 for the three phases respectively.
4. The generator internal voltages and angles are calculated from the specified real power.
5. For the slack machine the real power is estimated as the difference between total load and total generation plus a small percentage (say 8%) of the total load to allow for losses
Form the system admittance model from the raw data for each system component.
Constant Jacobians B' and B" are formed from the system admittance matrix.
Each equation are then solved using the iterative technique.
The iterative solution process yields the values of the system voltages which satisfy the specified system conditions of load, generation and system configuration.
The three-phase bus bar voltages, the line power flows and the total system losses are calculated.
