Computation of available transfer capability using neural network.
Rao, K. Narasimha ; Kumar, K. Kiran ; Amarnath, J. 等
Introduction
Transition of electric industry from its vertically integrated
structure to horizontal structure poses many problems to power system
engineers and researchers. In the environment of open transmission
access, US Federal Energy Regulatory Commission (FERC) requires that
Available Transfer Capability (ATC) information be made available on a
publicly accessible Open Access Same Time Information System (OASIS)
[1]. Utility engineers must continuously compute and update hourly ATC
values to be made available in the web.
Due to Deregulation, Power-wheeling transactions have become a very
important issue [2]. Generally Power wheeling is defined as the power
transmitted from a power producer to a customer through transmission
systems and distribution facilities of third party. Since the
transmission facilities have their physical limitation, not all of the
power wheeling transaction can be accepted and carried out in the power
market. Thermal limits of transmission facilities, voltage limits at
each bus, reactive power constraints of generating units and net
interchange constraints do limit the feasibility of power transfer.
Power transaction between a specific seller bus/area and a buyer
bus/area can be committed only when sufficient ATC is available for that
interface to ensure the system security [3], [4]. The information about
the ATC is to be continuously updated and made available to the market
participants through the Internet-based system such as Open Access Same
time Information System (OASIS).
Method based on linear sensitivity factors offer a great potential
for real time calculation of ATC [8]. Use of these factors offers an
approximate but extremely fast model for the static ATC determination.
Proposed the new set of AC Power Transfer Distribution Factors
(ACPTDF) to determine static ATC more accurately [5].
In deregulated power systems, Available Transfer Capability
analysis is presently a critical issue either in the operating or
planning because of increased area interchanges among utilities. A new
model employing artificial neural networks to calculate available
transfer capability is developed in this paper. Based on the AC power
transfer distribution factor formulation for calculating available power
transfer capability and the strong generalizing ability of the neural
networks[11],[12], the new model can calculate multi-area available
transfer capabilities quickly for a given power system status.
This paper is organized as follows. Section II provides a review of
ATC. In Section III provides a brief review of the artificial neural
network and the back propagation feed forward algorithm [13], which is
used in this paper to train this neural network. In section IV, the
problem of available transfer capability calculations is formulated. The
proposed methodology is implemented in section V and a case study is
given in section VI to demonstrate the effectiveness of the presented
method. Finally, a conclusion is made in section VII.
Available Transfer Capability
In a deregulated power system structure, power producers and
customers share a common transmission network for wheeling power from
the point of generation to the point of consumption. All parties in this
open access environment may try to produce the energy from the cheaper
source for greater profit margin, which may lead to overloading and
congestion of certain corridors of the transmission network. This may
result in violation of line flow, voltage and stability limits and
thereby undermine the system security. Utilities therefore need to
determine adequately their "Available Transfer Capability
(ATC)" to ensure that system reliability is maintained while
serving a wide range of bilateral and multilateral transactions. The
electric transmission utilities are required to post the information of
ATC of their transmission network through the open access same time
information system (OASIS).
Power transactions between a specific seller bus/area can be
committed only when sufficient ATC is available for that interface.
Thus, such transfer capability can be used for reserving transmission
services, scheduling firm and non-firm transactions and for arranging
emergency transfers between seller bus/areas or buyer bus/areas of an
interconnected power system network. ATC among areas of an
interconnected power system network and also for critical transmission
paths between areas are required to be continuously computed, updated
and posted to OASIS following any change in the system conditions.
Transfer Capability
Transfer capability [1] is the measure of the ability of
interconnected electric systems to reliably move or transfer power from
one area to another over all transmission lines (or paths) between those
areas under specified system conditions. In this context,
"area" may be an individual electric system, power pool,
control area, sub region, or North American Electric Reliability Council
(NERC) Region, or a portion of any of these. Transfer capability is also
directional in nature. That is, the transfer capability from Area A to
Area B is not generally equal to the transfer capability from Area B to
Area A.
ATC Definitions
Available Transfer Capability (ATC) [3] is a measure of the
transfer capability remaining in the physical transmission network for
further commercial activity over and above already committed uses.
Mathematically, ATC is defined as the Total Transfer Capability (TTC)
less the Transmission Reliability Margin (TRM), less the sum of existing
transmission commitments (which includes retail customer service) and
the Capacity Benefit Margin (CBM).
