May 26, 2020
METHODS OF PATEL LOADFLOW COMPUTATION FOR ELECTRICAL POWER
SYSTEM
FIELD OF THE INVENTION
[0001] The present invention relates to a method of loadflow computation in
power flow control
and voltage control for an electrical power system.
BACKGROUND OF THE INVENTION
[0002] The present invention relates to power-flow/voltage control in
utility/industrial power
networks of the types including many power plants/generators interconnected
through
transmission/distribution lines to other loads and motors. Each of these
components of the power
network is protected against unhealthy or alternatively faulty, over/under
voltage, and/or over
loaded damaging operating conditions. Such a protection is automatic and
operates without the
consent of power network operator, and takes an unhealthy component out of
service by
disconnecting it from the network. The time domain of operation of the
protection is of the order
of milliseconds.
[0003] The purpose of a utility/industrial power network is to meet the
electricity demands of its
various consumers 24-hours a day, 7-days a week while maintaining the quality
of electricity
supply. The quality of electricity supply means the consumer demands be met at
specified voltage
and frequency levels without over loaded, under/over voltage operation of any
of the power
network components. The operation of a power network is different at different
times due to
changing consumer demands and development of any faulty/contingency situation.
In other words
healthy operating power network is constantly subjected to small and large
disturbances. These
disturbances could be consumer/operator initiated, or initiated by overload
and under/over voltage
alleviating functions collectively referred to as security control functions
and various optimization
functions such as economic operation and minimization of losses, or caused by
a fault/contingency
incident.
[0004] For example, a power network is operating healthy and meeting quality
electricity needs of
its consumers. A fault occurs on a line or a transformer or a generator which
faulty component
gets isolated from the rest of the healthy network by virtue of the automatic
operation of its
protection. Such a disturbance would cause a change in the pattern of power
flows in the network,
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which can cause over loading of one or more of the other components and/or
over/under voltage at
one or more nodes in the rest of the network. This in turn can isolate one or
more other
components out of service by virtue of the operation of associated protection,
which disturbance
can trigger chain reaction disintegrating the power network.
[0005] Therefore, the most basic and integral part of all other functions
including optimizations in
power network operation and control is security control. Security control
means controlling power
flows so that no component of the network is over loaded and controlling
voltages such that there
is no over voltage or under voltage at any of the nodes in the network
following a disturbance
small or large. As is well known, controlling electric power flows include
both controlling real
power flows which is given in MWs, and controlling reactive power flows which
is given in
MVARs. Security control functions or alternatively overloads alleviation and
over/under voltage
alleviation functions can be realized through one or combination of more
controls in the network.
These involve control of power flow over tie line connecting other utility
network, turbine
steam/water/gas input control to control real power generated by each
generator, load shedding
function curtails load demands of consumers, excitation controls reactive
power generated by
individual generator which essentially controls generator terminal voltage,
transformer taps control
connected node voltage, switching in/out in capacitor/reactor banks controls
reactive power at the
connected node.
[0006] Control of an electrical power system involving power-flow control and
voltage control
commonly is performed according to a process shown in Fig. [[5]]7, which is a
method of
forming/defining and solving a loadflow computation model of a power network
to affect control
of voltages and power flows in a power system comprising the steps of:
Step-10: obtaining on-line/simulated data of open/close status of all switches
and circuit breakers
in the power network, and reading data of operating limits of components of
the power
network including maximum power carrying capability limits of transmission
lines,
transformers, and PV-node, a generator-node where Real-Power-P and Voltage-
Magnitude-V are given/assigned/specified/set, maximum and minimum reactive
power
generation capability limits of generators, and transformers tap position
limits, or stated
alternatively in a single statement as reading operating limits of components
of the power
network,
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Step-20: obtaining on-line readings of given/assigned/specified/set Real-Power-
P and Reactive-
Power-Q at PQ-nodes, Real-Power-P and voltage-magnitude-V at PV-nodes, voltage
magnitude and angle at a reference/slack node, and transformer turns ratios,
wherein said
on-line readings are the controlled variables/parameters,
Step-30: performing loadflow computation to calculate, depending on loadflow
computation model used, complex voltages or their real and imaginary
components or
voltage magnitude corrections and voltage angle corrections at nodes of the
power
network providing for calculation of power flow through different components
of the
power network, and to calculate reactive power generation and transformer tap-
position
indications,
Step-40: evaluating the results of Loadflow computation of step-30 for any
over loaded power
network components like transmission lines and transformers, and over/under
voltages at
different nodes in the power system,
Step-50: if the system state is acceptable implying no over loaded
transmission lines and
transformers and no over/under voltages, the process branches to step-70, and
if
otherwise, then to step-60,
Step-60: correcting one or more controlled variables/parameters set in step-20
or at later set by the
previous process cycle step-60 and returns to step-30,
Step-70: affecting a change in power flow through components of the power
network and voltage
magnitudes and angles at the nodes of the power network by actually
implementing the
finally obtained values of controlled variables/parameters after evaluating
step finds a
good power system or stated alternatively as the power network without any
overloaded
components and under/over voltages, which finally obtained controlled
variables/parameters however are stored for acting upon fast in case a
simulated event
actually occurs or stated alternatively as actually implementing the corrected
controlled
variables/parameters to obtain secure/correct/acceptable operation of power
system.
[0007] Overload and under/over voltage alleviation functions produce changes
in controlled
variables/parameters in step-60 of Fig.7. In other words controlled
variables/parameters are
assigned or changed to the new values in step-60. This correction in
controlled
variables/parameters could be even optimized in case of simulation of all
possible imaginable
disturbances including outage of a line and loss of generation for corrective
action stored and
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made readily available for acting upon in case the simulated disturbance
actually occurs in the
power network. In fact simulation of all possible imaginable disturbances is
the modern practice
because corrective actions need be taken before the operation of individual
protection of the power
network components.
[0008] It is obvious that loadflow computation consequently is performed many
times in real-time
operation and control environment and, therefore, efficient and high-speed
loadflow computation
is necessary to provide corrective control in the changing power system
conditions including an
outage or failure of any of the power network components. Moreover, the
loadflow computation
must be highly reliable to yield converged solution under a wide range of
system operating
conditions and network parameters. Failure to yield converged loadflow
solution creates blind spot
as to what exactly could be happening in the network leading to potentially
damaging operational
and control decisions/actions in capital-intensive power utilities.
[0009] The power system control process shown in Fig. 7 is very general and
elaborate. It includes
control of power-flows through network components and voltage control at
network nodes.
However, the control of voltage magnitude at connected nodes within reactive
power generation
capabilities of electrical machines including generators, synchronous motors,
and
capacitor/inductor banks, and within operating ranges of transformer taps is
normally integral part
of loadflow computation as described in "LTC Transformers and MVAR violations
in the Fast
Decoupled Load Flow, IEEE Trans., PAS-101, No.9, PP. 3328-3332, September
1982." If
under/over voltage still exists in the results of loadflow computation, other
control actions, manual
or automatic, may be taken in step-60 in the above and in Fig.7. For example,
under voltage can be
alleviated by shedding some of the load connected.
[0010] The prior art and present invention are described using the following
symbols and terms:
Ypq Gpq+ jBpq : (p-q) th element of nodal admittance matrix without
shunts
Ypp Gpp+ B pp : p-th diagonal element of nodal admittance matrix without
shunts
yp = gp + jbp : total shunt admittance at any node-p
VP = ep + jfp = VpZ0p : complex voltage of any node-p
V, = e + jfs = VsZO, : complex slack-node voltage
AO, AVp : voltage angle, magnitude corrections
Afp, Aep : imaginary, real part of complex voltage corrections
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Sp = Pp + j Qp : net nodal injected power, calculated
APp + j AQp : nodal power residue or mismatch
RPp + jRQp : modified nodal power residue or mismatch
RIp + j IIp : net nodal injected current, calculated
ARID + j Allp : nodal injected current residue or mismatch
RRIp + jRIIp : modified nodal current residue or mismatch
SSHp = PSHp + jQSHp: net nodal injected power, scheduled/specified
Cp = 1/T= Cos(top+ j Sint: Unitary rotation/transfoimation
: number of PQ-nodes
: number of PV-nodes
n=m+k+1 : total number of nodes
ep : node-q is connected to node-p excluding the case of q=p
[ : indicates enclosed variable symbol to be a vector or
matrix
LRA : Limiting Rotation Angle, -48 for invented models
PQ-node: load-node, where, Real-Power-P and Reactive-Power-Q are specified
PV-node: generator-node, where, Real-Power-P and Voltage-Magnitude-V are
specified
Vs Vi VN : Slack-node voltage magnitude, Base value, and Nominal value
of voltage
magnitude are very closely similar, and therefore, they can be used
interchangeably. However, in
the following development only Vs will be used. Particularly, in the treatment
of loadflow problem
with distributed slack-node, there is no specific slack-node and VB or VN can
be used.
Loadflow Computation: Each node in a power network is associated with four
electrical quantities,
which are voltage magnitude, voltage angle, real power, and reactive power.
The loadflow computation involves calculation/deteimination of two unknown
electrical quantities for other two given/specified/scheduled/set/known
electrical quantities for each node. In other words the loadflow computation
involves deteimination of unknown quantities in dependence on the
given/specified/scheduled/set/known electrical quantities.
Loadflow Model: a set of equations describing the physical power network and
its operation for
the purpose of loadflow computation. The term loadflow model' can be
alternatively referred to as 'model of the power network for loadflow
computation'. The process of writing Mathematical equations that describe
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physical power network and its operation is called Mathematical Modeling. If
the equations do not describe/represent the power network and its operation
accurately the model is inaccurate, and the iterative loadflow computation
method could be slow and unreliable in yielding converged loadflow
computation. There could be variety of Loadflow Models depending on
organization of set of equations describing the physical power network and its
operation, including Decoupled Loadflow Models, Super Decoupled Loadflow
Models, Fast Super Decoupled Loadflow (FSDL) Model, and Super Super
Decoupled Loadflow (SSDL) Model.
Loadflow Method: sequence of steps used to solve a set of equations describing
the physical
power network and its operation for the purpose of loadflow computation is
called Loadflow Method, which telin can alternatively be referred to as
`loadflow computation method' or 'method of loadflow computation'. One
word for a set of equations describing the physical power network and its
operation is: Model. In other words, sequence of steps used to solve a
Loadflow Model is a Loadflow Method. The loadflow method involves
definition/founation of a loadflow model and its solution. There could be
variety of Loadflow Methods depending on a loadflow model and iterative
scheme used to solve the model including Decoupled Loadflow Methods,
Super Decoupled Loadflow Methods, Fast Super Decoupled Loadflow (FSDL)
Method, and Super Super Decoupled Loadflow (SSDL) Method. All
decoupled loadflow methods described in this application use either successive
(10, 1V) iteration scheme or simultaneous (1V, 10) iteration scheme, defined
in the following.
[0011] Prior art method of loadflow computation of the kind carried out as
step-30 in Fig. 7,
include a class of methods known as decoupled loadflow. This class of methods
consists of
decouled loadflow and super decoupled loadflow methods including Super Super
Decoupled
Loadflow method all foimulated involving Power Mismatch computation and polar
coordinates.
Prior-art Loadflow Computation Methods are described in details in the
documents of Research
Publications and granted patents cited in Infoimation Disclosure Statement
(IDS) by this inventor.
Therefore, prior art methods will not be described here.
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SUMMARY OF THE INVENTION
[0012] It is a primary object of the present invention to improve convergence
and efficiency of the
prior art Super Super Decoupled Loadflow computation method under wide range
of system
operating conditions and network parameters for use in power flow control and
voltage control in
the power system. A further object of the invention is to reduce computer
storage/memory or
calculating volume requirements.
[0013] The above and other objects are achieved, according to the present
inventions, Patel
Loadflow (PL-1 & PL-2), Patel Super Decoupled Loadflow (PSDL-YY1 & PSDL-YY2),
Y matrix
based ¨ Patel Loadflow (CPL-1 & CPL2), Sparse Z or C-1 matrix - Patel Loadflow
(SZPL or
SCIPL), Accurate and Reliable Guass-Seidel Loadflow (ARGSL) Methods and their
many
variants, for loadflow calculation for Electrical Power System. In context of
voltage control, one
of the inventive systems of PSDL-YY2 and others listed in the above methods of
loadflow
computation is used for Electrical Power system consisting of plurality of
electromechanical
rotating machines, transformers and electrical loads connected in a network,
each machine having
a reactive power characteristic and an excitation element which is
controllable for adjusting the
reactive power generated or absorbed by the machine, and some of the
transformers each having a
tap changing element, which is controllable for adjusting turns ratio or
alternatively terminal
voltage of the transformer, said system comprising:
means defining and solving one of the loadflow models of the power network
listed in the
above for providing an indication of the quantity of reactive power to be
supplied
by each generator including the reference/slack node generator, and for
providing
an indication of turns ratio of each tap-changing transformer in dependence on
the
obtained-online or given/specified/set/known controlled network variables/
parameters, and physical limits of operation of the network components,
machine control means connected to the said means defining and solving
loadflow model
and to the excitation elements of the rotating machines for controlling the
operation
of the excitation elements of machines to produce or absorb the amount of
reactive
power indicated by said means defining and solving loadflow model in
dependence
on the set of obtained-online or given/specified/set controlled network
variables/parameters, and physical limits of excitation elements,
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transformer tap position control means connected to the said means defining
and solving
loadflow model and to the tap changing elements of the controllable
transformers
for controlling the operation of the tap changing elements to adjust the turns
ratios
of transformers indicated by the said means defining and solving loadflow
model in
dependence on the set of obtained-online or given/specified/set controlled
network
variables/parameters, and operating limits of the tap-changing elements.
