Note: Descriptions are shown in the official language in which they were submitted.
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METHOD AND APPARATUS FOR PARALLEL LOADFLOW COMPUTATION
FOR ELECTRICAL POWER SYSTEM
TECHNICAL FIELD
[001] The present invention relates to methods of loadflow computation in
power flow
control and voltage control in an electrical power system. It also relates to
the parallel
computer architecture and distributed computing architecture.
BACKGROUND OF THE INVENTION
[002] 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.
[003] 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.
[004] 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
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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, 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.
[0051 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.
[0061 Control of an electrical power system involving power-flow control and
voltage
control commonly is performed according to a process shown in Fig. 4, which is
a method of
forming/defining 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-
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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,
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.
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[007] Overload and under/over voltage alleviation functions produce changes in
controlled
variables/parameters in step-60 of Fig. 5. 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
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.
[008] 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.
[009] The power system control process shown in Fig. 5 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.
5. For
example, under voltage can be alleviated by shedding some of the load
connected.
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[010] 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 + jBpp : p-th diagonal element of nodal admittance matrix without
shunts
yp = gp + jbp : total shunt admittance at any node-p
VP = ep + jfp = VpZOp : complex voltage of any node-p
Abp, AVp : voltage angle, magnitude corrections
Aep, Afp : real, imaginary components of voltage corrections
Pp + jQp : net nodal injected power calculated
APP + jAQp : nodal power residue or mismatch
RPp + jRQp : modified nodal power residue or mismatch
(DP rotation or transformation angle
[RP] : vector of modified real power residue or mismatch
[RQ] : vector of modified reactive power residue or mismatch
[Y8] : gain matrix of the P-8 loadflow sub-problem defined by eqn. (1)
[YV] : gain matrix of the Q-V loadflow sub-problem defined by eqn. (2)
in : number of PQ-nodes
k : number of PV-nodes
n=m+k+l : total number of nodes
q>p : q is the node adjacent to node-p excluding the case of q=p
[ ] : indicates enclosed variable symbol to be a vector or a matrix
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
Bold lettered symbols represent complex quantities in description.
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/determination 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 determination of
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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 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 (SDL) Models, Fast Super
Decoupled Loadflow (FSDL) Model, and Novel Fast Super Decoupled
Loadflow (NFSDL) 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 term 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/formation 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 (SDL) Methods, Fast
Super Decoupled Loadflow (FSDL) Method, and Novel Fast Super
Decoupled Loadflow (NFSDL) Method. All decoupled loadflow methods
described in this application use either successive (10, IV) iteration
scheme or simultaneous (IV, 10), defined in the following.
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[0111 Prior art methods of loadflow computation of the kind carried out as
step-30 in Fig. 5,
are well known Gauss-Seidel Loadflow (GSL) and Super Super Decoupled Loadflow
(SSDL) methods. The GSL method is well known to be not able to converge to
high
accuracy solution because of its iteration scheme that lacks self iterations,
which realization
led to the invention of Gauss-Seidel-Patel Loadflow (GSPL) method. The prior
art methods
will now the described.
GAUSS-SEIDEL LOADFLOW: GSL
[012] The complex power injected into the node-p of a power network is given
by the
following relation,
~1 n
Pp - jQP = VP* Y. YpqVq = Vp* YppVp + Vp* E YpqVq (1)
q=1 q >p
Where,
n
Pp = Re {Vp* EIYPgVq } (2)
n
Qp = - Im{Vp* E YpgVq } (3)
q=1
Where, words "Re" means "real part of' and words "Im" means "imaginary part
of'.
[013] The Gauss-Seidel (GS) method is used to solve a set of simultaneous
algebraic
equations iteratively. The GSL-method calculates complex node voltage from any
node-p
relation (1) as given in relation (4).
Vp = [{(PSHp - jQSHP )/ VP* } - Y, YpqVq I / Ypp (4)
q>p
Iteration Process
[014] 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
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voltage value calculation at a time process is iterated over (n-1)-nodes in an
n-node network,
the reference/slack node voltage being specified not required to be
calculated.
Now, for the iteration-(r+l), the complex voltage calculation at node-p
equation (4) and
reactive power calculation at node-p equation (3), becomes
f/ 1 p-1 n
Vp( 1) = [{(PSHp - JQSHp )/ (Vp* )r} -q YpgVq(r+l) 9 Ypq Vq rl / Ypp (5)
P-1 n
Qp(' i) Im {(Vp* )r Ypq q(r+1) - (VP* )r I Ypq Vq r } (6)
q=1 q-p
Convergence
[0151 The iteration process is carried out 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 - c, as shown in relations (7) and (8). The lower the
value of the specified
tolerance for convergence check, the greater the solution accuracy.
Aep(r+1) I = I ep(r+1) - epr I < s (7)
Ofp(r+1' I = I fp(r+1) - fpr I < (8)
Accelerated Convergence
10161 The GS-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+l) is given by
Vp(r+1) (accelerated) = Vpr + [3 (Vp(r+l) - Vpr) (9)
Where, (3 is the real number called acceleration factor, the value of which
for the best
possible convergence for any given network can be determined by trial
solutions. The GS-
method is very sensitive to the choice of (3, causing very slow convergence
and even
divergence for the wrong choice.