Total Transfer Capability (TTC) is defined as the amount of
electric power that can be transferred over the interconnected
transmission network in a reliable manner while meeting all of a
specific set of defined pre and post-contingency system conditions.
Transmission Reliability Margin (TRM) is defined as that amount of
transmission transfer capability necessary to ensure that the
interconnected transmission network is secure under a reasonable range
of uncertainties in system conditions.
Capacity Benefit Margin (CBM) is defined as that amount of
transmission transfer capability reserved by load serving entities to
ensure access to generation from interconnected systems to meet
generation reliability requirements.
Limits to Transfer Capability
The ability of interconnected transmission networks to reliably
transfer electric power may be limited by the physical and electrical
characteristics of the systems including any one or more of the
following:
Thermal LimitsThermal limits establish the maximum amount of
electrical current that a-- transmission line or electrical facility can
conduct over a specified time period before it sustains permanent damage
by overheating or before it violates public safety requirements.
Voltage Limits--System voltages and changes in voltages must be
maintained within the range of acceptable minimum and maximum limits. A
widespread collapse of system voltage can result in a blackout of
portions or the entire interconnected network.
Stability Limits--The transmission network must be capable of
surviving disturbances through the transient and dynamic time periods
(from milliseconds to several minutes, respectively) following the
disturbance.
Neural Networks
Artificial Neural Networks
Artificial neural networks [14] are biologically inspired; that is,
they are composed of elements that perform in a manner that is analogous
to the most elementary functions of biological neuron. The artificial
neural networks are organized in a way that may be related to the
anatomy of the brain. Despite this superficial resemblance, an
artificial neural network exhibits a surprising number of brain's
characteristics. For example: they learn from experience, generalize
from previous example to new one, and abstract characteristics from
inputs containing irrelevant data. The key attributes for the
application of ANN to Power System as:
* Power system may need to be repeatedly solved in the hour to hour
operation and control of power systems. The frequency of solution
depends on the operational sophistication of the particular utility.
* Conventional solution techniques may be computationally intensive
and time consuming. They may use up excessive time on the EMS computers
resulting in high computational cost.
* Explicit mathematical modeling may not be feasible due either to
the complexity or to the lack of available information regarding the
problem.
* Available knowledge may not be in a functional form, but rather
in the form of historical input/output examples.
* Operating conditions could be noisy.
Artificial neural network is massive interconnection of neurons.
Basically the artificial neural networks can be of two types. They are
* Layered artificial neural networks.
* Homogeneous artificial neural networks.
In a layered model the neurons are arranged in layers and the
neurons receive information from the neurons of the previous layer and
give the information to the neurons of the next year. In this type of
network the neurons of the same layer are not connected. The layers can
be classified into three groups viz. Input layer, output layer and the
hidden layer. The input layer receives external inputs while the output
layer provides the output of the system. These layers are the interface
of the network with external world.
In homogeneous model, the layer concept is forgotten. Every neuron
is connected to every other neuron and inter faced with the external
world. Hopfield network is an example of this kind. Artificial neural
networks can again classified as
* Feed forward networks
* Feedback networks
Layered network is an example for feed forward network, while
Hopfield network is an example of feedback network.
Learning is a process of achieving the required net work
computation by determination of two types
* Supervised training
* Unsupervised training
Supervised Training
Supervised training requires the pairing of each input vector with
a target vector representing the desired output together these are
called a training pair. Unsupervised Training: It is difficult to
conceive of a training mechanism in the brain that compares desired and
actual outputs. Feeding the processed corrections back through the
network. The training set consists solely of input vectors.
Whatever kind of learning process is used, an essential
characteristic of any network is its learning rule, which specifies how
weights adapt in response to a learning example. Often learning requires
supplying a network with many examples for several thousand times. To
reduce the computational effort by the conventional method,
Back-Propagation Algorithm (BPA) based on Feed forward Neural Network
has been utilized to compute the ATC.
Back propagation Algorithm
The back propagation network (BP) is one of the most commonly used
types of neural networks. The BP networks are widely used because of
their robustness, which allows them to be applied in a wide range of
tasks. The Back Propagation is the way of using known input-output pairs
of a target function to find the coefficients that make a certain
mapping function approximate the target function as closely as possible.