[0014] The method and system of voltage control according to the preferred
embodiment of the
present invention provide voltage control for the nodes connected to PV-node
generators and tap
changing transformers for a network in which real power assignments have
already been fixed.
The said voltage control is realized by controlling reactive power generation
and transformer tap
positions.
[0015] One of the inventive methods of defining and solving loadflow
computation models PL-1,
PL-2, PSDL-YY1, PSDL-YY2, CPL-1, CPL-2, SZPL or SCIPL, or ARGSL can be used
for
voltage control in Electrical power System. For this purpose real and reactive
power assignments
or settings at PQ-nodes, real power and voltage magnitude assignments or
settings at PV-nodes
and transformer turns ratios, open/close status of all circuit breaker, the
reactive capability
characteristic or curve for each machine, maximum and minimum tap positions
limits of tap
changing transformers, operating limits of all other network components, and
the impedance or
admittance of all lines are supplied. A decoupled loadflow system of equations
1(28) and (29)1 or
1(30) and (31)1 is solved by an iterative process until convergence. During
this solution the
quantities which can vary are the real and reactive power at the
reference/slack node, the reactive
power set points for each PV-node generator, the transformer transformation
ratios, and voltages
on all PQ-nodes nodes, all being held within the specified ranges. When the
iterative process
converges to a solution, indications of reactive power generation at PV-nodes
and transformer
turns-ratios or tap-settings are provided. Based on the known reactive power
capability
characteristics of each PV-node generator, the determined reactive power
values are used to adjust
the excitation current to each generator to establish the reactive power set
points. The transformer
taps are set in accordance with the turns ratio indication provided by the
system of loadflow
computation.
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[0016] For voltage control, system of PSDL-YY2 or others and many variants
listed in the above
Methods of Loadflow computation can be employed either on-line or off-line. In
off-line
operation, the user can simulate and experiment with various sets of operating
conditions and
determine reactive power generation and transformer tap settings requirements.
A general-purpose
computer can implement the entire system. For on-line operation, the loadflow
computation
system is provided with data identifying the current real and reactive power
assignments and
transformer transformation ratios, the present status of all switches and
circuit breakers in the
network and machine characteristic curves in steps-10 and -20 in Fig. 7, and
steps 12, 14, 18, 22,
24, 32, 34, and 38 in Fig. 8 described below. Based on this information, a
model of the system
based on coefficient matrices of invented loadflow computation systems provide
the values for the
corresponding node voltages, reactive power set points for each machine and
the transformation
ratio and tap changer position for each transformer.
[0017] The present inventive system of loadflow computation for Electrical
Power System
consists of, one of the Patel Super Decoupled Loadflow: YY2-version (PSDL-YY2)
or PSDL-
B'B', or others listed in the above Methods characterized in that 1) single
decoupled coefficient
matrix solution requiring only 50% of memory used by prior art methods, 2) the
presence of
transformed values of known/given/specified/scheduled/set quantities in the
diagonal elements of
the gain matrices [Yf] and [Ye] of the decoupled loadflow sub-problems, and 3)
transformation
angles are restricted to maximum of ¨0 to ¨90 (say, -48 ) to be determined
experimentally, 4)
PV-nodes being active in both RI-f and The sub-problems, PQ-node to PV-node
and PV-node to
PQ-node switching is simple to implement, and these inventive loadflow
computation steps
together yield some processing acceleration and consequent efficiency gains,
and are each
individually inventive, and 5) modified real and imaginary current mismatches
at PV-nodes in
case of PSDL-YY1, SSDL-YY, HSSDL-YY, ESSDL-YY or their generalized variations
PSDL-
B'B', SSDL-B'B', HSSDL-B'B', ESSDL-B'B', are determined as RRIp = (Lpfp+epAPp
)/[Kp(ep2 +
f2)] and RIIp = (Lep - fpAPp)/[Kp(ep2 + 42)] in order to keep gain matrices
[Yf] and [Ye]
symmetrical. If the value of factor Kp=1, the gain matrices [Yf] and [Ye]
becomes unsymmetrical
in that elements in the rows corresponding to PV-nodes are defined without
transformation or
rotation applied, as Yfpq= Yepq= -Bpq. It is possible that Patel Super
Decoupled methods can be
formulated in polar coordinates by simply replacing correction vectors [Af]
and [Ae] in equations
(28) and (29) and subsequently followed equations by correction vectors [AO]
and [AV].
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However, it will not be easy to have single gain matrix model, because [AV]
for PV-nodes is zero
and absent.
BRIEF DESCRIPTION OF DRAWINGS
[0018] Fig. 1 is a flow-chart embodiment of the invented PSDL-YY1 computation
method.
[0019] Fig. 2 is a flow-chart embodiment of the invented PSDL-YY2 computation
method.
[0020] Fig. 3 is a flow-chart embodiment of the invented Y matrix based Patel
Loadflow (CPL-1)
computation method using complex algebra.
[0021] Fig. 4 is a flow-chart embodiment of the invented Y matrix based Patel
Loadflow (CPL-2)
computation method using complex algebra.
[0022] Fig. 5 is a flow-chart embodiment of the invented method of sparse [Z]
or [C]-1 based
Patel Loadflow (SZPL) or (SCIPL) computation method using complex algebra.
[0023] Fig. 6 is a flow-chart embodiment of the invented ARGSL computation
method.
[0024] Fig. 7 is a flow-chart of the overall controlling method for an
electrical power system
involving loadflow computation as a step which can be executed using one of
the loadflow
computation methods embodied in Figs. 1, 2, 3, 4, 5 or 6.
[0025] Fig. 8 is a flow-chart of the simple special case of voltage control
system in overall
controlling system of Fig.7 for an electrical power system.
[0026] Fig. 9 is a one-line diagram of an exemplary 6-node power network
having a
reference/slack/swing node, two PV-nodes, and three PQ-nodes.
[0027] Fig. 10a is a parallel computer organization/architecture that can be
applied for parallel
solution of SZPL or SCIPL and ARGSL methods (US Patent No.: 7788051 of year
2010).
00281 Fig. 10b is a parallel computer organization/architecture with wireless
interconnect that can
be applied for parallel solution of SZPL or SCIPL and ARGSL methods (US Patent
No.:
9891827 of year 2018).
DESCRIPTION OF A PREFERED EMBODYMENT
[0029] A loadflow computation is involved as a step in power flow control
and/or voltage control
in accordance with Fig. 7 or Fig. 8. A preferred embodiment of the present
invention is described
with reference to Fig. 8 as directed to achieving voltage control.
[0030] Fig. 9 is a simplified one-line diagram of an exemplary utility power
network to which the
present invention may be applied. The fundamentals of one-line diagrams are
described in section
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6.11 of the text ELEMENTS OF POWER SYSTEM ANALYSIS, fourth edition, by William
D.
Stevenson, Jr., McGrow-Hill Company, 1982. In Fig. 9 each thick vertical line
is a network node.
The nodes are interconnected in a desired manner by transmission lines and
transformers each
having its impedance, which appears in the loadflow models. Two transformers
in Fig.9 are
equipped with tap changers to control their turns ratios in order to control
terminal voltage of
node-1 and node-2 where large loads are connected.
[0031] Node-6 is a reference/slack-node alternatively referred to as the slack
or swing -node,
representing the biggest power plant in a power network. Nodes-4 and ¨5 are PV-
nodes where
generators are connected, and nodes-1, -2, and ¨3 are PQ-nodes where loads are
connected. It
should be noted that the nodes-4, -5, and ¨6 each represents a power plant
that contains many
generators in parallel operation. The single generator symbol at each of the
nodes-4, -5, and ¨6 is
equivalent of all generators in each plant. The power network further includes
controllable circuit
breakers located at each end of the transmission lines and transformers, and
depicted by cross
markings in one-line diagram of Fig. 9. The circuit breakers can be operated
or in other words
opened or closed manually by the power system operator or relevant circuit
breakers operate
automatically consequent of unhealthy or faulty operating conditions. The
operation of one or
more circuit breakers modify the configuration of the network. The arrows
extending certain nodes
represent loads.
[0032] A goal of the present invention is to provide a reliable and
computationally efficient
loadflow computation that appears as a step in power flow control and/or
voltage control systems
of Fig. 7 and Fig. 8. However, the preferred embodiment of loadflow
computation as a step in
control of terminal node voltages of PV-node generators and tap-changing
transformers is
illustrated in the flow diagram of Fig. 8 in which present invention resides
in function steps 42 and
44.
[0033] Short description of other possible embodiment of the present invention
is also provided
herein. The present invention relates to control of utility/industrial power
networks of the types
including plurality of power plants/generators and one or more motors/loads,
and connected to
other external utility. In the utility/industrial systems of this type, it is
the usual practice to adjust
the real and reactive power produced by each generator and each of the other
sources including
synchronous condensers and capacitor/inductor banks, in order to optimize the
real and reactive
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power generation assignments of the system. Healthy or secure operation of the
network can be
shifted to optimized operation through corrective control produced by
optimization functions
without violation of security constraints. This is referred to as security
constrained optimization of
operation. Such an optimization is described in the United States Patent
Number: 5,081,591 dated
Jan. 13, 1992: "Optimizing Reactive Power Distribution in an Industrial Power
Network", where
the present invention can be embodied by replacing the step nos. 56 and 66
each by a step of
constant gain matrices [Yf] and [Ye], and replacing steps of "Exercise Newton-
Raphson
Algorithm" by steps of "Exercise PSDL-YY1 or PSDL-YY2 or CPL-1 or CPL-2 or
SZPL or
SCIPL or ARGSL Computation" in places of steps 58 and 68. This is just to
indicate the possible
embodiment of the present invention in optimization functions like in many
others including state
estimation function. However, invention is being claimed through a simplified
embodiment
without optimization function as in Fig. 8 in this application. The inventive
steps-42 and ¨44 in
Fig.8 are different than those corresponding steps-56, and ¨58, which
constitute a well known
Newton-Raphson loadflow method, and were not inventive even in United States
Patent Number:
5,081,591.
[0034] In Fig. 8, function step 12 provides stored impedance values of each
network component in
the system. This data is modified in a function step 14, which contains stored
information about
the open or close status of each circuit breaker. For each breaker that is
open, the function step 14
assigns very high impedance to the associated line or transformer. The
resulting data is than
employed in a function step 16 to establish an admittance matrix for the power
network. The data
provided by function step 12 can be input by the computer operator from
calculations based on
measured values of impedance of each line and transformer, or on the basis of
impedance
measurements after the power network has been assembled.
[0035] Each of the transformers Ti and T2 in Fig. 9 is a tap changing
transformer having a
plurality of tap positions each representing a given transformation ratio. An
indication of initially
assigned transformation ratio for each transformer is provided by function
step 18 in Fig. 8.
[0036] The indications provided by function steps 14, and 22 are supplied to a
function step 42 in
which constant gain matrices [Yf] and [Ye], or [Y] or sparse [Z] or sparse [C]-
1 of any of the
invented PSDL-YY1 or PSDL-YY2 or CPL-1 or CPL-2 or SZPL or SCIPL or ARGSL
models are
constructed, factorized or inverted and stored. The coefficient matrices [Yf]
and [Ye], or [C] or
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sparse [C]-1 or sparse [Z] are conventional tools employed for solving PSDL-
YY1 or PSDL-YY2
or CPL-1 or CPL-2 or SZPL or SCIPL models defined by equations 1(28) and (29)1
or 1(30) and
(31)1 or 1(56) or (58)1 or 1(69) or (70)1 of a power system. [C] is the most
general representation
of all possible matrices involved in the solution of linear and non-linear
simultaneous algebraic
equations. [C] could be the Jacobian, approximated Jacobian, constant
Jacobian, approximated
constant decoupled Jacobian in case of Newton-Raphson based approaches. It
could be the
coefficient matrix, approximated coefficient matrix, constant coefficient
matrix, approximated
constant decoupled coefficient matrix in case of Patel Numerical Method (PNM)
based approaches
described as preferred embodiments in this application. [CI' when fully
inverted is the full matrix.
However, it can be made sparse by storing and processing only selected
elements, and it becomes
approximation of fully inverted [CI'.