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Scheduled or specified voltage at a PV-node
[0171 Of the four variables, real power PSHp and voltage magnitude VSHp are
specified 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
relation (5) by
using actually calculated reactive power Qp in place of QSHp is adjusted to
specified voltage
magnitude by relation (10).
Vp(r+l) _ (VSHp Vp(r+1))/ I Vp(r+l)I (10)
Calculation steps of Gauss-Seidel Loadflow (GSL) method
10181 The steps of loadflow computation method, GSL method are shown in the
flowchart of
Fig. I a. Referring to the flowchart of Fig. l a, different steps are
elaborated in steps marked
with similar numbers in the following. The words "Read system data" in Step-1
correspond
to step-10 and step-20 in Fig. 5, and step-14, step-20, step-32, step-44, step-
50 in Fig. 6. All
other steps in the following correspond to step-30 in Fig. 5, and step-60,
step-62, and step-64
in Fig. 6.
1. 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.
2. Form nodal admittance matrix, and Initialize iteration count r= 1
3. 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
4. Test for the type of a node at a time. For the slack-node go to step-12,
for a PQ-node go
to the step-9, and for a PV-node follow the next step.
5. Compute Qp(r+1) for use as an imaginary part in determining complex
schedule power at
a PV-node from relation (6) after adjusting its complex voltage for specified
value by
relation (10)
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6. If Qp(r+1) is greater than the upper reactive power generation capability
limit of the PV-
node generator, set QSHp = the upper limit Qpmax for use in relation (5), and
go to step-
9. If not, follow the next step.
7. If Qp(r+l) is less than the lower reactive power generation capability
limit of the PV-
node generator, set QSHp = the lower limit Qpm'n for use in relation (5), and
go to step-
9. If not, follow the next step.
8. Compute V P("-') from relation (5) using QSHp = Qp(`+1), and adjust its
value for
specified voltage at the PV-node by relation (10), and go to step-10
9. Compute V P("") from relation (5)
10. Compute changes in the real and imeginary parts of the node-p voltage by
using
relations (7) and (8), and replace current value of DEMX and DFMX respectively
in
case any of them is larger.
11. Calculate accelerated value of V p(r+l) by using relation (9), and update
voltage by V pr =
V p(r+1) for immediate use in the next node voltage calculation.
12. Check for if the total numbers of nodes - n are scanned. That is if p< n ,
increment
p=p+1, and go to step-4. Otherwise follow the next step.
13. If DEMX and DFMX both are not less than the convergence tolerance (s)
specified for
the purpose of the accuracy of the solution, advance iteration count r=r+1 and
go to
step-3, otherwise follow the next step
14. 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, and reactive power generation at PV-nodes.
Decoupled Loadflow
[019] In a class of decoupled loadflow methods, each decoupled method
comprises a
system of equations (11) and (12) differing in the definition of elements of
[RP], [RQ], and
[Y0] and [YV]. It is a system of equations for the separate calculation of
voltage angle and
voltage magnitude corrections.
[RP] _ [Y0] [A9] (11)
[RQ] _ [YV] [AV] (12)
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Successive (10,1V) Iteration Scheme
[020] In this scheme (11) and (12) are solved alternately with intermediate
updating. Each
iteration involves one calculation of [RP] and [AO] to update [0] and then one
calculation of
[RQ] and [AV] to update [V]. The sequence of relations (13) to (16) depicts
the scheme.
[A0] = [Y0] -' [RP] (13)
[0] = [0] + [DO] (14)
[AV] = [YV] -' [RQ] (15)
[V]=[V]+[AV] (16)
[021] The scheme involves solution of system of equations (11) and (12) in an
iterative
manner depicted in the sequence of relations (13) to (16). This scheme
requires mismatch
calculation for each half-iteration; because [RP] and [RQ] are calculated
always using the
most recent voltage values and it is block. Gauss-Seidal approach. The scheme
is block
successive, which imparts increased stability to the solution process. This in
turn improves
convergence and increases the reliability of obtaining solution.
Super Super Decoupled Loadflow: SSDL
[022] This method is not very sensitive to the restriction applied to nodal
transformation
angles; SSDL restricts transformation angles to the maximum of -48 degrees
determined
experimentally for the best possible convergence from non linearity
considerations, which is
depicted by relations (19) and (20). However, it gives closely similar
performance over wide
range of restriction applied to the transformation angles say, from -36 to -90
degrees.
RPp = (zPpCos4p + AQpSin(Dp)/ Vp2 -for PQ-nodes (17)
RQp = (AQpCos'p - APpSin(Dp )/Vp -for PQ-nodes (18)
Cos1ip = Absolute (Bpp / (Gpp2 + Bpp2)) >_ Cos (-48 ) (19)
Sinop = -Absolute (Gpp / (Gpp2 + Bpp2)) >_ Sin (-48 ) (20)
RPp = APp / (KpVp2) -for PV-nodes (21)
Kp = Absolute (Bpp/YOpp) (22)
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YOpq = -Ypq -for branch r/x ratio <_ 3.0
- (Bpq + 0.9(Ypq-Bpq)) -for branch r/x ratio > 3.0
-Bpq -for branches connected between two PV-
nodes or a PV-node and the slack-node (23)
YVpq = -YPq -for branch r/x ratio <_ 3.0
L (Bpq + 0.9(Ypq-Bpq)) -for branch r/x ratio > 3.0 (24)
YOpp = E-YOpq and YVpp = bp' + E-YVpq (25)
q>p q>p
by = (QSHpCos D - PSHpSin4Dp / VS2) - bpCos b or
by = 2(QSHpCos(Dp - PSHpSin(Dp )/ VS2 (26)
where, Kp is defined in relation (22), 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.