[FIGURE 1 OMITTED]
A back propagation network typically starts out with a random set
of weights. The network adjusts its weights each time it sees an
input-output pair. Each pair requires two stages: a forward pass and a
backward pass. The forward pass involves presenting a sample input to
the network and letting activations flow until they reach the output
layer. During the backward pass, the network's actual output (from
the forward pass) is compared with the target output and error estimates
are computed for the output units. The weights connected to the output
units can be adjusted in order to reduce those errors. We can then use
the error estimates of the output units to derive error estimates for
the units in the hidden layers. Finally, errors are propagated back to
the connections stemming from the input units.
The back propagation algorithm usually updates its weights
incrementally, after seeing each input-output pair. After it has seen
all the input-output pairs (and adjusted its weights that many times),
we say that one iteration has been completed. Training a back
propagation network usually requires much iteration.
A set of weights for a two-layer network that maps inputs onto
corresponding outputs.
1. Let [N.sub.i] be the number of units in the input layer, as
determined by the length of the training input vectors. Let Nk be the
number of units in the output layer. Now, choose [N.sub.j], the number
of units in the hidden layer. Weights connecting the input layer to the
hidden layer are denoted by [W.sub.ji]. Likewise, weights connecting the
hidden layer to the output layer are denoted by [W.sub.kj].
2. Initialize the weights in the network. Each weight should be set
randomly to a number between -0.1 and 0.1.
3. Choose an input-output pair. Suppose the input vector is
[I.sub.i] and the target output vector is [T.sub.k].
4. Propagate the input vector from the units in the input layer to
the units in the hidden layer using the activation function.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)
Where,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
5. Propagate the activations from the units in the hidden layer to
the units in the output layer.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)
Where,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
6. Compute the errors of the units in the output layer, denoted
[[sigma].sub.k]. Errors are based on the network's actual output
([O.sub.k]) and the target ([T.sub.k]).
[[sigma].sub.k] = [O.sub.k] (I-[O.sub.k])(T.sub.k]-[O.sub.k] (3)
7. Compute the errors of the units in the hidden layer, denoted
[[sigma],sub.j].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)
8. Adjust the weights between the hidden layer and output layer.
The learning rate is denoted by [eta]; [eta] tells us how far to move in
the direction of the gradient. A small [eta] will lead to slower
learning, but a large [eta] may cause move through weight space that
overshoots the solution vector.
[DELTA][W.sub.jk] = [[eta][sigma].sub.k][O.sub.j] (5)
9. Adjust the weights between the input layer and the hidden layer.
[DELTA][W.sub.ij] = [[eta][sigma].sub.k][O.sub.j] (6)
10. Go to step 3 and repeat. When all the input-output pairs have
been presented to the network, one iteration has been completed. Repeat
steps 3 to 9 for as many iterations as desired.
[FIGURE 2 OMITTED]
Problem Formulation
A 1996 report by North American Electric Reliability Council (NERC)
establishes a framework for determining ATC of the interconnected
transmission networks for a commercially viable wholesale market. The
report defines ATC principles under which ATC values are to be computed
and it permits individual systems, power pools, regions and sub regions
to develop their procedure for determining ATC in accordance with these
principles. [10] discussed some theoretical aspects of ATC and the
problem of its evaluation under open access environment. In order to
consider line flow (MW) limits for static AC determination under the
system intact, AC power transfer distribution factors is used.
Referring to Fig. 2, a simple interconnected power system can be
divided into three kinds of areas: receiving area, sending areas and
external areas. "Area" can be defined in an arbitrary fashion.
It may be an individual electric system, power pool, control area, sub
regions, etc. The objective is to determine the available transfer
capability from sending areas to receiving area through the transfer
path. The calculation of available transfer capability can be formulated
as follows:
Consider a bilateral transaction [t.sub.p] between a seller bus, m
and buyer bus, n. Further consider a line, 'l' carrying a part
of the transaction power. Let the line be connected between a bus-i and
a bus-j. For a change in real power transaction between the above seller
and buyer say by [DELTA][t.sub.p] MW, if the change in transmission line
quantity ql is * ql, the AC power transfer distribution factors can be
defined as:
[(ACPTDF).sub.ql-tp] = [DELTA][q.sub.1]/[DELTA][t.sub.p] (7)
In this paper, the transmission quantity [q.sub.1] is taken as real
power flow from bus-i to bus-j. This can be utilized to compute new
values of transmission quantity ql and thus the change in the quantity
[DELTA][q.sub.1] from the base case. Once [DELTA][q.sub.1] is known for
all the lines is computed at all the buses corresponding to a
transaction [DELTA][t.sub.p], the ACPTDFs for each line and buses,
respectively, can be obtained from (7).