[0037] Indications of initial reactive power, or Q on each node, based on
initial calculations or
measurements, are provided by a function step 22 and these indications are
used in function step
24, to assign a Q level to each generator and motor. Initially, the Q assigned
to each machine can
be the same as the indicated Q value for the node to which that machine is
connected.
[0038] An indication of measured real power, P, on each node is supplied by
function step 32.
Indications of assigned/specified/scheduled/set generating plant loads that
are constituted by
known program are provided by function step 34, which assigns the real power,
P, load for each
generating plant on the basis of the total P, which must be generated within
the power system. The
value of P assigned to each power plant represents an economic optimum, and
these values
represent fixed constraints on the variations, which can be made by the system
according to the
present invention. The indications provided by function steps 32 and 34 are
supplied to function
step 36 which adjusts the P distribution on the various plant nodes
accordingly. Function step 38
assigns initial approximate or guess solution to begin iterative method of
loadflow computation,
and reads data file of operating limits on power network components, such as
maximum and
minimum reactive power generation capability limits of PV-nodes generators.
[0039] The indications provided by function steps 24, 36, 38 and 42 are
supplied to function step
44 where inventive PSDL-YY1 or PSDL-YY2 or CPL-1 or CPL-2 or SZPL or SCIPL or
ARGSL
model solution is carried out, the results of which appear in function step
46. The loadflow
computation yields voltage magnitudes and voltage angles at PQ-nodes, real and
reactive power
13
Date Recue/Date Received 2020-05-27
May 26, 2020
generation by the reference/slack/swing node generator, voltage angles and
reactive power
generation indications at PV-nodes, and transfoimer turns ratio or tap
position indications for tap
changing transfaimers. The system stores in step 44 a representation of the
reactive capability
characteristic of each PV-node generator and these characteristics act as
constraints on the reactive
power that can be calculated for each PV-node generator for indication in step
46. The indications
provided in step 46 actuate machine excitation control and transfoimer tap
position control. All the
loadflow computation methods using inventive PSDL-YY1 or PSDL-YY2 or CPL-1 or
CPL-2 or
SZPL or ARGSL computation models can be used to affect efficient and reliable
voltage control in
power systems as in the process flow diagram of Fig. 8.
[0040] Particularly inventive PSDL-YY1 or PSDL-YY2 or CPL-1 or CPL-2 or SZPL
or ARGSL
models in teims of equations for deteimining elements of vectors [RI'], [II'],
[ARP], [MI'1, [I],
[AI] and elements of coefficient matrices [Yf] and [Ye], or [C] or [Z] or
sparse {[Z] or [CV} are
described followed by computation steps of corresponding methods are
described.
[0041] The presence of values of known/given/specified/scheduled/set
quantities in the diagonal
elements of the coefficient matrix [Yf] and [Ye], or [C] or [Z] or sparse {[Z]
or [C]'}, which
takes different faun for different methods, is brought about by such
foimulation of loadflow
equations. The said quantities in the diagonal elements in the coefficient
matrices improved
convergence and the reliability of obtaining converged loadflow computation.
[0042] The slack-start is to use the same voltage magnitude and angle as those
of the
reference/slack/swing node as the initial guess solution estimate for
initiating the iterative
loadflow computation. With the specified/scheduled/set voltage magnitudes, PV-
node voltage
magnitudes are adjusted to their known values after the first P-0 iteration.
This slack-start saves
almost all effort of mismatch calculation in the first P-f iteration. It
requires only shunt flows from
each node to ground to be calculated at each node, because no flows occurs
from one node to
another because they are at the same voltage magnitude and angle.
Patel Numerical Method
[0043] All inventions of this application are based on newly propounded Patel
Numerical Method
(PNM), which is applicable to both linear and nonlinear simultaneous algebraic
equations first of
its kind in about 200 years.
[0044] Propounding Statement of Patel Numerical Method (PNM)
14
Date Recue/Date Received 2020-05-27
May 26, 2020
1. Organize linear or nonlinear equations as mismatch functions equated to
zero.
2. In each of the mismatch functions, club any teim with known quantities
or value into a
diagonal teim with simple algebraic manipulations.
3. Express a vector of the mismatch functions as a product of a coefficient
matrix and a vector
of unknown variables, which can sometimes be treated as a correction vector of
unknown
variables.
4. Equate the vector of mismatch functions to the product of the
coefficient matrix and the
vector of unknown variables or the correction vector of unknown variables to
be calcula-
ted.
5. Solve such a matrix equation by iterations for the vector of unknown
variables or the
correction vector of unknown variables using evaluation of the vector of
mismatch
functions with guess values of unknown variables to begin with, and inverting
or factoring
the coefficient matrix.
Alternatively the PNM can be propounded as:
1. Let yp = f(xq) be a system of n linear or non-linear simultaneous algebraic
equations in n
variables, where subscripts p and q are 1, 2, ..., n. It could also be a
single equation in
single variable. yp are "constant terms" or "known value terms", and xq are
"unknowns" or
"variables" to be deteimined or calculated or solved for.
2. Subtract from each side the other side as: yp - f(xq) = f(xq) - yp
transfaiming functional
equations into equations of mismatches.
3. Express usually a right hand side as a product of the coefficient matrix
and the vector of
unknown variables or the correction vector of unknown variables to be
deteimined after
clubbing yp teims into diagonal terms with simple algebraic manipulations.
4. Solve such a matrix equation by iterations for the vector of unknown
variables or the
correction vector of unknown variables using evaluation of usually the left
hand side
vector of mismatch functions with guess values of unknown variables to begin
with, and by
inverting or factoring the coefficient matrix.
5. It is also possible to replace the coefficient matrix of right hand side by
the Jacobian or its
many possible variants by taking first order differentiations with respect to
relevant real or
complex valued variables of the right hand side, and the resulting numerical
method may be
referred to as Newton-Raphson-Patel (NRP) numerical method.
Date Recue/Date Received 2020-05-27
May 26, 2020
6. Such a matrix equation involving either coefficient matrix or Jacobian and
its many
possible variants can also be solved by Gauss or Gauss-Seidel iterations not
requiring
coefficient matrix factorization or inversion, and the resulting numerical
methods may be
referred to as Gauss-Patel (GP) and Gauss-Seidel-Patel (GSP) methods
respectively.
7. Patel Numerical Method (PNM) is the first known formalized updating attempt
to the
classical numerical methods, and they can be referred to as Gauss-Patel (GP),
Gauss-
Seidel-Patel (GSP), and Newton-Raphson-Patel methods, in addition to its being
the new
numerical method on its own.
[0045] Preliminary investigations suggest that Patel Numerical Method may
potentially produce
monotonous convergence, and therefore may be amenable to acceleration factors
unlike Newton-
Raphson method.
[0046] Patel Loadflow ¨ 1 (PL-1)
The PL-1 Model comprises eqns. (1) to (9)
[R; 1 e
[ C 1 [ f 1
(1)
I
[f 1 (r+1) [ ci -1[ Ri (r)
(2)
II
Where,
RIp = (epPSHp + fpQSHp)/(ep2 + fp2) = - [(Bpp bp)fp + 1 Bpqfq] + [(Gpp +
gp)e + 1 GPq eq ] (3)
cFp P cl>1)
Hp ¨ (epQSHp - fpPSHp)/(ep2 + fp2) ¨ -[(Gpp + gp)fn + 1 Gnnfn'] - [(Bpp
bp)ep + 1 BPq eq ] (4)
r cFp " cl>1)
[ l [Bf Ge
C = l
(5)
Gf Be j
Bfpq ¨ Bepq ¨ -BIN Bfpp= Bepp= - (Bpp bp)
(6)
Gfpq ¨ -Gepq ¨ 'Gm Gfpp= -Gepp= - (Gpp gp)
(7)
[0047] The equations (1) to (7) represents linearized global solution of the
nonlinear loadflow
equations. Local nonlinearity can be handled by introduction of self-
iterations as per equations (8)
to (9).
16
Date Recue/Date Received 2020-05-27
May 26, 2020
[fp(sr-(1)] (r+1) pup/Bfpo (sr)] (r)
(8)
[ep(sr+1)] (r+1) = [(Ilp/Bepp)(s1)] (I)
(9)
[0048] Equations (8) to (9) are solved independently for each node, and can be
performed
simultaneously in parallel for all the nodes. Equations (2) and {(8) and (9)1
are solved in
sequence. In other words linear global solution followed by non-linear local
(nodal) solution by
self-iterations, or non-linear local (nodal) solution by self-iterations
followed by linear global
solution.
[0049] It should be noted that equations (3) and (4) can be organized for
Decoupled-Gauss-Seidel-
Patel formulation involving factor Lp defined in the following equation (68).
Factor Lp takes care
of diagonal dominance issue and ensures that all the Loadflow Computation
Models developed in
this application almost always provide converged solution, particularly when
diagonal dominance
issue arises in the presence of capacitive series branch or an excessive shunt
capacitive
compensation at a node. This is being achieved for the first time since the
development of
Loadflow Computation Models began in 1950s, and for first the time in about
200 years of use
and application of Gauss-Seidel numerical method in different subject areas.
Patel Loadflow ¨2 (PL-2)
[0050] The PL-2 model comprises eqns. {(11) and (12)1 or (14), (5), (15) to
(20), and {(21) to
(24)1 or 425) to (26)1.
ARI Af
(10)
[AII 1 [ CI [ Ael
rAfi (r+1) [ Cl -1 ARP (I)
(11)
L. Ae j (AII .
[ f 1 (r+1) if I (I) [ [ Afl (r+1) e [e +
Ae J
(12)
[AR1 [ CI [ f 1
(13)
AII e
17
Date Recue/Date Received 2020-05-27
May 26, 2020
i fi (r+1) -1
1 [ARP (r)
[ i = [ C
.,
(14)
e AII
Where,
ARID = (epPSHp + fpQSHp)/(ep2 + fp2) + [(Bpp bp)fp + Icillyqfq] - [(Gpp
gp)ep 1 q9rpqed (15)
ARID = [ {(Bpp+bp)+Q SHp/(ep2+fp2)} fp+ ilp3pqfq] - [ {(Gpp+gp)-
PSHp/(ep2+fp2)} ep+ V, meg] (15)
ARID = (epAPp + fpAQp) / (ep2+ fp2)
(15)
ARID ,,--f RepPSHp + fpQSHp)/(ep2 + 42)] - [(epPSHp + fpQSHp)/(es2 + fs2)]
(15)
Allp = (epQSHp - fpPSHp)/(ep2 + fp2) + [(Gpp + gp)fp + Igypqfq] + [(Bpp
bp)ep + 1 qB,req] (16)
Allp = [ {(Gpp+gp)-PSHp/(ep2+fp2)} fp+ wpc,fd + [ {(Bpp+bp)+Q SHp/(ep2+fp2)}
ep+ Ilbeq] (16)
Allp = (epAQp - fpAPp) / (ep2+ fp2)
(16)
Allp ,,--f RepQSHp - fpPSHp)/(ep2 + 42)] - [(epQSHp - fpPSHp)/(es2 + fs2)]
(16)
Bfpq = Bepq = - Bpq
(17)
Gfpq ¨ -Gepq ¨ - Gpq
(18)
Bfpp = Bepp = - [Bpp bp] - QSHp/(ep2 + fp2) ,c--f -[B bp] - QS}Ipi(es2
fs2) (19)
Gfpp= -Gepp= - [Gpp gp] PS}Ipi(ep2 fp2) ':"-- - [Gpp gp] P Slipi(eS2
t2) (20)
[0051] Equations (15) and (16) provides alternative expressions of real and
imaginary current
mismatches where AQp = 0.0 at PV-nodes. An alternative definition of PL-2
model can be
provided by defining ARID of (15) and Allp of (16) as the subtraction of the
temis containing
specified values from the calculated values that would make ARID and Allp
defined by eqns. (15)
and (16) and elements of [C] defined by eqns. (17) to (20) negative.
[0052] It can be seen that diagonal elements of the coefficient matrix [C] are
changing with
changing values of (ep2 + fp2), and therefore, requiring time consuming re-
factorization of [C] in
each iteration. To avoid re-factorization, it is proposed to make [C] constant
by using (es2 + fs2),
the slack-node voltage values, instead of (ep2 + fp2) in equations (19) and
(20) requiring
factorization or full inversion of [C] only once in the beginning of the
iteration process.
[0053] The equations (10) to (20) represents linearized global solution of the
nonlinear loadflow
18
Date Recue/Date Received 2020-05-27
May 26, 2020
equations. Local nonlinearity can be handled by introduction of self-
iterations as per equations
421) to (24)1 or 1(25) to (26)1. It is possible to expand in detail all
equations involving self
iterations as in equations (21), (23), (25), (26), (54), (55), (66), (67) etc.
in the following.