Restrictions to the factor Kp as stated in the above is system independent.
However it can be
tuned for the best possible convergence for any given system. In case of
systems of only
PQ-nodes and without any PV-nodes, equations (23) and (24) simply be taken as
YOpq =
YVpq = -Ypq .
[0231 Branch admittance magnitude in (23) and (24) is of the same algebraic
sign as its
susceptance. Elements of the two gain matrices differ in that diagonal
elements of [YV]
additionally contain the b' values given by relations (25) and (26) and in
respect of elements
corresponding to branches connected between two PV-nodes or a PV-node and the
slack-
node. Relations (19) and (20) with inequality sign implies that transformation
angles are
restricted to maximum of -48 degrees for SSDL. The method consists of
relations (11) to
(26). In two simple variations of the SSDL method, one is to make YVpq YOpq
and the other
is to make YOpq YVpq.
Calculation steps of Super Super Decoupled Loadflow (SSDL) method
[024] The steps of loadflow computation method, SSDL method are shown in the
flowchart
of Fig. 1 b. Referring to the flowchart of Fig. I b, different steps are
elaborated in steps marked
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with similar letters in the following. The words "Read system data" in Step-1
correspond to
step-10 and step-20 in Fig. 5, and step-14, step-20, step-32, step-44, step-50
in Fig. 6. All
other steps in the following correspond to step-30 in Fig. 5, and step-60,
step-62, and step-64
in Fig. 6.
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. This is referred to as the slack-start.
b. Form nodal admittance matrix, and Initialize iteration count ITRP = ITRQ= r
= 0
c. Compute Cosine and Sine of nodal rotation angles using relations (19) and
(20), and
store them. If they, respectively, are less than the Cosine and Sine of -48
degrees,
equate them, respectively, to those of -48 degrees.
d. Form (m+k) x (m+k) size matrices [YO] and [YV] of (11) and (12)
respectively each
in a compact storage exploiting sparsity. The matrices are formed using
relations (23),
(24), (25), and (26). In [YV] matrix, replace diagonal elements corresponding
to PV-
nodes by very large value (say, 10.0**10). In case [YV] is of dimension (m x
m), this
is not required to be performed. Factorize [YO] and [YV] using the same
ordering of
nodes regardless of node-types and store them using the same indexing and
addressing information. In case [YV] is of dimension (m x m), it is factorized
using
different ordering than that of [Y9].
e. Compute residues [OP] at PQ- and PV-nodes and [OQ] at only PQ-nodes. If all
are
less than the tolerance (6), proceed to step-n. Otherwise follow the next
step.
f. Compute the vector of modified residues [RP] as in (17) for PQ-nodes, and
using (21)
and (22) for PV-nodes.
g. Solve (13) for [06] and update voltage angles using, [0] = [0] + [08].
h. Set voltage magnitudes of PV-nodes equal to the specified values, and
Increment the
iteration count ITRP=ITRP+1 and r=(ITRP+ITRQ)/2.
i. Compute residues [AP] at PQ- and PV-nodes and [iQ] at PQ-nodes only. If all
are
less than the tolerance (c), proceed to step-n. Otherwise follow the next
step.
j. Compute the vector of modified residues [RQ] as in (18) for only PQ-nodes.
k. Solve (15) for [AV] and update PQ-node magnitudes using [V] = [V] + [AV].
While
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solving equation (15), skip all the rows and columns corresponding to PV-
nodes.
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.
m. Increment the iteration count ITRQ=ITRQ+1 and r=(ITRP+ITRQ)/2, and Proceed
to
step-e.
n. From calculated and known values of voltage magnitude and voltage angle at
different power network nodes, and tap position of tap changing transformers,
calculate power flows through power network components, and reactive power
generation. at PV-nodes.
SUMMARY OF THE INVENTION
[0251 It is a primary object of the present invention to improve solution
accuracy,
convergence and efficiency of the prior art GSL and SSDL computations method
under wide
range of system operating conditions and network parameters for use in power
flow control
and voltage control in the power system.
[0261 The above and other objects are achieved, according to the present
invention, with
Gauss-Seidel-Patel loadflow (GSPL), the prior art Super Super Decoupled
Loadflow (SSDL)
and their parallel versions PGSPL, PSSDL loadflow computation methods for
Electrical
Power System. In context of voltage control, the inventive system of parallel
loadflow
computation 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 for defining and solving loadflow model of the power network
characterized
by inventive PGSPL or PSSDL model for providing an indication of the
quantity of reactive power to be supplied by each generator including the
14
CA 02564625 2010-05-18
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,
means for machine control connected to the said means for 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 for 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,
means for transformer tap position control connected to said means for
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 for
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.
10271 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.
10281 The inventive system of parallel loadflow computation can be used to
solve a model
of the Electrical Power System for voltage control. 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. GSPL or
SSDL
CA 02564625 2010-05-18
loadflow equations are solved by a parallel 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.
[0291 For voltage control, system of parallel GSPL or SSDL 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. An invented parallel computer System can implement
any of the
parallel loadflow computation methods. 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. 5, and
steps 12, 20, 32,
44, and 50 in Fig 6 described below. Based on this information, a model of the
system
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.