The above factors have also been calculated for multilateral
transactions in which a group of sellers have a bilateral contract with
group of buyers. In this paper, a change in the multilateral transaction
is assumed to be shared equally by each of the sellers and the buyers.
However, the transaction amount can be shared in a predicted ratio in a
deregulated market. The mismatch vector for the multilateral
transactions will have nonzero entries corresponding to the buyer and
seller buses between which the transactions are taking place.
Implementation Algorithm
From the formulation in Section IV, a neural network approach to
solve the available transfer capability problem is presented in this
section for the system topology and generation availability.
1) Input Vector: Generation status, load level and line status
define a specified power system state. Therefore the input vector
consists of following three parts:
* Generation status 1--Generator is available 0--Generator is
unavailable
* Line status 1--Line is available 0--Line is unavailable
* Load conditions
It is assumed that each bus changes its load at the same rate
within the area, but the rate may differ for different areas. The number
of input neurons representative of load conditions is equal to the
number of system areas. For each area, if the load is equal to base
load, the input is 1.0; if the load is 115% of the base load, the input
value is 1.15, etc.
For a large power system, since the number of generators and lines
is large, it is important to find those critical generators and lines
whose unavailability will have the largest effect on available transfer
capability. Contingency screening and ranking techniques are used to
find those critical generators and lines. Only the status of these
elements are taken as inputs, thus the number of input neurons can be
reduced, which is advantageous for the training of the neural network.
2) Output Vector: Only one output signal is used here, the
available transfer capability between the sending areas and the
receiving area.
3) Network Architecture and Training: The complexity of a neural
network is characterized by the number of neurons. There is no general
rule for selection of these parameters. The critical issue for
developing a neural network is generalization.
The neural networks can suffer from either underfitting or
overfitting. A neural network with a small number of neurons may not be
sufficiently powerful to model a complex function. On the other side, a
neural network with too many neurons may lead to overfitting the
training sets and lose its ability to generalize which is the main
desired characteristic of a neural network. Here, we need to select an
optimal number of neurons.
[FIGURE 3 OMITTED]
Test Cases And Results
Test System
For the purpose of testing, the proposed method was applied to a
modified IEEE 30-bus system. The modified system includes 6 generators,
21 load buses and 41 transmission lines. Single line diagram of this
system is shown in Fig. 3. The system is divided into 3 areas and the
available transfer capability to be calculated is from areas 2 to area
1.
Training Patterns of the Neural Network
Training sets provided to the neural network are representative of
the whole state space of concern so that the trained neural network has
the ability of generalization. We assume only one line is on outage at a
time because the outage probability of a line is very small. The outage
probabilities of generators are larger than those of transmission lines,
so we assume that it is possible for two generators to be on outage at
the same time. Training patterns for the IEEE 30-bus system are composed
of:
* Load levels for each area from 1% to 195% of base load while all
lines and generators remain in operation.
* Generator outages (including one and two generators on outage) at
70%, 105% of the base load of area 1.
* Single line outages at 70%, 105% of the base load of area 1.
* Joint outage (one generator and one line) at 65%, 115% of the
base load of area 1.
There are 250 training patterns in total. This may not be an ideal
set of training patterns, but it covers the range of load levels and the
outage of generators and transmission lines.
Test Patterns of the Neural Network
The trained neural network was tested using 80 test cases which are
composed of load variations and generator and line outages. None of
these test cases were used in the training of the neural network.
Architecture of the Neural Network
1) Input Layer: The input layer is composed of the neurons which
are representative of the load conditions, generator and line status.
For this modified IEEE 30-bus system, 3 input neurons are taken to
represent the load conditions in each of the 3 areas and 6 neurons are
taken for the status of each of the 6 generators. Since there are a
total of 41 lines, a line contingency screening technique was used to
find those critical lines which have the greatest effect on maximum
transfer capability. Here we identified 11 lines as critical: 4-12,
9-10, 10-17, 12-15, 6-9, 2-5, 4-6, 15-18, 2-4, 2-6 and 12-16. Thus 11
input neurons represent the critical line status. The total number of
input neurons is thus 20.