[Afp(sr+1)] (r+1) RARIp/Bfpp) (sr)] (r)
(21)
[fp(sr+1)] (r+1) = [fp(sr)] (r)
[Afp (sr+1)] (r+1) (22)
[Aep(sr+1)] (r(1)= [(AIIp/Bepp) (sr)] (r)
(23)
[ep(sr+1)] (r+1) [ep(sr)] (r)
[Aep (sr+1)] (r+1) (24)
[0054] Equations 421) to (24)1 or 1(25) to (26)1 are solved independently for
each node, and can
be performed simultaneously in parallel for all the nodes. Equations 411) and
(12)1 or (14), and
421) to (24)1 or 1(25) and (26)1 are solved in sequence. In other words linear
global solution
followed by non-linear local (nodal) solution by self-iterations, or non-
linear local (nodal)
solution by self-iterations followed by linear global solution.
[fp(sr+1)] (r+1) RARIp/Bfpp) (sr)] (r)
(25)
[ep(sr+1)] (r+1) RAIIp/Bepp) (sr)] (r)
(26)
Patel Super Decoupled Loadflow (PSDL)
[0055] In a class of super decoupled loadflow models, each super decoupled
loadflow model
comprises a system of equations 1(28) and (29)1 or 1(30) and (31)1 differing
in the definition of
elements of [ART'], [All'], [RI'], [II'], and [Yfl and [Ye]. It is a system of
equations for the separate
calculation of imaginary part of and real part of complex voltage or its
corrections. [C] is the
transformed coefficient matrix.
rYf
C'l =
(27)
0 Ye
[ART'] = [Yfl [Af]
(28)
[All'] = [Ye] [Ae]
(29)
([ART'] or [RI']} = [Yfl [fl
(30)
1[AII1 or [II']} = [Ye] [e]
(31)
Successive (1f, le) Iteration Scheme
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Date Recue/Date Received 2020-05-27
May 26, 2020
[0056] In this scheme {(28) and (29)1 or {(30) and (31)1 are solved
alternately with inteitnediate
updating. Each iteration involves one calculation of HARP] or [MT and {[Af] or
[f] } to update
[f] and then one calculation of HAM or [II']} and {[Ae] or [e]} to update [e].
The sequence of
relations {(32) to (35)1 or {(36) to (37)1 depicts the scheme.
[Af] = [Yf] -1 [AU]
(32)
[fl = [fl [Af]
(33)
[Ae] = [Ye] -1 [All']
(34)
[e] = [e] + [Ae]
(35)
[f] = [Yf] -1 {[ARI1 or [RI']}
(36)
[e] = [Ye]' {[AII1 or [II']}
(37)
[0057] The scheme involves solution of system of equations {(28) and (29)1 or
{(30) and (31)1 in
an iterative manner depicted in the sequence of relations {(32) to (35)1 or
{(36) to (37)1. This
scheme requires calculation for each half iteration because HARP] and [All']
or {[RI1 and [II']
is calculated always using the most recent imaginary part of and real part of
complex voltage
values, and it is block Gauss-Seidel approach. The scheme is block successive,
which imparts
increased stability to the solution process, and it in turn improves
convergence and increases the
reliability of obtaining solution.
Patel Super Decoupled Loadflow - 1 (PSDL-YY1)
[0058] The PSDL-YY1 model comprises equations{(32) to (35)1or {(36) to (37)1&
(38) to (50).
Where,
Yfpq¨ Yepq¨ - Ypq : for branch r/x ratio < 3.0
- (Bpq 0.9(Ypq-Bpq)) : for branch
r/x ratio > 3.0
Brtn : for branches connected between two PV-
nodes or a PV-node and the slack-node
(38)
Yfpp= Yepp= bp' + Lp + Yfpq
(39)
ep
bp' = - (QS}IpCos(top - PS}IpSin(top)/(es2+ fs2) - bpCos(top : at PQ-node
(40)
bp' = - Qpo/(es2 + fs2) - bp : at PV-node
(41)
(QO- calculated at initial estimate solution)
ARID' = Lpfp + ARIpCos(top + AIIpSin(top : for PQ-nodes
(42)
Date Recue/Date Received 2020-05-27
May 26, 2020
ARID' = Lpfp + (epAPp' + fpAQp') /(ep2 + fp2) : for PQ-nodes
(42)
AIIp' = Lep + AIIpCos(top - ARIpSin(top : for PQ-nodes
(43)
AIIp' = Lep + (epAQp' - fpAPp')/(ep2+ fp2) : for PQ-nodes
(43)
APR' = APpCos(top + AQpSin(top : for PQ-nodes
(44)
AQp' = AQpCos(top - APpSin(top : for PQ-nodes
(45)
ARID = (Lpfp +epAPp ) / [Kp(ep2+ 42)] : for PV-nodes
(46)
Allp = (Lep - fpAPp) / [Kp(ep2+ 42)] : for PV-nodes
(47)
Costo = I [ Bpp/A/(Gpp2 +Bpp2)] I > COS (00 to -90 ) : to be determined
experimentally (48)
Sinto = - I [Gpp/A/(Gpp2 +Bpp2)] I > Sin (0 to -90 ) : to be determined
experimentally (49)
Kp = I ((Lp +Bpp)/ (Lp + I-Yfpq) I
(50)
ep
Lp in all equations is defined by equation (68) in the following.
Super Super Decoupled Loadflow (SSDL-YY)
[0059] Two new versions of SSDL-YY are provided. One is Hybrid SSDL-YY (HSSDL-
YY) and
the other is Efficient SSDL-YY (ESSDL-YY). The HSSDL model comprises eqns.
(32) to (35),
(38a), (38b), (39a), (39b), (40a), (40b), (41a), (41b), {(42) to (45)}, (46a),
(47a), and (48) to (50).
The ESSDL-YY model comprises eqns. (32) to (35), (38c), (39a), (39b), (40a),
(40b), {(41c) and
(41d)} where QSHp replaced by Qpo calculated value at initial estimate at PV-
nodes, {(42) and
(43)} with approximate versions of {(15) and (16)1 where QSHp replaced by Qp
(calculated value)
at PV-nodes, and {(48) and (49)1.
Yfpq= t- -YPq : for branch r/x ratio < 3.0
-(Bpq 0.9(Ypq-Bpq)) : for branch r/x ratio > 3.0
-13,-F4 : for branches connected between two PV-
nodes or a PV-node and the slack-node
(38a)
Yepq- [ -Ypq : for branch r/x ratio < 3.0
-(Bpq 0.9(Ypq-Bpq))
: for branch r/x ratio > 3.0 (38b)
Yfpq = Yepq= [-Ypq : for branch r/x ratio < 3.0
-(Bpq 0.9(Ypq-Bpq)) : for branch r/x ratio > 3.0
(38c)
Yfpp = bfp' + Lp + I-Yfpq
(39a)
ep
21
Date Recue/Date Received 2020-05-27
May 26, 2020
Yepp = bee' + Lp + I-YePq
(39b)
ep
bfp' = -(QS}IpCos(top - PSHpSin(top)/(es2+ fs2) - bpCos(top
: at PQ-node (40a)
bee' = +(QS}IpCos(top - PSHpSin(top)/(es2+ fs2) - bpCos(top
: at PQ-node (40b)
bfp' = 0.0 : at PV-node
(41a)
bee' = 10.010 (say, it is chosen very large value) : at PV-node
(41b)
bfp' = -(QS}IpCos(top - PSHpSin(top)/(es2+ fs2) - bpCos(top
: at PV-node (41c)
bee' = +(QS}IpCos(top - PSHpSin(top)/(es2+ fs2) - bpCos(top :
at PV-node (41d)
ARID = APp / (KpVp2) : at PV-node
(46a)
Allp = 0.0 : at PV-node
(47a)
[0060] Branch admittance magnitude in (38), (38a), (38b), (38c) is of the same
algebraic sign as
its susceptance. Rotation angles are to be determined as per (48) and (49),
and could be restricted
to the maximum anywhere ¨0 to ¨90 degrees to be determined experimentally.
There can be
many possible variations of PSDL, HSSDL, and ESSDL models, and the one
variation being their
generalized versions PSDL-B'B', HSSDL-B'B', and ESSDL-B'B' where B' symbolizes
suceptance
matrix transformed, B'pq= Bpq+Gpqtan(topq and tan(topq= Gpq/Bpq. Also, the two
versions PSDL-YY
and PSDL-B'B', HSSDL-YY and HSSDL-B'B', and ESSDL-YY and ESSDL-B'B' can be
mixed
in any possible combination. Corresponding transformed diagonal elements Bpp'
and transformed
mismatches can easily be determined.
Slack-start
[0061] Slack-Start is use of the same voltage magnitude and angle as those of
the slack-node for
all nodes as an initial guess solution. With the specified magnitudes, PV-
nodes voltage
magnitudes are adjusted to their known values after the first half iteration.
This start procedure
referred to as the slack-start, saves almost all effort of mismatch
calculation in the first P-f
iteration as it requires only shunt flows to be calculated at each node.
[0062] where, Kp is defined in equation (50) which is initially restricted to
the minimum value
of 0.75 determined experimentally; however its restriction is lowered to the
minimum value
of 0.6 when its average over all less than 1.0 values at PV nodes is less than
0.6.
[0063] In super decoupled loadflow models [Yf] and [Ye] are real, sparse,
symmetrical and built
only from network elements. Since they are constant, they need to be
factorized once only at the
22
Date Recue/Date Received 2020-05-27
May 26, 2020
start of the solution. Equations 1(28) and (29)1 or 1(30) and (31)1 are to be
solved repeatedly by
forward and backward substitutions. [Yf] and [Ye] are of the same dimensions
(m+k) x (m+k)
when only a row/column of the reference/slack-node is excluded and both are
triangularized using
the same ordering regardless of the node-types.
[0064] Unlike the HSSDL and the prior art SSDL (Super Super Decoupled
Loadflow, presented
at Toronto International Conference ¨ Science and Technology for Humanity-
2009, pages: 652-
659) methods, the PSDL methods are single matrix loadflow computations
substantially reducing
memory requirements, and since all nodes are active in the iterative process
implementations of
PQ-node to PV-node and PV-node to PQ-node switching is simple. The best
possible convergence
from non-linearity consideration could be achieved by restricting rotation
angle to maximum of -0
to ¨90 degrees (say, -48 degrees) to be deteimined experimentally.
[0065] The steps of loadflow computation method, PSDL-YY1 method are shown in
the flowchart
of Fig. 1. Computation steps of HSSDL method are similar, therefore, they are
not given
explicitly. Referring to the flowchart of Fig.1, different steps are
elaborated in steps marked with
similar letters in the following. Double lettered steps are the characteristic
steps of PSDL-YY1
method. The words "Read system data" in Step-a correspond to step-10 and step-
20 in Fig. 7, and
step-16, step-18, step-24, step-36, step-38 in Fig. 8. All other steps in the
following correspond to
step-30 in Fig.7, and step-42, step-44, and step-46 in Fig. 8.
a. Read system data and assign an initial approximate solution. If better
solution estimate is
not available, set voltage magnitude and angle of all nodes equal to those of
the slack-node,
referred to as the slack-start.
b. Form nodal admittance matrix, and Initialize iteration count ITRF =
ITRE= r = 0
c. Compute Cosine and Sine of nodal rotation angles using equations (48),
(49), and store
them. If Costo < Cos (0 to -90 degrees, to be determined experimentally), set
Costo = Cos
(say, 0 to -90 degrees to be determined experimentally) and Sinto = Sin (say,
0 to -90
degrees to be determined experimentally).
dd. Form, factorize, and store (m+k) x (m+k) matrix [Yf] or [Ye] of {(28)
and (29)1 or {(30)
and (31)1 in a compact storage exploiting sparsity, using equations (38) to
(41).
e. Compute residues [AP] at PQ- and PV-nodes and [AQ] at only PQ-nodes. If
all are less
than the tolerance (6), proceed to step-n. Otherwise follow the next step.
23
Date Recue/Date Received 2020-05-27
May 26, 2020
ff. Compute the vector of transformed residues [ARP] as in (42) for PQ-
nodes, and using (46)
and (50) for PV-nodes.
gg. Solve 1(32) for [Af]} or 1(36) for [f]} and update using, [f] = [f] +
[Af].
h. Set voltage magnitudes of PV-nodes equal to the specified values, and
Increment the
iteration count ITRF=ITRF+1 and r=(ITRF+ITRE)/2.
i. Compute residues [AP] at PQ- and PV-nodes and [AQ] at PQ-nodes only. If
all are less
than the tolerance (6), proceed to step-n. Otherwise follow the next step.
Compute the vector of transformed residues [All'] as in (43) for PQ-nodes, and
using (47)
and (50) for PV-nodes.
kk. Solve 1(34) for [Ae]} or 1(37) for [e]} and update using [e] = [e] +
[Ae].
1. Calculate reactive power generation at PV-nodes and tap positions of tap-
changing
transformers. If the maximum and minimum reactive power generation capability
and
transformer tap position limits are violated, implement the violated physical
limits and
adjust the loadflow solution by the method like one described in "LTC
Transformers and
MVAR violations in the Fast Decoupled Load Flow, IEEE Trans., PAS-101, No.9,
PP.