10301 Inventions include Gauss-Seide-Patel Loadflow (GSPL) method for the
solution of
complex simultaneous algebraic power injection equations or any set of complex
simultaneous algebraic equations arising in.any other subject areas. Further
inventions are a
technique of decomposing a network into sub-networks for the solution of sub-
networks in
parallel referred to as Suresh's diakoptics, a technique of relating solutions
of sub-networks
into network wide solution, and a best possible parallel computer architecture
ever invented
to carry out solution of sub-networks in parallel.
16
CA 02564625 2010-05-18
BRIEF DESCRIPTION OF DRAWINGS
[0311 Fig. 1 is a flow-charts of the prior art GSL and SSDL methods
[0321 Fig. 2 is a one-line diagram of IEEE 14-node network and its
decomposition into
level-1 connectivity sub-networks
10331 Fig. 3 is a flow-charts embodiment of the invented GSPL, PGSPL methods
[0341 Fig. 4 is a block diagram of invented parallel computer
architecture/organization
[0351 Fig. 5 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 invented loadflow computation method of Fig. 3.
[0361 Fig. 6 is a flow-chart of the simple special case of voltage control
system in overall
controlling system of Fig. 5 for an electrical power system
10371 Fig. 7 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
DESCRIPTION OF A PREFERED EMBODYMENT
[0381 A loadflow computation is involved as a step in power flow control
and/or voltage
control in accordance with Fig. 5 or Fig. 6. A preferred embodiment of the
present invention
is described with reference.to Fig. 7 as directed to achieving voltage
control.
10391 Fig. 7 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 6.11 of the text ELEMENTS OF POWER SYSTEM ANALYSIS, forth
edition, by William D. Stevenson, Jr., McGrow-Hill Company, 1982. In Fig. 7,
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.7 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.
10401 Node-6 is a reference 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
17
CA 02564625 2010-05-18
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. 7.
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.
[0411 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. 5 and Fig. 6. 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. 6 in which present
invention resides in
function steps 60 and 62.
10421 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 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
[YO] and [YV],
and replacing steps of "Exercise Newton-Raphson Algorithm" by steps of
"Exercise parallel
GSPL or SSDL Computation" in places of steps 58 and 68. This is just to
indicate the
18
CA 02564625 2010-05-18
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. 6 in this application. The
inventive
steps-60 and -62 in Fig.6 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.
[043J In Fig. 6, function step 10 provides stored impedance values of each
network
component in the system. This data is modified in a function step 12, which
contains stored
information about the open or close status of each circuit breaker. For each
breaker that is
open, the function step 12 assigns very high impedance to the associated line
or transformer.
The resulting data is than employed in a function step 14 to establish an
admittance matrix
for the power network. The data provided by function step 10 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.
[0441 Each of the transformers Ti and T2 in Fig. 7 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 20.
[045J The indications provided by function steps 14, and 20 are supplied to a
function step
60 in which constant gain matrices [Y@] and [YV] of any of the invented super
decoupled
loadflow models are constructed, factorized and stored. The gain matrices [YO]
and [YV] are
conventional tools employed for solving Super Decoupled Loadflow model defined
by
equations (1) and (2) for a power system.
[0461 Indications of initial reactive power, or Q on each node, based on
initial calculations
or measurements, are provided by a function step 30 and these indications are
used in
function step 32, 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.
19
CA 02564625 2010-05-18
[0471 An indication of measured real power, P, on each node is supplied by
function step
40. Indications of assigned/specified/scheduled/set generating plant loads
that are constituted
by known program are provided by function step 42, 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 40 and
42 are supplied to function step 44 which adjusts the P distribution on the
various plant nodes
accordingly. Function step 50 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.
10481 The indications provided by function steps 32, 44, 50 and 60 are
supplied to function
step 62 where inventive Fast Super Decoupled Loadflow computation or Novel
Fast Super
Decoupled Loadflow computation is carried out, the results of which appear in
function step
64. The loadflow computation yields voltage magnitudes and voltage angles at
PQ-nodes,
real and reactive power generation by the reference/slack/swing node
generator, voltage
angles and reactive power generation indications at PV-nodes, and transformer
turns ratio or
tap position indications for tap changing transformers. The system stores in
step 62 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 64. The indications provided in step 64
actuate machine
excitation control and transformer tap position control. All the loadflow
computation
methods using SSDL models can be used to effect efficient and reliable voltage
control in
power systems as in the process flow diagram of Fig. 6.
[0491 Inventions include Gauss-Seide-Patel Loadflow (GSPL) method for the
solution of
complex simultaneous algebraic power injection equations or any set of complex
simultaneous algebraic equations arising in any other subject areas. Further
inventions are a
technique of decomposing a network into sub-networks for the solution of sub-
networks in
CA 02564625 2010-05-18
parallel referred to as Suresh's diakoptics, a technique of relating solutions
of sub-networks
into network wide solution, and a best possible parallel computer architecture
ever invented
to carry out solution of sub-networks in parallel.
Gauss-Seidel-Patel Loadflow: GSPL
10501 Gauss-seidel numerical method is well-known to be not able to converge
to the high
accuracy solution, which problem has been resolved fqr the first-time in the
proposed Gauss-
Seidel-Patel (GSP) numerical method.