2) Output Layer: The output layer has only one neuron here, whose
output is the available transfer capability from area 2 to area 1.
3) Data Scaling: Scaling either input or target variables tends to
make the training process better behaved by improving the numerical
condition of the optimization problem and ensuring that various default
values involved in initialization and termination are appropriate. Here
in our study, the values of the input vectors are between 0 and 1.5,
thus there is no need to scale them. The output vector, which is the
value of available transfer capability, varies a lot. Therefore, we
scale the output value and make it between 0 and 1.
4) Network Topology: Table I shows the average error at different
numbers of iterations for four different neural network topologies. The
Table I Average error comparison of four different architectures
(err/unit) structure 20/40/1 means that there are 20 input neurons, 40
neurons in the hidden layer and one neuron for output. From Table I, we
can see the structure 20/40/1 converges quickly and is more accurate
than the others. Hence we have chosen it as the network architecture in
our example. Sigmoid transfer functions are used for both the hidden
layer and the output layer.
Analysis of Results
The simulation programme is developed by Object oriented
programming using C++. Based on the selected training and testing
patterns and the chosen neural network topology, the Back propagation
algorithm is used to train the neural network.
Table II shows the training and testing statistics for the chosen
neural network as applied to the test system.
Tables III.1 - III.3 give the relative error list of ACDF outputs
(exact available transfer capability) and neural network output
(approximate available transfer capability) for different test cases.
Fig. 4 shows graphically the neural network estimates for available
transfer capability as compared to exact values as determined from ACDF
calculations. The base value in Fig. 4 is 100MW. Relative error is
defined as follows:
Relative Error = [absolute value of [b.sub.i] - [a.sub.i]/[a.sub.i]
* 100% (8)
[FIGURE 4 OMITTED]
Where, '[a.sub.i]'is the exact value from ACDF and
'[b.sub.i]'is the output of Neural Network
Tables III.1-III.3 shows that of the total 90 testing patterns, 44
are between -8% and zero, 41 errors are between 0-5% and 4 cases have
errors greater than 5%. The greatest error occurred in case 37(Generator
outages with area 1 loads at 105% of the base and area 2 & 3 loads
at the base values) where the error is 8.804%. Also from Fig. 4 and
Tables III.1-III.3, we can see that for the load variation test cases
(cases 190) neural network results are very close to those of ACDF
results. This indicates that the neural network can accurately estimate
available transfer capabilities for varying load levels.
Similarly, errors are small for test cases 1-30 in which load is
varied in one area by considering the loads in the other areas is at
base values only were studied. Thus, the neural network can also
accurately estimate available transfer capabilities for varying
generator or line status. Errors are seen to be larger, though still
generally acceptable, in test cases 31-40 where Generator outages with
area 1 loads at 105% of the base and area 2 & 3 loads at the base
values were studied.
Conclusion
The Available Transfer Capability calculation method proposed in
this paper is capable of reflecting variations in load levels and in the
status of generation and transmission lines. Using the IEEE 30-bus
system, the method is shown to accurately estimate available transfer
capabilities between system areas with variations in load levels, in the
status of generation, and in the status of lines.
We believe that the proposed BPA method may have important
applications in power system operation, planning and reliability
assessment. The method would allow a system operator to immediately
update available transfer capabilities as loads and the statuses of
generation units and transmission lines change. This should enhance the
economy and security of a system. Similarly, because the method can very
quickly calculate available transfer capabilities than ACDF when system
conditions change, the method should be useful in planning and
reliability studies where a wide range of system conditions must be
considered and evaluated.
Hence, Available Transfer Capability determination using Artificial
Neural Network can be used real-time in the deregulated electricity
market.
Acknowledgment
The authors would like to thank SACET, Chirala & CVRCE,
Hyderabad for providing the Computer lab facility with necessary
softwares and support during the work also thank to the JNTUCE,
Hyderabad for providing the necessary facilities.