3328-3332, September 1982".
m. Increment the iteration count ITRE=ITRE+1 and r=(ITRFATRE)/2, & Proceed
to step-e.
n. From calculated values of real and imaginary components of nodal
voltages, reactive
power generation at PV-nodes, and tap position of tap changing transformers,
calculate
power flows through power network components.
Patel Super Decoupled Loadflow ¨2 (PSDL-YY2)
[0066] The Patel Super Decoupled Loadflow-2 (PSDL-YY2) model comprises
equations 1(32) to
(35)1 or 1(36) to (37)1, 1(3), (4), (51), (52), and (40c) or 1(42), (43) with
approximate versions of
(15) and (16), and (40)1, (39), and 1(53), (54) and (55)1. In (3), (4), (15),
and (16): QSHp is
replaced by Qp (calculated) for PV-nodes, and in (40) QSHp is replaced by Qpo
(calculated at
initial estimate) for PV-nodes.
Where,
RI'p = RIp Costo + IIp Sin(top + Lpfp
(51)
II', = IIp Costo - RIp Sinto + Lep
(52)
24
Date Recue/Date Received 2020-05-27
May 26, 2020
Yfpq¨ Yfpq¨ -Ypq [ : for branch r/x ratio < 3.0
-(3pq 0.9(Ypq-Bpq)) : for branch r/x ratio > 3.0
(53)
bp' = - bpCos(DP : at PV-nodes
(40c)
[0067] All equations other than (54) and (55) of the model PSDL-YY2 represents
linearized global
solution of the nonlinear loadflow equations. Local nonlinearity can be
handled by introduction of
self-iterations as per equations (54) to (55).
[fp(sr+1)] (r+1) [{(RI' or ARI'p)/Yfpp} (sr)] (r)
(54)
[ep(sr+1)] (r+1)
[{(II'p or Affp)/Yepp} (Sr)]
(r)
(55)
Equations (54) to (55) are solved independently for each node, and can be
perfoitned
simultaneously in parallel for all the nodes. Super Decoupled equations {(32)
or (36), and (54)1
and {(34) or (37), and (55)1 are solved in sequence. In other words linear
global solution followed
by non-linear local (nodal) solution by self-iterations.
[0068] It should be noted that in addition to being derived from a new Patel
Numerical Method
(PNM) first of its kind in about 200 years, the models PSDL-YY1 and PSDL-YY2
with their
relevant defining equations {(38), (39), (40), (41)} and [{(53a), (39), (40)}
or {(53b), (39),
(40c)}] respectively indicate single decoupled matrix models. The models
require factorization
and storage of single decoupled matrix for its solution reducing computer
storage requirements
and factorization time to almost 50%. It is a significant reduction in
computational resources
(computer time, storage, and electricity) for achieving the same result, in
addition to being highly
reliable in providing converged solution because of application of factor L.
There is a huge
literature on the subject of Loadflow Computation Models and their solution
methods developed
since 1950s, but a single decoupled matrix model has never been achieved.
[0069] The steps of loadflow computation method, PSDL-YY2 method are shown in
the
flowchart of Fig. 2. Computation steps of ESSDL method are similar, therefore,
they are not given
explicitly. Referring to the flowchart of Fig.2, different steps are
elaborated in steps marked with
similar letters in the following. Triple lettered steps are the characteristic
steps of PSDL-YY2
method. The words "Read system data" in Step-a correspond to step-10 and step-
20 in Fig. 7, and
step-16, step-18, step-24, step-36, step-38 in Fig. 8. All other steps in the
following correspond to
Date Recue/Date Received 2020-05-27
May 26, 2020
step-30 in Fig.7, and step-42, step-44, and step-46 in Fig. 8.
a. Read system data and assign an initial approximate solution vectors
[fO], [e0], and store it.
If better solution estimate is not available, set voltage magnitude to 1.0 pu
at load nodes
and specified values at PV-nodes, and angle of all nodes equal to that of the
slack-node,
referred to as the flat-start.
b. Form nodal admittance matrix, and Initialize iteration count ITRF =
ITRE= r = 0 and
MXDF=MXDE=0.0
c. Compute Cosine and Sine of nodal rotation angles using equations (48),
(49), and store
them. If Costo < Cos (0 to -90 degrees , to be determined experimentally), set
Costo =
Cos (say, 0 to -90 degrees to be determined experimentally) and Sinto = Sin
(say, 0 to -90
degrees to be deteimined experimentally).
ddd. Form, factorize, and store (m+k) x (m+k) size matrix [Yf] or [Ye] of
{(28) and (29)1 or
{(30) and (31)1 in a compact storage exploiting sparsity, using equations
{(53a), (39), (40)
and (41)1 or {(53b), (39), and (40c)1 .
eee. Compute the vector of transformed residues [ART'] using (42) {with
approximated values
of [ART] and [All] from (15) and (16)1 or [RI'] using (51). Use calculated
values of Qp in
place of QSHp for PV-nodes, and implement Q-limit violations at PV-node
generators.
fff. Solve {(32) for [Af]} or {(36) for [f]}, perform Self-Iterations for
each node using (54),
and update using, [f] = [f] + [Af].
g. Take a difference of vectors {[f ] ¨ [f0]}, find the maximum of the
element difference and
store it in variable 'MXDF', and perfoim [f0] = [f] and Increment the
iteration count
ITRF=ITRF+1 and r=(ITRFATRE)/2.
hl. Are MXDF and MXDE both less than specified tolerance? If it is, Proceed
to step-n or else
follow the next step.
Compute the vector of modified residues [ATP] using (43) {with approximated
values
[ART] and [All] from (15) and (16)1or [II'] using (52). Use calculated values
of Qp in place
of QSHp for PV-nodes, and implement Q-limit violations at PV-node generators.
Solve {(34) for [Ae]} for {(37) for [e]1, perfaim Self-Iterations for each
node using (55),
and update using [e] = [e] + [Ae].
k. Take a difference of vectors {[e] ¨ [e0]}, find the maximum of the
element difference and
store it in variable 'MXDE', and perfoim [e0] = [e] and Increment the
iteration count
26
Date Recue/Date Received 2020-05-27
May 26, 2020
ITRE=ITRE+ 1 and r=(ITRFATRE)/2 .
11. Are MXDF and MXDE both less than specified tolerance? If it is not,
Proceed to step-eee
or else follow the next step.
n. From calculated values of real and imaginary components of nodal
voltages, reactive
power generation at PV-nodes, and tap position of tap changing transformers,
calculate
power flows through power network components.
Coefficient matrix [C] based Patel Loadflow (CPL)
[0070] Patel Loadflow model can be organized in coefficient matrix [C] based
complex form,
because it is not involved with any partial differentiation of original or
mismatch functions. The
model constitutes eqns. {(57) or (59)1, {(60) to (62)1 or {(63) to (65)1 or
{(60a), (61), and (62a)1,
{(66) or (67)1, and (68). It involves one solution of {(57) or (59)1 followed
by one solution of
{(66) or (67)1, or one solution of {(66) or (67)1 followed by one solution of
{(57) or (59)1.
However, {(66) or (67)1 constitutes one equation for each node except the
Slack-node, and
equations for all the nodes can be solved in parallel, just like Gauss
numerical method.
[AI] = [C] [AV]
(56)
[AV] = [C]-1 [AI]
(57)
OR
{[AI] or [I]} = [C] [V]
(58)
[V] = [C]-1 {[AI] or [I]}
(59)
Where, components of vectors [I] and [AI], and matrix [C] are defined in the
following:
Ip = (PSHp - jQSHp)/(ep - jfp) = (SSHp*Np*) = [(Ypp+ yp)Vp + ii typNci]
(60a)
AS (SSHp*¨ Sp* )= [(PSHp ¨ j Q SHp) ¨ (Pp ¨ j Qp )] = (AP_ j AQp )
(60)
Alp= (SSHp*¨ Sp* )Np* = [(PSHp ¨ j QSHp) ¨ (Pp ¨ jQp )]/Arp*= (APp ¨ j AQp
)/Vp* (60)
AIp= [ISSIlp*/(ep2 + 42)1 - (Ypp+yp)]Vp ¨ 1 YpqVci
(60)
cl>P
AIp;---=-= [ISSIlp*/(ep2 +42)1 - {Lp ssiip*i(es2 +f,2)}wp= SSHp*Np*- Lp
smip*viivs2 (60)
AIp z [ISSIlp*/(ep2 +42)1 - Lp]Vp= SSHp*Np* - LpVp
(60)
Cpq Ypq
(61)
Cpp=[(Ypp+yp) ¨ ISSHp*/(ep2 + 42)1] ,=,' [(Ypp+yp) ¨ {Lp ssHp*/(es2 +t2)}]
(62)
27
Date Recue/Date Received 2020-05-27
May 26, 2020
Cpp = [(Ypp+Yp) ¨ Lid
(62)
CPP = (YPICEYP)
(62a)
OR
AI= (Sp*- SSIIp*)Np* = [(Pp ¨ j Qp) - (P SHp - j QSHp)]/Vp* = [(-APp ) - j (-
AQp )]Np* (63)
AIp = [(Ypp+yp) ¨ {SSHp*/(ep2 fp2)}wp ypqvq
(63)
AI [{Lp SSHp*/(e52 +f52)} {SSHp*/(ep2 fp2)1]¨p
v Lp SSHp* Vp/Vs2¨ SSHp*Np*
(63)
Alp [Lp - {SSHp*/(ep2 fp2)}]¨p
v LpVp ¨ SSHp*Np*
(63)
Cpq -Ypq
(64)
Cpp= [ {Lp SSHp*/(ep2 42)}
(Ypp-Fyp) [{Lp SSHp*/(e52 f52) (ypp+3,)]
(65)
CPP = 1LP ¨ (YPICEYP)1
(65)
[AVp(sr+1)] (r+1) [(mpicpp) (sr)] (r)
(66)
[vp(sr+1)] (r+1) [((AIp or I)/C) (sr)] (r)
(67)
Lp = - co, ..., -1, 0, +1, + co (including fractions)
(68)
[0071] The equations (62), (65), and (68) provide elegant formulation for
diagonal elements of
the coefficient matrix [C] that suggest a mechanism for their numerical
manipulations particularly
useful when diagonal dominance issue arise in the presence of a capacitive
series branch or an
excessive capacitive compensation at a node. The factor Lp of different value
can be applied
separately to real and imaginary components of a diagonal element of [C].
Similar developments
can be provided for Patel Super Decoupled Loadflow models and other loadflow
models.
Equations (66) and (67) and their expanded versions can also be written with
factor L.
[0072] It can be seen that diagonal elements of the coefficient matrix [C] are
changing with
changing values of Vp, and therefore, values of (ep2 fp2) during iteration
process requiring time
consuming re-factorization of [C] in each iteration. To avoid re-
factorization, it is proposed to
make [C] constant by using (e52 + f52), the slack-node voltage values, instead
of (ep2 fp2) in
equations (62) and (65) requiring factorization of [C] only once in the
beginning of the iteration
process.
[0073] It should be noted that equations (61), (62), (63), and (66), (67),
(68) can be organized as
28
Date Recue/Date Received 2020-05-27
May 26, 2020
ARGSL formulations of equations (76), (80) and (82) for solution of individual
node equation.
The factor Lp in equations {(39), (42), (43), (46), (47), (50), (51), (52),
(60), (62), (63), (65)} and
in equations {(76), (80), (82)} provides an experimentation opportunity to
study the effect of
degree of diagonal dominance on the convergence of Patel Super Decoupled
Loadflow (PSDL),
CPL, ARGSL iterative methods and similar application of numerical methods for
solution of
similar problems in different subject areas. It is at this stage the factor Lp
can be finalized and
used in formation of matrices [Yf] and [Ye] of {(28), (29)} or {(30), (31)},
and matrix [C] of (56)
or (58) before inverting and solving by Patel Sparse [C] Inverse Solver (PSIS)
of the next section.
Coefficient matrix [C] based Patel Loadflow-1 (CPL-1)
[0074] The steps of loadflow calculation by CPL-1 method are shown in the
flowchart of Fig. 3.
Referring to the flowchart of Fig.3, different steps are elaborated in steps
marked with similar
numbers in the following. Double numbered steps are the inventive steps. The
words "Read
system data" in Step-a correspond to step-10 and step-20 in Fig. 7, and step-
16, step-18, step-24,
step-36, step-38 in Fig. 8. All other steps in the following correspond to
step-30 in Fig.7, and step-
42, step-44, and step-46 in Fig. 8.
1. Read system data and assign an initial approximate solution. If better
solution estimate is
not available, set voltage magnitude and angle of all nodes equal to those of
the slack-node,
referred to as the slack-start.
2. Form nodal admittance matrix [Y], and Initialize iteration count ITR = 0
33. Form, factorize, and store (m+k) x (m+k) size complex coefficient
matrix [C] of {(56) or
(58)1 in a compact storage exploiting sparsity, using equations 461) and (62)}
or {(64)
and (65)1.