10511 The GSP method introduces the concept of self-iteration of each
calculated variable
until convergence before proceeding to calculate the next. This is achieved by
replacing
relation (5) by relation (27) stated in the following where self-iteration-
(sr+1) over a node
variable itself within the global iteration-(r+l) over (n-1) nodes in the n-
node network is
depicted. During the self-iteration process only vp changes without affecting
any of the terms
involving Vq. At the start of the self-iteration VP" = Vpr , and at the
convergence of the self-
iteration Vp(r4 l) = Vp(sr+l)
f p=1 n
l
(VP(sr+l))(r+l) = [{(PSHp - JQSHp )/ ((Vp* )s) r I - r+Ypgvq(r+l) q~ YPq Vq r]
/ YPp (27)
Self-convergence
[0521 The self-iteration process 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.
I Dep(sr+l) I ep(sr+l) - epsrI < 106 (28)
Ofp(sr+1) I fp(sr+1) - fpsr I < 10s (29)
[0531 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
21
CA 02564625 2010-05-18
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.
Network Decomposition Technique: Suresh's Diakoptics
[0541 A network decomposition technique referred to as Suresh's diakoptics
involves
determining a sub-network for each node involving directly connected nodes
referred to as
level-1 nodes and their directly connected nodes referred to as level-2 nodes
and so on. The
level of outward connectivity for local solution of a sub-network around a
given node is to be
determined experimentally. This is particularly true for gain matrix based
methods such as
Newton-Raphson(NR), SSDL methods. Sub-networks can be solved by any of the
known
methods including Gauss-Seidel-Patel Loadflow (GSPL) method.
[0551 In the case of GSPL-method only one level of outward connectivity around
each node
is found to be sufficient for the formation of sub-networks equal to the
number of nodes.
Level-1 connectivity sub-networks for IEEE 14-node network is shown in Fig.
2b. The local
solution of equations of each sub-network could be iterated for experimentally
determined
two or more iteration. However, maximum of two iterations are fond to be
sufficient. In case
of GSPL-method, processing load on processors can be attempted equalization by
assigning
two or more smaller sub-networks to the single processor for solving
separately in sequence.
10561 Sometimes it is possible that a sub-network around any given node could
be a part of
the sub-network around another node making it redundant needing local solution
of less than
(m+k) sub-networks in case of gain matrix based methods like SSDL. Level-1
connectivity
sub-networks for IEEE 14-node for parallel solution by say, SSDL-method is
shown in Fig,
2b. The local solution iteration over a sub-network is not required for gain
matrix based
methods like SSDL. Fig. 2c shows the grouping of the non-redundant sub-
networks in Fig.
2b in an attempt to equalize size of the sub-networks reducing the number of
processors
without increasing time for the solution of the whole network.,
[0571 It should be noted that no two decomposed network parts contain the same
set of
nodes, or the same set of lines connected to nodes, though some same nodes
could appear in
22
CA 02564625 2010-05-18
two or more sub-networks.
10581 Decomposing network in level-1 connectivity sub-networks provides for
maximum
possible parallelism, and hopefully fastest possible solution. However,
optimum outward
level of connectivity around a node sub-network can be determined
experimentally for the
solution of large networks by a gain matrix based method like SSDL.
Relating Sub-network Solutions to get the Network-wide Global Solution
10591 Suresh's decomposition subroutine run by server-processor decomposes the
network
into sub-networks and a separate processor is assigned to solve each sub-
network
simultaneously in parallel. A node-p of the network could be contained in two
or more sub-
networks. Say a node-p is contained in or part of `q' sub-networks. If GSPL-
method is used,
voltage calculation for a node-p is performed by each of the `q' sub-networks.
Add `q'
voltages calculated for a node-p by `q' number of sub-networks and divide by
`q'to take an
average as given by relation (30).
v p(r+1) _ (V P1 (r+1) + V .2r-Fl) + V p3 (r+1) + ... + V pq(r+l))/q (30)
[0601 If a gain matrix based method like SSDL is used, voltage angle
correction and voltage
magnitude correction calculation for a node-p is performed by each of the `q'
sub-networks
in which node-p is contained. Add `q' voltage angle corrections and `q'
voltage magnitude
corrections calculated for the node-p by `q' sub-networks and divide by number
`q' to take
average as given by relations (31) and (32).
,&Op(r+l) (AOpl(r+l) + AOp2(r+l) + AOp3(r+]) + ... + &Opq(r+l)/q (31)
AVp(r+l) _ (QVpt(r+1) + AVp2(r+l) + 0Vp3(r+l) + ... + iVpq(r+1))/q (32)
[0611 Sometimes, gain matrix based methods can be organized to directly
calculate real and
imaginary components of complex node voltages or GSPL-method can be decoupled
into
calculating real (ep) and imaginary (fp) components of complex voltage
calculation for a
node-p, which is contributed to by each of the `q' sub-networks in which node-
p is contained.
Add `q' real parts of voltages calculated for a node-p by `q' sub-networks and
divide by
23
CA 02564625 2010-05-18
number `q'. Similarly, add `q' imaginary parts of voltages calculated for the
same node-p by
`q' sub-networks and divide by number `q' to take an average as given by
relation (33) and
(34).
e p(r+l) (e pl(r+]) + e p2(r+l) + e p3(r+]) + ... + e pq(r+]))/q (33)
fp(r+1) _ (fp1(r+1) + f p2(r+l)+ f p3(r+l) + ... + f pq(r+1)/q (34)
[0621 The relations (30) to (34), can also alternatively be written as
relations (35) to (39).