References
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[15] ftp://ftp.sas.com/pub/neural/
(1) K. Narasimha Rao, (2) K. Kiran Kumar, (3) J. Amarnath and (4)
S. Kamakshaiah
(1) Research Scholar in Dept of EEE at JNT University, Hyderabad,
India. Professor with St. Ann's College of Engineering &
Technology, Chirala, AP, India E-mail: raosimha_k@yahoo.co.in
(2) C.V.R College of Engineering & Technology, Hyderabad, AP;
India
(3) Professor in Electrical & Electronics Engineering
department, JNTU, Hyderabad, AP, India. E-mail: amarnathjinka@yahoo.com
(4) Professor and Head of EEE in CVR college of Engineering &
Technology.
Table1 : Average Error Comparison Of Four Different Architectures
(Err/Unit)
NN No of No of No of
struc- iteration:5000 iteration:15000 iteration:30000
ture
Training Testing Training Testing Training Testing
20/25/1 0.03057 0.08408 0.01233 0.0339 0.01842 0.05000
20/30/1 0.02820 0.07756 0.01456 0.0400 0.00227 0.00620
20/35/1 0.03389 0.09320 0.02207 0.0721 0.01290 0.03547
20/40/1 0.01287 0.03541 0.01238 0.0340 0.00672 0.01848
Table 2 : Neural Network Statistics
Input neurons 20
Output neurons 1
Neurons in hidden layer 40
Training patterns 250
Testing patterns 90
Eta 0.55
convergence 0.01
Total iteration number 40000
Training (error/unit) 0.00449
Testing (error/unit) 0.001615
Table 3(i) : Relative Error List Of Load Variation Test Cases
Case relative Case relative Case relative Case relative
error % error % error % error %
1 -1.29189 9 -1.72056 17 0.63063 25 -1.40288
2 1.90086 10 1.54801 18 0.73853 26 0.32665
3 7.362 11 0.41169 19 -0.28290 27 -0.27119
4 0.50910 12 0.84702 20 0.84698 28 -0.64453
5 0.85003 13 0.82574 21 -0.46663 29 -2.39384
6 -0.71519 14 -2.13149 22 -0.84833 30 -0.32959
7 0.30386 15 0.37581 23 0.90943
8 -1.08699 16 0.15643 24 1.31597
Case 1-10: Area 1 loads vary while area 2 & 3 loads are constant at
the base values
Case 11-20: Area 2 loads vary while area 1 & 3 loads are constant at
the base values
Case 21-30: Area 3 loads vary while area 1 & 2 loads are constant at
the base values
Table 3(ii) : Relative Error List Of Single Outage Test Cases
Case relative Case relative Case relative Case relative
error % error % error % error %
31 7.96 41 -1.90187 51 -0.04600 61 -0.34328
32 1.52204 42 1.039 52 -0.02131 62 -0.60699
33 1.37074 43 0.10675 53 0.41292 63 -0.47116
34 -5.47458 44 0.02227 54 -0.19772 64 -0.43943
35 -4.039 45 -1.23527 55 1.13655 65 0.62891
36 6.73350 46 0.02807 56 -1.3205 66 -0.33312
37 8.804 47 0.03211 57 -0.44200 67 -0.61092
38 -1.83452 48 0.02690 58 -2.13006 68 -0.57351
39 -6.201 49 -0.96735 59 1.26520 69 -0.63450
40 1.37074 50 0.35056 60 0.61838 70 -0.47116
Case 31-40: Generator outages with area 1 loads at 105% of the base
and area 2 & 3 loads at the base values
Case 41-50: Line outages with area 1 loads at 105% of the base and
area 2 & 3 loads at the base values
Case 51-60: Generator outages with area 1 loads at 70% of the base
and area 2 & 3 loads at the base values
Case 61-70: Line outages with area 1 loads at 70% of the base and
area 2 & 3 loads at the base values
Table 3(iii) : Relative Error List Of Joint Outage Test Cases
Case Relative Case Relative Case Relative Case Relative
error % error % error error %
71 0.447 76 0.78450 81 -2.19993 86 -3.02652
72 -1.65 77 0.15066 82 2.77320 87 2.24160
73 0.437 78 1.082 83 -1.50291 88 0.62060
74 -0.12440 79 3.634 84 -0.97840 89 1.16525
75 -0.643 80 -0.22719 85 1.12066 90 -0.26645
Case 71-80: Joint outrages with area 1 loads at 115% of the base
and area 1 & 3 loads at the base values
Case 81-90: Joint outrages with area 1 loads at 65% of the base
and area 1 & 3 loads at the base values