44. Compute complex residues [AS] and [AI] using (60) or (63). However, use
calculated
value Qp instead of QSHp for PV-nodes, and implement violated Qmax or ()imp
limits of PV-
node generators, and change the status of violated PV-node to PQ-node.
5. Find residues [AP] at PQ-nodes and PV-nodes and [AQ] at PQ-nodes from
complex vector
[AS], and if all are less than the tolerance (6), proceed to step-9. Otherwise
follow the next
step.
66. Solve (57) for [AV] or (59) for [V], perform Self-Iterations for only
violated PV-nodes
using (66) or (67). Update voltage using, [V] = [V] + [AV]. Adjust real and
reactive
29
Date Recue/Date Received 2020-05-27
May 26, 2020
components of PV-node complex voltage for specified voltage magnitude.
7. Increment the iteration count ITR=ITR+1, and Go to step-4
10. From calculated values of complex nodal voltages, output reactive power
generation at PV-
nodes, tap position of tap changing transformers, and calculate power flows
through
power network components.
Recently, it has been found out that when Q-limit violations are enforced at
PV-node generators,
real and reactive power mismatch convergence check may not work to produce
converged
solution and continue to oscillate to the maximum number of iterations for the
solution
process to stop. However, it provides converged solution if Q-limit violations
not enforced,
and PV-nodes are not present such as in some distribution power networks.
Therefore, it is
preferred to have two stages of convergence check by both real and reactive
components of
complex voltage change from iteration to iteration, and also by real and
reactive power
mismatch after convergence by the first criterion particularly when Q-limit
violations need
to be enforced as in CPL-2 method of the following.
Coefficient matrix [C] based Patel Loadflow-2 (CPL-2)
[0075] The steps of loadflow calculation by CPL-2 method are shown in the
flowchart of Fig. 4.
Referring to the flowchart of Fig.4, different steps are elaborated in steps
marked with similar
numbers in the following. Triple numbered steps are the inventive steps. The
words "Read system
data" in Step-a correspond to step-10 and step-20 in Fig. 7, and step-16, step-
18, step-24, step-36,
step-38 in Fig. 8. All other steps in the following correspond to step-30 in
Fig.7, and step-42, step-
44, and step-46 in Fig. 8.
1. Read system data and assign an initial approximate solution vector [V0],
and store it. If
better solution estimate is not available, set voltage magnitude to 1.0 pu at
load nodes and
specified values at PV-nodes, and angle of all nodes equal to that of the
slack-node,
referred to as the flat-start.
2. Form nodal admittance matrix [Y], and Initialize iteration count ITR = 0
333. Form, factorize, and store (m+k) x (m+k) size complex coefficient matrix
[C] of {(56) or
(58)1 in a compact storage exploiting sparsity, using equations 461) and
(62)}, or {(64)
and (65)} or 461) and (62a)}.
444. Compute the vector {[I] using (60a)} or {[AI]} using (60) or (63)1.
However, use
calculated value Qp instead of QSHp for PV-nodes, and implement violated Qmax
or Qmm
Date Recue/Date Received 2020-05-27
May 26, 2020
limits of PV-node generators, and change the status of violated PV-node to PQ-
node.
66. Solve {(57) for [AV]} or {(59) for [V]1, perform Self-Iterations only
for violated PV-
nodes using (66) or (67), update voltage using [V] = [V] + [AV], and adjust
real and
reactive components of PV-node complex voltage for specified voltage
magnitude.
777. Take a difference of vectors {[V] ¨ [V0]}, find the maximum of the real
and imaginary
components differences, and store them as DEMX and DFMX, perform [VO] = [V],
and
increment iteration count ITR=ITR+1.
8. Are both DEMX and DFMX less than specified tolerance (EV)? If they are not,
Proceed to
step-444 or else follow the next step.
9. Compute real power mismatch APp at PQ+PV nodes, compute reactive power
mismatch
AQp at PQ nodes, and if all (or both DPMX and DQMX) are not less than
specified
tolerance (EM), go to step 444, otherwise follow the next step.
10. From calculated values of complex nodal voltages, reactive power
generation at PV-nodes,
and tap position of tap changing transformers, calculate power flows through
power
network components.
Patel Sparse Inverse Solver (PSIS) and Sparse [C]-1 (pronounced as 'C
inverse',
which is sparse [Z]) based Patel Loadflow (SCIPL or SZPL):
[0076] Matrix complex [C] or real [C] is the coefficient matrix based on
original equations
(functions) or organized as mismatch equations (functions) in the solution of
linear or non-linear
simultaneous algebraic equations. Inverses complex [C]-1 and real [C]-1 are
sometimes
correspondingly referred to as complex [Z]-matrix and real [Z]-matrix in this
application. The
complex [C]-1 and real [C]-1 can also represent admittance matrices
respectively complex [Ye and
real [Y]-1. Complex ICI' and real [C]-1 are generalized representations,
wherein real [C]-1 can be
Newton-Raphson approach based Jacobian [J]-1 and its simplified approximations
including
inverses of decoupled or Super Decoupled matrices.
100771 The complex [C] and real [C] are generally sparse matrices wherein many
of its off
diagonal elements are zeros. In order to save computation time and computer
storage, processing
of off-diagonal elements that are zeros can be avoided by sparsity preserving
programing
techniques. However, fully inverted complex [C]-matrix and real [C]-matrix are
full matrices
wherein no off-diagonal elements are zeros. Therefore, it is proposed to make
complex ICI' -
31
Date Recue/Date Received 2020-05-27
May 26, 2020
matrix and real [C]-1-matrix sparse by selectively choosing off-diagonal
elements that need to be
stored and processed, and thereby introducing approximations. There are two
extremes, one is to
store and process only one element in each raw corresponding to the diagonal
element introducing
maximum approximation and the other is to store and process the diagonal and
all the off-
diagonal elements in each row introducing zero approximation. And there are
many situations in
between the two extremes stated in the above to be determined experimentally
depending on the
nature of the problem for optimal use of computational resources (computer
time and computer
storage). In electrical circuits (networks), one situation for Complex ICI'
and real [C]-1 is to store
and process off-diagonal elements only corresponding to directly connected
nodes (level-1
connectivity) to a given node, which is the same sparsity of the matrix
complex [C] or real [C].
For a given node, other situations are to store off-diagonal elements
corresponding to directly
connected nodes (level-1), and directly connected nodes to level-1 nodes
(level-2 nodes), and
directly connected nodes to level-2 nodes (level-3 nodes), and so on. For a
given node, the level of
outward connectivity is to be determined experimentally to determine number of
off-diagonal
element required to be stored and processed in the complex [C]-1-matrix and
the real [C['-matrix
for efficient and reliable computation.
[0078] In equations (69) and (70): vectors [V] and [I] are of complex voltage
and complex current
element components respectively. Vectors [AV] and [AI] are composed of complex
voltage
correction and complex current mismatch components respectively. Voltage and
current quantities
appear in electrical circuits. Equation (69) corresponds to equation (59),
Equation (70)
corresponds to equation (57), and relevant quantities are defined in equations
(60) to (65), and
(68), wherein complex [Z] becomes complex [C]-1.
[0079] Equation (69) corresponds to two super decoupled equations (36) and
(37), and equation
(70) corresponds to two super decoupled equations (32) and (34). Relevant
quantities are defined
in equations (38) to (50), and (51) to (53). It should be noted that all
quantities involved are real
and not complex, wherein real [Z] becomes equivalent to two super decoupled
real matrices [Yr]-1
and [Ye]-1, which are the same and identical in case of PSDL-YY1 and PSDL-YY2
models.
[0080] Application of Newton-Raphson approach to solution of simultaneous non-
linear algebraic
equations involves calculation of correction vector in each iteration and
requires updating as in
equations (33) and (35) in case of decoupled models.
32
Date Recue/Date Received 2020-05-27
May 26, 2020
All the computation models and their solution methods developed in this
application are for
electrical power network. However, similar computation models and their
solution methods can be
developed using techniques developed in this application for all possible
areas of study and
application that requires solution of linear or non-linear simultaneous
algebraic equations.
Computation models and their solution methods could be for a system, a
circuit, a machine, an
apparatus, a device, a material etc.
[0081] It should be noted that ZPL [[or]]and SZPL or CIPL and SCIPL are
embarrassingly parallel
problems, and readily amenable to parallel processing. This inventor believes,
an approach
outlined in the above is likely to work. If it works subject to verification
by this inventor, it can
produce grand simplifications in the sense that no need for specialized
triangulation and back-
substitution or factorization software, and no need for storing indexing and
addressing information
required for processing elements of factorized matrix. It appears the next
numerical wonder is
brewing. This inventor is humbled listening words Guardians of Galaxy
chanting: "Indeed,
Mr. Patel, You are the one, chosen".
[0082] Fifteen different variant of invented models constitute: {(69) and
(72a)}, and {(73a) or
(73b) or ((71a), (71b), and (73c)) or ((71c), (71d), and (73d)) or (73e)} OR
{(69) or (70) and
(72b)}, and {(74a) or (74b) or ((71a), (71b), and (74c)) or ((71c), (71d), and
(74d)) or (74e)} .
However, {(73a) or (73c) or (73d) or (730 or (74a) or (74c) or (74d) or (7401
constitutes one
equation for each node except the Slack-node, and equations for all the nodes
can be solved in
parallel, just like Gauss numerical method with self-iteration for each-node
to handle local non-
linearity. The name of self-iteration, formulation of self-iterations in an
equation, and its analogy
to planet earth spinning on its own axis while making rounds around the Sun
Self iterations were
introduced by this inventor the first-time in the year 2005 in his patent # US
7788051 and
Canadian Patent # 2548096 issued January 5, 2011. This is a grand
Gaussification of all the
possible classical numerical methods. Equation {(73b) or (73e) or (73g) or
(74b) or (74e) or
(74g)} constitutes one equation for each node except the slack-node, and
equations for all the
nodes can be solved in sequence like Gauss-Seidel numerical method with self-
iteration for each-
node to handle local non-linearity. This is a grand Gauss-Seidelization of all
the possible
classical numerical methods. Gauss numerical method is Raflembarrassingly
parallel. However,
the best approach seems to solve nodal equation for each node and nodal
equations of its directly
33
Date Recue/Date Received 2020-05-27
May 26, 2020
connected nodes in sequence like Gauss-Seidel numerical method with self-
iteration for each-node
to handle local non-linearity on separate processor simultaneously in parallel
by the technique
introduced by this inventor in his patent # US 7788051 and Canadian Patent #
2548096 issued
January 5, 2011. The parallelization technique of patent # US 7788051 has
produced 10-times
speed-up in Ybus formulation of Gauss-Seidel loadflow method involving self-
iterations. The same
parallelization technique applied to sparse [Z] based models (69) and (70),
could potentially
produce 20-to-40 times speed-ups and could be of the order of 100 times speed-
up in case of large
problems. It looks like a revolution (Patelution) in numerical computation.
The real matrix [Z]
or the complex matrix [Z] in equations (69) and (70) can also be created by
using building
algorithm to create real or complex inverted coefficient matrix [C] or complex
inverted admittance
matrix [Y] or inverted real Jacobian matrix [J] and [[its]]their different
variations.