V p(r+l) = (Re ((V pl(r+1) )2) + Re ((V p2(r-]) )2) + ... + Re ((V pq(r+])
)2)/q
+ J F(Inl((NT pl(r+l) )2) + Im((V p2(r+1) )2) + ... + Im((V P9 (r+) )2))/q
(35)
A0 p(r+l) _ f(AOpr+l) )2 + (D8 p2(t+1) )2 + ... + (AS pq(r+]) )2)/q (36)
AV p(r+]) = 4(OV p1(r+]) )2 + (AV p2(r+1) )2 + ... + (AV pq(r+]) )2)/q (37)
ep(r+])- f((ePpi,(r+1))2 + (ep2(r+l))2 + ... + (epq(r+t))2)/q (38)
fp(r+])= ((fp1(r+]))2 + (fp2(r+l))2 + ... + (f pq(r+l))2)/q (39)
[0631 Mathematically, square of any positive or negative number is positive.
Therefore, in
the above relations if the original not-squared value of any number is
negative, the same
algebraic sign is attached after squaring that number. Again if the mean of
squared values
turns out to be a negative number, negative sign is attached after taking the
square root of the
unsigned number.
Parallel Computer Architecture
[0641 The Suresh's diakoptics along with the technique of relating sub-network
solution
estimate to get the global solution estimate does not require any
communication between
processors assigned to solve each sub-network. All processors access the
commonly shared
memory through possibly separate port for each processor in a multi-port
memory
organization to speed-up the access. Each processor access the commonly shared
memory to
24
CA 02564625 2010-05-18
write results of local solution of sub-network assigned to contribute to the
generation of
network-wide or global solution estimate. The generation of global solution
estimate marks
the end of iteration. The iteration begins by processors accessing latest
global solution
estimate for local solution of sub-networks assigned. That means only
beginning of the
solution of sub-networks by assigned processors need to be synchronized in
each iteration,
which synchronization can be affected by server-processor.
10651 There are two possible approaches of achieving parallel processing for a
problem. The
first is to design and develop a solution technique for the best possible
parallel processing of
a problem and then design parallel computer organization/architecture to
achieve it. The
second is to design and develop parallel processing of solution technique that
can best be
carried out on any of the existing available parallel computer. The inventions
of this
application follow the first approach. The trick is in breaking the large
problem into small
pieces of sub-problems, and solving sub-problems each on a separate processor
simultaneously in parallel, and then relating solution of sub-problems into
obtaining global
solution of the original whole problem. That exactly is achieved by the
inventions of
Suresh's diakoptics of breaking the large network into small pieces of sub-
networks, and
solving sub-networks each on a separate processor simultaneously in parallel,
and then by the
technique of relating solutions of sub-networks into obtaining network wide
global solution
of the original whole network.
[0661 Invented technique of parallel loadflow computation can best be carried
out on
invented parallel computer architecture of Fig. 4. The main inventive feature
of the
architecture of Fig. 4 is that processors are not required to communicate with
each other and
provision of private local main memory for each processor for local solution
of sub-problem
for contribution to the generation of network wide global solution in commonly
shared main
memory. Other applications can be developed that can best be carried out using
the parallel
computer architecture of Fig. 4.
[0671 Fig. 4 is the generalized and simplified block diagram of a
multiprocessor computer
system comprising few to thousands of processors meaning the value of number
`n' in
`processor-n' could be small to in the range of thousands. The invention of
the server
CA 02564625 2010-05-18
processor-array processor architecture of the computer of Fig.4 comprises a
multiprocessor
system with -processors and input/output (I/O) adapter coupled, by a common
bus
arrangement, to the commonly shared main memory. One of the processors . is
the
main/server processor coupled to the I/O adapter, which is only one for the
system and
coupled to the I/O devices. The I/O adapter and I/O devices are not explicitly
shown but are
considered to have been included in the block marked I/O unit. Similarly,
dedicated cache
memories if required for each processor and I/O adapter are considered to have
been included
in each processor block and the block of I/O unit. Each processor is also
provided with its
private local main memory for local processing, and it is not visible or
accessible to any other
processor or I/O unit. The Fig. 4 also explicitly depicts that no
communication of any short is
required between processors except that each processor communicates only with
the
main/server processor for control and coordination purposes. All connecting
lines with
arrows at each end indicates two way asynchronous communication.
Distributed Computing Architecture
[0681 The parallel computer architecture depicted in Fig. 4 land itself into
distributed
computing architecture. This is achieved when each processor and associated
memory
forming a self-contained computer in itself is physically located at each
network node or a
substation, and communicates over communication lines with commonly shared
memory and
server processor both located at central station or load dispatch center in a
power network. It
is possible to have an input/output unit with a computer at each network node
or substation,
which can be used to read local sub-network data in parallel and communicate
over
communication line to commonly shared memory for the formation and storage of
network
wide global data at the central load dispatch center in the power network.
Conclusion
[0691 The inventions of Suresh's diakoptics, technique of relating local
solution of sub-
networks into network-wide global solution, and parallel computer architecture
depicted in
Fig. 4 afford an opportunity for the maximum possible parallelism with minimum
possible
communication and synchronization requirements. Also parallel computer
architecture and
parallel computer program are scalable, which is not possible with most of the
parallel
computers built so far. Moreover, these inventions provide bridging and
unifying model for
26
CA 02564625 2010-05-18
parallel computation.
[0701 A multiprocessor computing apparatus, wherein each sub-power-network
gets
assigned to a different processor, for performing parallel loadflow
computation as
defined in paragraphs [072] and [073] comprising in combination:
a plurality of processing units adapted to receive and process data,
instructions and
control signals, and connected to common system bus in parallel asynchronous
fashion;
a plurality of local private main memory means for storing data, instructions
and
control signals, each said local private main memory means being directly and
asynchronously connected to each said processing unit;
common shared memory coupled directly to said common system bus for
sending/receiving data, instructions and control signals asynchronously
to/from
each said processing unit, without providing inter-processor communications;
I/O adapter/control unit coupled directly to a main/server processor, which is
one of
the said plurality of processing units;
also a different I/O adapter/control unit coupled directly to each of the said
plurality
of processing units physically located at far distances in case of said
multiprocessor computing apparatus organized for distributed processing.
[711 Suresh's diakoptics involves decomposing the network for performing said
loadflow
computation in parallel by a method referred to as Suresh's diakoptics that
involves determining a sub-power-network for each node involving directly
connected nodes referred to as level-1 nodes and directly connected nodes to
level-1 nodes referred to as level-2 nodes, and a level of outward
connectivity for
local solution of a sub-power-network around a given node is determined
experimentally,
initializing at the beginning of each new iteration, a vector of dimension
equal to the
number of nodes in the power network with each element value zero,
solving all sub-networks in parallel using available solution estimate at the
start of
each iteration,
adding newly calculated solution estimates or corrections to available
solution
estimate for a node resulting from different sub-networks, `q' number of sub-
27
CA 02564625 2010-05-18
networks, in which a node is contained, in a corresponding vector element that
gets initialized zero at the beginning of each new iteration,
counting the number of additions and calculating new solution estimate or
corrections
to the available solution estimate by taking the average or root mean square
value
using any relevant relations (30) to (39) in the above depending on the
loadflow
computation method used, and
storing the new solution estimate at the end of the current iteration as
initial available
solution estimate at the start of the next iteration.
Calculation steps for Parallel Gauss-Seidel-Patel Loadflow Method
[0721 The steps of parallel Gauss-Seidel-Patel loadflow (PGSPL) computation
method,
using invented parallel computer of Fig. 4 are shown in the flowchart of Fig.
3b. Referring to
the flowchart of Fig.3b, different steps are elaborated in steps marked with
similar numbers
in the following. The words "Read system data" in Step-I correspond to step-10
and step-20
in Fig. 5, and step-14, step-20, step-32, step-44, step-50 in Fig. 6. All
other steps in the
following correspond to step-30 in Fig. 5, and step-60, step-62, and step-64
in Fig. 6.
21. 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, which
is referred to as the flat-start. The solution guess is stored in complex
voltage vector
say, V (I) where "I" takes values from 1 to n, the number of nodes in the
whole
network.
22. All processors simultaneously access network-wide global data stored in
commonly
shared memory, which can be under the control of server-processor, to form and
locally
store required admittance matrix for each sub-network.
23. Initialize complex voltage vector, say VV (I) =CMPLEX (0.0, 0.0) that
receives
solution contributions from sub-networks.
24. All processors simultaneously access network-wide global latest solution
estimate
vector V (I) available in the commonly shared memory to read into the local
processor
memory the required elements of the vector V (I), and perform 2-iterations of
the
GSPL-method in parallel for each sub-network to calculate node-voltages.
28
CA 02564625 2010-05-18
25. As soon as 2-iterations are performed for a sub-network, its new local
solution estimate
is contributed to the vector VV (I) in commonly shared memory under the
control of
server processor without any need for the synchronization. It is possible that
small sub-
network finished 2-iterations and already contributed to the vector VV (I)
while 2-
iterations are still being performed for the larger sub-network.
26. Contribution from a sub-network to the vector VV (I) means, the complex
voltage
estimate calculated for the nodes of the sub-network are added to the
corresponding
elements of the vector VV (I). After all sub-networks finished 2-iterations
and
contributed to the vector VV (I), its each element is divided by the number of
contributions from all sub-networks to each element or divided by number of
nodes
directly connected to the node represented by the vector element, leading to
the
transformation of vector VV (I) into the new network-wide global solution
estimate.
This operation is performed as indicated in relation (30) or (35). This step
requires
synchronization in that the division operation on each element of the vector
VV(I) can
be performed only after all sub-networks are solved and have made their
contribution
to the vector VV(I).
27. Find the maximum difference in the real and imaginary parts of [VV(I)-
V(I)]
28. Calculate accelerated value of VV(I) by relation (9) as VV(I) = V(I) + R
[VV(I)-V(I)]
and perform V(I)=VV(I)
29. If the maximum difference calculated in step-27 is not less than certain
solution
accuracy tolerance specified as stopping criteria for the iteration process,
increment
iteration count and go to step-23, or else follow the next step.
30. From calculated values of complex voltage, at different power network
nodes, and tap
position of tap changing transformers, calculate power flows through power
network
components, and reactive power generation at PV-nodes.
[073] It can be seen that steps-22, -24, and -25 are performed in parallel.