[V] = [Z] {[AI] or [I]} OR
(69)
[AV] = [Z] [Al]
(70)
[0083] Wherein, though it is possible to write equations (69) and (70) in
complex form or real
form in terms of real and imaginary components, of involved
variables/parameters relevant to
problem being solved, development in the following is given only for complex
versions of
equations (69) and (70) involving variables/parameter (voltage, current, and
admittance) relevant
to an electrical circuit or a network where,
components of vectors [V], [I], [AV], [Al], and special Symbols are defined in
the following:
": means node q is directly connected to node-p
q<p : means node-q among directly connected are processed prior to the current
node-p
q>p : means node-q among directly connected are yet to be processed after the
current node-p
nq : No. of off-diagonal elements in a row-p of [Z] that correspond to
directly connected nodes
to a node-p
nk : No. of off-diagonal elements in a row-p of [Z] that correspond to not
directly connected
nodes to a node-p = (n-1) ¨ nq
n : No. of total elements in a row-p of [Z] that corresponds to total no.
of nodes or equations
p-1 n
ZKp = {IZpk +IZpk} /(n-1)
(71a)
k=1 k=p+1
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Date Recue/Date Received 2020-05-27
May 26, 2020
p-1 n p-1 n
IKp ={IIk +DO / (n-1) OR AIKp / (n-1)
(71b)
k=1 k=p+1 k=1 k=p+1
p-1 n
ZKp = {IZpk +IZpk} /(nk)
(71c)
k=1 k=p+1
k#q k#q
p-1 n p-1 n
IKp ={IIk +DO / (nk) OR AIKp / (nk)
(71d)
k=1 k=p+1 1=1 k=p+1
k#q k#q k#q k#q
Ip = SSHp*Np* = (PSHp - jQSHp ) / (ep - jfp)
(72a)
Alp = SSHp*Np*- (Ypp Yp)Vp YpqVg
(72b)
q¨>p
Sparse Complex Matrix-Z formulation:
[Vp(sr+11(r+1) = zpp[ {(Ip)(sr)} (I)]
IZpqici( (73a)
q¨>p
q<p q>p
[Vp(sr+11(r+1) = zpp[Ip)(sr)} (r) [IZpq(Ig)(r+1)) +IZpq(Ig)(11
(73b)
q¨>p q¨>p
[vp(sr+1)] (r+1) zpp [ {(4)(sr)}(r)] (n_i)(zKpoKie
:from (71a), (71b)
(73c)
[V1 i(1 zpp [ {(4)(sr)}(r)] (r)s
(1)) +(nk)(ZKp)(IKp)(r) :from (71c), (71d)
(73d)
q¨>p
q<p q>p
[vp(sr+1)] (r+1) zpp[ip)(sr)}(r)] [Izpoo(r+1)) Izpq(ion
+(nk)(ZKp)(IKp)(r)
(73e)
q¨>p q¨>p
Full Complex Matrix-Z fotmulation:
p-1
[vp(sr+11(r+1)
Zpp[SSHp* /{(V*))} (r)] IZpq(SSHq* /(Vq*)(r))+ IZpq(SSHq* /(Vq*)(r))
(730
cr1 q=p+1
P-1
[vp(sr+1)] (r+1)
Zpp [SRI; /{(Vp*)(sr)} { IZpq(SSHq* /(Vq*)(r+1)) +IZpq(SSHq*
/(Vq*)(r))} (73g)
q1q=p+1
Sparse Complex Matrix-Z fotmulation:
Date Re cue/Date Received 2020-05-27
May 26, 2020
[AVp(sr+i)](r+l) = Zpp[{(AIp)(sr)}(r)] + IZpq(Alq)(r))
(74a)
q¨>p
q<p q>p
[AVp(sr+1)](r+1) = Zpp[ {(j
4)(sr)} (r)]+ IZpq(Alq)(1* 1)) +IZpq(Alq))
(74b)
q¨>p q¨>p
[AVp(sr+1)](r+1) = Zpp[ {(411"11)(sr)} (r)] +(n-1)(ZKI)(IKI,)(r)
:from (71a), (71b) (74c)
[AVp(sr+1)](r+1) =Zpp [ [ (Alp*)(sr)} (I)] +IZpq(Alq*)(r)HIlk)(ZKI,)(AlKp)(r):
from (71c), (71d) (74d)
q¨>p
q<p q>p
[AVp(sr+1)](r+1) =Z[ {(AI *)(sr)} (r)]+ql_>Zppci q>
Alq*P+1)) +ZppciAlq*P)) +(11k)(ZKO(AlKp)(r)
P
(74e)
Full Complex Matrix-Z formulation:
p-1 n
[Avp(sr+l)](r+1) ¨ zpp[ 441111)00} (1)] Izpq(ALIII*11)(0) IzpodqR*11)(0)
(740
q=1 q=p+1
P-1 n
[Avp(sr+l)](r+1) ¨ zpp[ 441111)00} (1)] Izpq(ALIII*11)(1+1))
Izpq(AIqR*11)(0) (74g)
q=1 q=p+1
I VP(r+1) - VP I < E
(75)
[0084] Matrix [Z] can also be made-up of real or complex components, and it is
an inverse of
coefficient matrix of linear and non-linear equations organized in different
possible ways
including in super-decoupled form, or an inverse of the Jacobian [J]-1 or its
different constant or
approximated variations including decoupled or super decoupled versions. It
should be noted that
equations (72a) and (72b) are the same as (60a) and (60) respectively, and
(60s) and (63s) are
different variations of (60). Equations (71a) to (74g) are provided for the
case matrix [Z] is built
with network ground as reference and equation for slack-node current Is and AI
s based on all other
calculated nodal I's and AI's and known value of slack-node voltage can be
written. However, for
slack-node as reference requires deletion of a row and a column corresponding
to the slack-node
from matrix [Z] built from ground as reference, and simple algebraic
manipulation of equations
(72a) and (72b). In Newton-Raphson (NR) approach based real matrix [Z] which
is inverted
Jacobian, or its constant Jacobian and Super Decoupled variations, the
dimension of real matrix
[Z] with a row and column corresponding to the slack-node removed is the same
as the complex
matrix [Z] with the slack-node as reference. Therefore, for both NR-approach
based inverted
36
Date Recue/Date Received 2020-05-27
May 26, 2020
Jacobian or its variants as real [Z], and real [Z] or complex [Z] built with
the slack-node as
reference, the definition of "n" becomes "(n-1)" and "nk" becomes nk=(n-2)-nq
in the above
equations (71a) to (74g).
[0085] The steps of loadflow calculation by ZPL or SZPL method are shown in
the flowchart of
Fig. 5. It should be noted that Fig.5 and corresponding calculation steps in
the following are for
complex inverted matrix based, which is Gauss method without immediate
updating as in Gaus-
Seidel method. Referring to the flowchart of Fig.5, different steps are
elaborated in steps marked
with similar numbers in the following. Four numbered steps are the inventive
steps. The words
"Read system data" in Step-a correspond to step-10 and step-20 in Fig. 7, and
step-16, step-18,
step-24, step-36, step-38 in Fig. 8. All other steps in the following
correspond to step-30 in Fig.7,
and step-42, step-44, and step-46 in Fig. 8.
1. Read system data and assign an initial approximate solution vector [VO],
and store it. If
better solution estimate is not available, set voltage magnitude to 1.0 pu at
load nodes and
specified values at PV-nodes, and angle of all nodes equal to that of the
slack-node,
referred to as the flat-start.
2. Faun nodal impedance matrix [Y], and Initialize iteration count ITR = 0.
3333. Faun and store (m+k) x (m+k) size constant sparse matrix [Z] of {(69) or
(70)}, using an
algorithm or by inverting a coefficient matrix [C] or the Jacobian matrix [J]
or its different
variants
4444. Compute the vector of {[I] or [AI]} using equations {(72a) or (72b)}.
However, use
calculated value Qp instead of QSHp for PV-nodes, implement violated Qmax or
Qmm limit
of PV-node generators, and change the status of violated PV-node to PQ-node.
5555. Solve {(69) for [AV]} or {(70) for [V]}, using Gauss method as per (73a)
or (73c) or (73d)
or (730 or (74a) or (74c) or (74d), or (740, or using Gauss-Seidel as per
(73b) or (73e) or
(73g) or (74b) or (74e) or (74g). Self-iterations need to be perfoitned only
at violated PV-
nodes. Update voltage using [V] = [V] + [AV], and adjust real and reactive
components of
PV-node complex voltage for specified voltage magnitude.
777. Take a difference of vectors {[V] ¨ [V0]}, find the maximum of the real
and imaginary
components differences, and store them as DEMX and DFMX, perfonn [VO] = [V],
and
increment iteration count ITR=ITR+1.
8. Are both DEMX and DFMX less than specified tolerance (EV)? If they are not,
Proceed to
37
Date Recue/Date Received 2020-05-27
May 26, 2020
step-444 or else follow the next step.
9. Compute real power mismatch APp at PQ+PV nodes, compute reactive power
mismatch
AQp at PQ nodes, and if all (or both DPMX and DQMX) are not less than
specified
tolerance (EM), go to step 444, otherwise follow the next step.
10. From calculated values of complex nodal voltages, reactive power
generation at PV-nodes,
and tap position of tap changing transfoimers, calculate power flows through
power
network components.
Accurate and Reliable GAUSS-SEIDEL LOADFLOW (ARGSL)
[0086] The complex conjugate power injected into the node-p of a power network
is given by the
following equation (76) and its other alternative organizations.
* n *
Pp - j Qp ¨ Vp YpqVg =Vp (Ypp+ Yp)Vp Vp YpqVg (76)
q=1 q>p
(PSHp - jQSHp)Np* - LpVp= (Ypp+ yp)Vp - LpVp+ YpqVg (76)
q >p
(SSHp*Np*) - LpVp= (Ypp+ yp - 1-ap)Vp YpqVg (76)
q >p
Vp = YpqVg [{ SRI; i(ep2 fp2)} (Ypp+ yp)] (76)
q>p
Vp = [(SSHp*Np* ) - LpVp - YpqVg (Ypp+ yp - Lp) (76)
q>p
(SSHp*Np*) - KYpp+ yp) 1vp - 2.0* YpqVg = i(Ypp+ Yp) - (ssHp*Np2 )1vp
(76)
q >p
Vp = [(SSHp*Np* ) - 1(Ypp+ yp) 1vp - 2.0* YpqVcd / i(Ypp+ yp) -
(ssHp*Np2 )] (76)
q >p
Vp ¨ [(16Sp*Np* ) Lp vp - YpqVg] / 10(pp+ yp) Lp - (ssHp*Np2 )] (76)
q >p
Where,
Lp = - co, ..., -1, 0, +1.....+ co (including fractions) (77)
Pp = Re {Vp YpqVg (78)
cr1
38
Date Recue/Date Received 2020-05-27
May 26, 2020
Qp = - Im{Vp YpqVg (79)
q=1
[0087] Where, Re means "real part of' and Im means "imaginary part of'. The
equation (76) can
also be written for complex power injected into the node-p, instead of complex
conjugate power
injected into the node-p for the purpose of the following development of ARGS
Loadflow method.
However, detailed generalized propounding statement of the Accurate and
Reliable Gauss-
Seidel[[-Patel]] numerical method will be provided in the proposed book
writing project.
[0088] The Accurate and Reliable Gauss-Seidel (ARGS) is for solving a set of
simultaneous linear
and nonlinear algebraic equations iteratively. The ARGSL-method calculates
complex node
voltage for any node-p as given in equation (76).
Iteration Process
[0089] Iterations start with the experienced/reasonable/logical guess for the
solution. The
reference node also referred to as the slack-node voltage being specified,
starting voltage guess is
made for the remaining (n-1)-nodes in n-node network. Node voltage value is
immediately
updated with its newly calculated value in the iteration process in which one
node voltage is
calculated at a time using latest updated other node voltage values. A node
voltage value
calculation at a time process is iterated over (n-1)-nodes in an n-node
network, the reference node
voltage being specified not required to be calculated.
[0090] Now, for the iteration-(r+1), the complex voltage calculation at node-p
equation (76) and
reactive power calculation at node-p equation (79), becomes:
p-1
vp(r+1) vpqvg(r+1) vpq vg r
) [I (PSHp - j Q Slip ) / (ep2 + fp2) r - (Yee+
ye)] (80)
cr1 q=p+1
p-1
Vp(r+1)= KSSIlp*/(Vp*)r) - LpVpr - (n(pqVq(r+1) n(pqVcir)) / (Yee+ ye - Le)
(80)
C1=1 crp+
p-1
Vp(r+1)= KASp"/(Vp")r)-LpVp - r (n(pqVg(r+1)+D(pqVcir)1/1Ypp-Fyp-Lp-SSII *N 21
p s
(80)
cr1 q=p+1
p-1
VP {(y*(r1) fr 1Pq v v r (r+1)
* 1Pq v r r (81)
P q P q
14=1 q=P
39
Date Recue/Date Received 2020-05-27
May 26, 2020
[0091] The well-known limitation of the Gauss-Seidel numerical method to be
not able to
converge to the high accuracy solution, was resolved by the introduction of
the concept of self-
iteration of each calculated variable until convergence before proceeding to
calculate the next. This
is achieved by replacing equation (80) by equation (82) stated in the
following where self-
iteration-(sr+1) over a node variable itself within the global iteration-(r+1)
over (n-1) nodes in the
n-node network is depicted. During the self-iteration process only Vp and its
real and imaginary
components change without affecting any of the terms involving Vq. At the
start of the self-
iteration Vpsr= Vpr, , and at the convergence of the self-iteration Vp(r+1)
vp(sr+1)
p-1
(vp(sr+1))(r+1) (µ , Pq V (r+1) vpq .. r
V )/[{(PSHp - j Q SHp )/((ep2 +42) sry _
(ypp+yp)) (82)
, (1
q=p+1
(vp(sr+1))(r+1)=
P-1
1(SSI1p*/(Vp*)sr)r ¨ Lp(Vp)sry _
(1VpqVq(r+1)+IYpqVqr)1/1(Ypp+yp)-(LpSSHp*/Vs2)]
(82)
crl q=p+1
(vp(sr+1))(r+1)=
P-1
i(ASp*/(Vp*)sr)r)-(Lp(vosTivs2)41ypqvq(r+i)+Iypqvgr))/Kypp yp_
+ Lp - SSHp*/Vs2)]
(82)
crl q=p+1
Self-convergence
[0092] The self-iteration process for a node is carried out until changes in
the real and imaginary
parts of the node-p voltage calculated in two consecutive self-iterations are
less than the specified
tolerance. It has been possible to establish a relationship between the
tolerance specification for
self-convergence and the tolerance specification for global-convergence. It is
found sufficient for
the self-convergence tolerance specification to be ten times the global-
convergence tolerance
specification.
Afp(sr+i) fp(sr+i) _ fpsrl < los
(83)
Aep(sr+1) = ep(sr+1) epsr < 06
(84)
[0093] For the global-convergence tolerance specification of 0.000001, it has
been found
sufficient to have the self-convergence tolerance specification of 0.00001 in
order to have the
Date Recue/Date Received 2020-05-27
May 26, 2020
maximum real and reactive power mismatches of 0.0001 in the converged
solution. However, for
small networks under not difficult to solve conditions they respectively could
be 0.00001 and
0.0001 or 0.000001 and 0.0001, and for large networks under difficult to solve
conditions they
sometimes need to be respectively 0.0000001 and 0.000001.
Convergence
[0094] There are indications that there is a need for two stages of solution
process stopping
criteria. The first stopping criterion is to carry out the iteration process
until changes in the real
and imaginary parts of the set of (n-1)-node voltages calculated in two
consecutive iterations are
all less than the specified tolerance as given by equations (85) and (86). In
the second stopping
criteria, iterations are continued until both the maximum real power mismatch
(DPMX) at PQ+PV
(Ni) nodes and the maximum reactive power mismatch (DQMX) at PQ (N2) nodes are
not less
than the specified tolerance as given by equations (85a) and (86a). A typical
value of
EV=0.000001, and that of 04=0.0001. The lower the value of the specified
tolerance for
convergence check, the greater the solution accuracy.
Afp(r+i)1 fp(r+i) _
r < EV : subscript P= 1, 2, ..., Ni (number of PQ+PV ¨ nodes)
(85)
Aeprr+1)1=1 eprr+1) - epr < EV : subscript P= 1, 2, ..., Ni (number of PQ+PV ¨
nodes) (86)
App(r+i)1 < Em
: subscript P= 1, 2, ..., Ni (number of PQ+PV ¨ nodes)
(85a)
AQp(r+i)1 < Em
: subscript P= 1, 2, ..., N2 (number of PQ ¨ nodes)
(86a)
Accelerated Convergence
[0095] The ARGS-method being inherently slow to converge, it is characterized
by the use of an
acceleration factor applied to the difference in calculated node voltage
between two consecutive
iterations to speed-up the iterative solution process. The accelerated value
of node-p voltage at
iteration-(r+1) is given by,
Vp(r+1) (accelerated) = Vpr +13 (Vprr+1) - Vpr)
(87)
Where, 13 is the real number called acceleration factor, the value of which
for the best possible
convergence for any given network can be deteitnined by trial solutions. The
ARGS-method is
very sensitive to the choice of 13, causing very slow convergence and even
divergence for the
41
Date Recue/Date Received 2020-05-27
May 26, 2020
wrong choice.
Scheduled or specified voltage at a PV-node
[0096] Of the four variables, real power PSHp and voltage magnitude VSHp are
scheduled/specified/set at a PV-node. If the reactive power calculated using
VSHp at the PV-node
is within the upper and lower generation capability limits of a PV-node
generator, it is capable of
holding the specified voltage at its terminal. Therefore the complex voltage
calculated by equation
(80) or (82) by using actually calculated reactive power Qp in place of QSHp
is adjusted to
specified voltage magnitude by equation (88). However, in case of violation of
upper or lower
generation capability limits of a PV-node generator, a violated limit value is
used for QSHp in (80)
and (82), meaning a PV-node generator is no longer capable of holding its
terminal voltage at its
scheduled voltage magnitude VSHp, and the PV-node is switched to a PQ-node
type.
vp(r+i) (VSHp Vp(r+1))/1Vp(r+1)1
(88)
Calculation steps of Accurate and Reliable Gauss-Seidel Loadflow (ARGSL)method
[0097] The steps of loadflow calculation by ARGSL method are shown in the
flowchart of Fig. 6.
Referring to the flowchart of Fig.6, different steps are elaborated in steps
marked with similar
numbers in the following. Steps marked with double numerals are the inventive
steps. The words
The words "Read system data" in Step-a correspond to step-10 and step-20 in
Fig. 7, and step-16,
step-18, step-24, step-36, step-38 in Fig. 8. All other steps in the following
correspond to step-30
in Fig.7, and step-42, step-44, and step-46 in Fig. 8.
71. Read system data and assign an initial approximate solution. If better
solution estimate is
not available, set specified voltage magnitude at PV-nodes, 1.0 p.u. voltage
magnitude at
PQ-nodes, and all the node angles equal to that of the slack-node angle, which
is referred
to as the flat-start.
72. Form nodal admittance matrix, and Initialize iteration count r= 1
73. Scan all the node of a network, except the slack-node whose voltage
having been specified
need not be calculated. Initialize node count p=1, and initialize maximum
change in real
and imaginary parts of node voltage variables DEMX=0.0 and DFMX=0.0
74. Test for the type of a node at a time. For the slack-node go to step-
82, for a PQ-node go
to the step-99, and for a PV-node follow the next step.
75. Compute Qp(r+1) for use as an imaginary part in determining complex
schedule power at a
42
Date Recue/Date Received 2020-05-27
May 26, 2020
PV-node from equation (81) after adjusting its complex voltage for specified
value by
equation (88)
76. If Qp(r+i) is
greater than the upper reactive power generation capability limit of the PV-
node
generator, set QSHp = the upper limit Qpn' for use in equation (82), and go to
step-99. If
not, follow the next step.
78. If Qprr+1) is less than the lower reactive power generation capability
limit of the PV-node
generator, set QSHp = the lower limit Qpnlin for use in equation (82), and go
to step-99. If
not, follow the next step.
88. Compute V IP" by equation (80) using QSHp = Qprr+1), and adjust for
specified voltage at
the PV-node by equation (88), and go to step-80.
99. Compute VIP" by equations (82), (83), (84) involving self-iterations.
80. Compute changes in the imaginary and real parts of the node-p voltage
by using
equations (85) and (86), and replace current value of DFMX and DEMX
respectively in
case any of them is larger.
81. Calculate accelerated value of Vprr+1) by using equation (87), and
update voltage by Vpr =
Vp(r+1) for immediate use in the next node voltage calculation.
82. Check if the total numbers of nodes - n are scanned. That is if p< n,
increment p=p+1,
and go to step-74. Otherwise follow the next step.
83. Advance iteration count r=r+1, and if DEMX and DFMX both are not less
than the
convergence tolerance (EV) specified for the purpose of the accuracy of the
solution, go to
step-73, otherwise follow the next step.
84. Compute real power mismatch APp at PQ+PV (Ni) nodes, compute reactive
power
mismatch AQp at PQ (N2) nodes, and if all (or both DPMX and DQMX) are not less
than
(EM), go to step 73, otherwise follow the next step.
85. From calculated and known values of complex voltage at different power
network nodes,
and tap position of tap changing transformers, calculate power flows through
power
network components, reactive power generation at PV-nodes, and real and
reactive power
generation at the slack node.
Parallel Processing and Parallel Computer Architecture
[0098] US Patent no. 7788051 and US Patent no. 9891827 of this inventor and
applicant are
43
Date Recue/Date Received 2020-05-27
May 26, 2020
incorporated here by reference as patented prior art. Fig. 10a is reproduced
Fig.4 of US 7788051,
and Fig. 10b reproduced Fig.2a of US 9891827 for easy reference to patented
best possible parallel
computer architectures. Fig.2a reproduced as Fig. 10b of US Patent no. 9891827
implements
wireless interconnections among components of Parallel Computer, wherein
different symbols are
defined as follows:
PU : a Processing Unit, which can be a Central Processing Unit (CPU), or a
Graphical processing
Unit (GPU), or a Field Programmable Gate Array (FPGA), or an Application
Specific Processing
Unit (ASPU) like Tensor Processing Unit (TPU).
PM : a Private Memory for each PU, which is not accessible by any other
component of Parallel
Computer.
SMU : Shared Memory Unit, and different SMUs can be accessed by different PUs
simultaneously that reduces the contention for memory access by different
processors unlike as in
case of large single Common Shared Memory Unit (SMU) of Fig. 10a Shared by all
processors.
TRA : Transmitter-Receiver-Antenna, wherein antenna is common for both
Transmitter and
Receiver.
[0099] Equations (73a) or (73b) or (73c) or (73d) or (73e) or (74a) or (74b)
or (74c) or (74d) or
(74e) are Sparse Inverse matrix based formulations of loadflow calculation
problem called SZPL
or SCIPL models. The ARGSL model is inherently solved by sequential steps.
Therefore,
sequential steps for the solution of (73b) or (73e) or (73g) or (74b) or (74e)
or (74g) are similar
to those of ARGSL model and corresponding Fig. 6, and they are not given
explicitly.
[0100] However, {(73b) or (73e) or (73g) or (74b) or (74e) or (74g)} or {(76)
or (80) or (82)1
can be solved in parallel by the technique of US patent no. 7788051 as per its
flow-chart of Fig.3b
using its the best possible parallel computer architecture of Fig. 4
(reproduced as Fig. 10a in this
application). The parallel solution technique of US Patent no. 7788051
involves decomposition of
a whole n-node problem into n-sub-problems each being comprised of a node and
its directly
connected nodes, and solving each sub-problem on different processor wherein
all different
processors are coordinated by a main or server processor. The solution of each
sub-problem (each
sub-network: a node and its directly connected nodes) on different processors
is then coordinated
(mapped) into network wide a whole n-node problem solution.
[0101] Similarly, (73a) or (73c) or (73d) or (730 or (74a) or (74c) or (74d)
or (740 for different
44
Date Recue/Date Received 2020-05-27
May 26, 2020
nodes can be solved on a single computer in sequence without immediate
updating of nodal
solution or said equations for different nodes can be solved simultaneously in
parallel on different
processors. Steps involving vectors {[I] or [AI]}, {[V] or [AV]}, and [VO] in
Fig. 5, wherein
each component of vector is computed in parallel on different processors, and
the rest of the steps
are performed in sequence on Server Processor (delegating and coordinating
processor of Fig. 10a
in similar corresponding steps of parallel computation in Fig.3b of US patent
no. 7788051.
Patel Loadflow (PL)Model
[[0102] Equations (3) and (4) can be organized in matrix form as per Patel
Numerical Method:
IR1 = i_B f
II j -B [ e
(89)
Patel Transformation Decoupled Loadflow Model
[IR] = [-Y] [f]
(90)
[II = [-Y] [e]
(91)
where,
IRp' = (epPSHp' + fpQSHp')/(ep2 + fp2)
(92)
II' = (epQSHp' - fpPSHp')/(ep2 + fp2)
(93)
[0103] This is the model where elements of equations (90) and (91) are defined
by following
equations.
[-Y] = [-B] + [G] [-B]-1 [G]
(94)
[1R1 = [IR] - [G] [-B]-1[II]
(95)
[II'] = [II] + [G] [-B]-1[RI]
(96)
Regular loadflow models can also be obtained by differentiating on both sides
of equations (89),
(90) and (91).
Generalized Gauss-Seidel-Patel Numerical method for Solution of System of
Simultaneous
Algebraic Equations both linear and nonlinear:
[0104] A linear system of equations Ax=b can be written for any equation-p as
equations (98) and
(97). They can also be written in alternative forms like equation (76)
including factor Lp of (77).
(r+1)
XP ¨ apqXq(1+1) apqXqr) / [ bp/(Xp)r - app]
(97)
q=1 q=p+1
Date Recue/Date Received 2020-05-27
May 26, 2020
p-1 n
(Xp(sr+1))(r+1) = (1 apqXcP+1) - 1 ax r) / [Ibp/ ((xp) sr)r} - app]
(98)
q1crp-Nq q
[0105] A nonlinear system of equations f(x)=y can be written for any equation-
p as equations (82),
which is specifically a nonlinear power flow equation of a power network
involving complex
variables and constant parameters.
[0106] Equations (98) and (82) are defining equations of Generalized Gauss-
Seidel-Patel
numerical method involving self-iterations. It should be noted that self-
iterations within global
iterations are analogous to the earth rotating on its own axis while making
rounds around the Sun.
This generalized approach for solution of both linear and nonlinear system of
simultaneous
algebraic equations could potentially be amenable to acceleration factors
greater than 2 unlike
original Gauss-Seidel numerical method subject to experimental numerical
verification. Further
verbal elaborations about the Generalized Gauss-Seidel-Patel numerical method
will be provided
as part of the proposed book writing project.
General Statements
[0107] The system stores a representation of the reactive capability
characteristic of each machine
and these characteristics act as constraints on the reactive power, which can
be calculated for each
machine.
[0108] While the description above refers to particular embodiments of the
present invention, it
will be understood that many modifications may be made without departing from
the spirit
thereof. The accompanying claims are intended to cover such modifications as
would fall within
the true scope and spirit of the present invention.
[0109] The presently disclosed embodiments are therefore to be considered in
all respect as
illustrative and not restrictive, the scope of the invention being indicated
by the appended claims
in addition to the foregoing description, and all changes which come within
the meaning and range
of equivalency of the claims are therefore intended to be embraced therein.
46
Date Recue/Date Received 2020-05-27