While other steps
are performed by the server-processor. However, with the refined programming,
it is possible
to delegate some of the server-processor tasks to the parallel-processors. For
example, any
assignment functions of step-21 and step-22 can be performed in parallel. Even
reading of
system data can be performed in parallel particularly in distributed computing
environment
where each sub-network data can be read in parallel by substation computers
connected to
29
CA 02564625 2010-05-18
operate in parallel.
Calculation steps for Parallel Super Super Decoupled Loadflow Method
[0741 The steps of Parallel Super Super Decoupled Loadflow (PSSDL) computation
method
using invented parallel computer of Fig. 4 are given in the following without
giving its
flowchart.
41. Read system data and assign an initial approximate solution. If better
solution estimate
is not available, set all node voltage magnitudes and all node angles equal to
those of
the slack-node, which is referred to as the slack-start. The solution guess is
stored in
voltage magnitude and angle vectors say, VM (I) and VA(I) where "I" takes
values
from 1 to n, the number of nodes in the whole network.
42. All processors simultaneously access network-wide global data stored in
commonly
shared memory, which can be under the control of server-processor to form and
locally
store required admittance matrix for each sub-network. Form gain matrices of
SSDL-
method for each sub-network, factorize and store them locally in the memory
associated with each processor.
43. Initialize vectors, say DVM (I)= 0.0, and DVA(I)=0.0 that receives
respectively
voltage magnitude corrections and voltage angle corrections contributions from
sub-
networks.
44. Calculate real and reactive power mismatches for all the nodes in
parallel, find real
power maximum mismatch and reactive power maximum mismatch by the server-
computer. If both the maximum values are less then convergence tolerance
specified,
go to step-49. Otherwise, follow the next step.
45. All processors simultaneously access network-wide global latest solution
estimate
VM(I) and VA(I) available in the commonly shared memory to read into the local
processor memory the required elements of the vectors VM(I) and VA(I), and
perform
1-iteration of SSDL-method in parallel for each sub-network to calculate node-
voltage-magnitudes and node-voltage-angles.
46. As soon as 1-iteration is performed for a sub-network, its new local
solution
corrections estimate are contributed to the vectors DVM (I) and DVA(I) in
commonly
shared memory under the control of server processor without any need for the
CA 02564625 2010-05-18
synchronization. It is possible that small sub-network finished 1-iteration
and already
contributed to the vectors DVM (I) and DVA(I) while 1-iteration is still being
performed for the larger sub-network.
47. Contribution from a sub-network to the vectors DVM (I) and DVA(I) means,
the
complex voltage estimate calculated for the nodes of the sub-network are added
to the
corresponding elements of the vectors DVM (I) and DVA(I). After all sub-
networks
finished 1-iteration and contributed to the vectors DVM (I) and DVA(I), its
each
element is divided by the number of contributions from all sub-networks to
each
element or divided by number of nodes directly connected to the node
represented by
the vector element, leading to the transformation of vectors DVM (I) and
DVA(I) into
the new network-wide global solution correction estimates. This operation is
performed
as indicated in relation (31) and (32) or (38) and (39). This step requires
synchronization in that the division operation on each element of the vectors
DVM (I)
and DVA(I) can be performed only after all sub-networks are solved and made
their
contribution to the vectors DVM (I) and DVA(I).
48. Update solution estimates VM(I) and VA(I), and proceed to step-43
49. From calculated values of complex voltage at different power network
nodes, and tap
position' of tap changing transformers, calculate power flows through power
network
components, and reactive power generation at PV-nodes.
[0751 It can be seen that steps-42, -44, and -45 are performed in parallel.
While other steps
ate performed by the server-processor. However, with the refined programming,
it is possible
to delegate some of the server-processor tasks to the parallel-processors. For
example, any
assignment functions such as in step-43 can be performed in parallel. Even
reading of system
data can be performed, in parallel particularly in distributed computing
environment where
each sub-network data can be read in parallel by substation computers
connected to operate
in parallel.
General Statements
[076] 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.
31
CA 02564625 2010-05-18
10771 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.
[0781 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.
32
CA 02564625 2010-05-18
REFERENCES
Foreign Patent Document
1. US Patent Number: 4868410 dated September 19, 1989: "System of Load Flow
Calculation for Electric Power System"
2. US Patent Number: 5081591 dated January 14, 1992: "Optimizing Reactive
Power
Distribution in an Industrial Power Network"
Published Pending Patent Applications
3. Canadian Patent Application Number: CA2107388 dated 9 November, 1993:
"System
of Fast Super Decoupled Loadflow Calcutation for Electrical Power System"
4. International Patent Application Number: PCT/CA/2003/001312 dated 29
August,
2003: "System of Super Super Decoupled Loadflow Computation for Electrical
Power
System"
Other Publications
5. Stagg G.W. and El-Abiad A.H., "Computer methods in Power System Analysis",
McGrow-Hill, New York, 1968
6. S.B.Patel, "Fast Super Decoupled Loadflow", IEE proceedings Part-C,
Vol.139, No.1,
pp. 13-20, January 1992
7. Shin-Der Chen, Jiann-Fuh Chen, "Fast loadflow using multiprocessors",
Electrical
Power & Energy Systems, 22 (2000) 231-236
8. S.B.Patel, "Super Super Decoupled Loadflow", IEEE Toronto International
Conference
on Science and Technology for Humanity (IEEE TIC-STH 2009), September 2009,
pp.
652-659.
33
