EE 521 Analysis of Power Systems
Fall 2012
C.C. Liu
Midterm Examination I
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SOLUTIONS
Instructions:
This is an open-book, 1-hour-15-min test. All problems will be graded. Describe the
procedures that you use to obtain the answer. Calculators are allowed but not laptops.
Grading:
Problem 1: System Security Concepts __________ (20 Points)
Problem 2: Update Zbus Matrix _______________ (20 Points)
Problem 3: Contingency Ranking ______________ (30 Points)
Problem 4: State Estimation ___________________ (30 Points)
Total: _____________________________________ (Max 100 Points)
Total Number of Pages Including the Cover Sheet:
1
Problem 1: System Security Concepts (20 Points)
Circle only the most appropriate item between parentheses.
Dispatcher Jane at the Control Center of Northwest Power Grid (NPG) regularly
performs security assessment of the system condition. For NPG, the critical operating
constraints are the bus voltage limits. At this time all bus voltages satisfy the operating
limits and all loads are served. Therefore, the power system is in a (secure, normal,
restorative) state. Dispatcher Jane uses a list of next contingencies and performs on-line
(economic dispatch, automatic generation control, power flow) for the next
contingencies. For one of the contingencies, Dispatcher Jane finds that the system
violates the low bus voltage limits at some substations. She concludes that the system is
in an (emergency state, insecure, secure, normal) state. Dispatcher Jane should take
(emergency control, preventive control, restorative control) actions to improve the
system operating condition.
NPG assigns John to analyze the power grid’s transmission capability to accommodate
the market transactions. John has to calculate the Available Transfer Capability (ATC)
between Area A and Area B linked by tie lines. He starts with a base case that already has
1,000 MW power transfer from Area A to Area B. John then identifies the critical
contingencies so that he can ensure that (Security Assessment, Automatic Generation
Control, Economic Dispatch) is incorporated in the ATC calculation. John performs
power flow calculations and check the (System Frequency, Total Area
Load, Operating Constraints) each time he increases the power transfer amount from A
to B by (Increasing Generation in A and Decreasing Load in B, Decreasing
Generation in A and Decreasing Load in B, Increasing Generation in A and
Increasing Load in B).
When John identifies the power transfer level where (Pre-Contingency System Meets
All Constraints and Post-Contingency System Hits At Least One Constraint, PreContingency System Violates Some Constraints but Post-Contingency System Meets
All Constraints, Pre-Contingency System Violates Only One Constraint and PostContingency System Meets All Constraints), he decides that the transfer level in
addition to the base case 1,000MW is the (First Contingency Total Transfer
Capability, First Contingency Incremental Transfer Capability, Optimal Dispatch).
John then reduces the amount by the Transmission Reliability Margin and the nonrecallable reservation amounts of the transmission capability to obtain the (Recallable
ATC, Non-recallable ATC, Capacity Benefit Margin).
2
Problem 2: Update Zbus Matrix (20 Points)
A 4-bus power system is shown below. It is desired to obtain the Z bus (inverse of Y bus )
matrix after the line from the reference node to bus 3 is eliminated.
(a) Find the Z bus before the line from the reference node to bus 3 is eliminated.
(a) Find the new Z bus from the old Z bus for the system before the line is eliminated. You
must use the rank 1 update formula.
1
2
Ref.
3
All impedances equal to
Zl=j0.2 pu
Solution:
a)
b)
3
4
Problem 3: Contingency Ranking (30 Points)
A 3-bus power system is shown below:
1
θ1 = 0.1 rad
jx = j1
3
jx = j1
θ 3 = 0 (ref. bus)
jx = j1
2
θ 2 = 0.3 rad
All quantities are in per unit. Use the DC load flow method for this problem. The angles
shown above are the base case values. The performance index for the contingencies is
defined by
1 P
PI = ∑  liml it
l =1 2  Pl
L



2
Where L is the total number of lines (=3). The line flow limit for all lines is Pl lim it = 1 p.u.
Bus 3 is the reference bus for the DC load flow. Note that the flow P l is a function of the
angle difference.
Suppose the line outages are being evaluated by the ‘Adjoint Network’ method.
5
Solution:
(a) Calculate the PI value for the base case.
b) Solve the adjoint network DC load flow to find θˆ .
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Problem 4: State Estimation (30 Points)
The one-line diagram of a power system is shown below.
G2
G1
M2
Bus1
Bus2
M1
Bus3
Bus4
M3
All transmission lines have the same reactance X = j0.1pu, and the standard deviation of
T
all meters is given by σ = 0.01. Assume that the state vector is (θ 2 θ3 θ 4 ) and θ 1 = 0.
a) Write down the measurement matrix H with the 3 meters.
Solution:
𝜃1 − 𝜃2 −𝜃2
=
𝑃12 =
𝑋12
𝑋12
𝑃2 =
𝑃3 =
𝜃2 − 𝜃1 𝜃2 − 𝜃4
1
1
−1
+
=�
+
� 𝜃2 + ( )𝜃4
𝑋21
𝑋24
𝑋21 𝑋24
𝑋24
𝜃3 − 𝜃1 𝜃3 − 𝜃4
1
1
−1
+
=�
+
� 𝜃3 + ( )𝜃4
𝑋31
𝑋34
𝑋31 𝑋34
𝑋34
−1
𝑋12
0
0
1
1
+
𝑋31 𝑋34
⎛
1
1
𝐻=⎜
⎜𝑋 + 𝑋
24
⎜ 21
⎝
0
0
⎞
−10 0
−1 ⎟
=
�
20
0
𝑋24 ⎟
⎟
0
20
−1
𝑋34 ⎠
0
−10�
−10
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b) Calculate the best estimates of bus angles. Assume that Z measure is given by:
 Z1 
 
Zmeasure =  Z 2  =
Z 
 3
 0.210 


 −0.028  pu
 −0.180 


Solution:
𝜃�2
−0.0210
𝜃� = �𝜃3 � = (𝐻 𝑇 𝑅−1 𝐻)−1 𝐻 𝑇 𝑅−1 𝑍𝑚𝑒𝑎𝑠𝑢𝑟𝑒 = �−0.0286�
−0.0392
𝜃�3
c) Due to the failure of a remote terminal unit at Bus 2, meter M2 is out of service. Is the
system observable with the two remaining meters? Justify your answer step-by-step.
Solution:
𝑃12 =
𝑃3 =
𝐻=
𝜃1 − 𝜃2 −𝜃2
=
𝑋12
𝑋12
𝜃3 − 𝜃1 𝜃3 − 𝜃4
1
1
−1
+
=�
+
� 𝜃3 + ( )𝜃4
𝑋31
𝑋34
𝑋31 𝑋34
𝑋34
−1
⎛𝑋12
⎝
0
1
1
+
𝑋31 𝑋34
0
0
⎞ = �−10 0
−1
0
20
𝑋34 ⎠
0
�
−10
Rank of H is 2 so H has full row rank.
However,
𝑅 = �0.01
0
2
0 �
0.012
1000000
0
0
𝐻 𝑇 𝑅−1 𝐻 = �
0
4000000 −2000000�
0
−2000000 1000000
Since 𝐻 𝑇 𝑅 −1 𝐻 is 3 by 3 matrix and has a rank of 2.
So it is singular, i.e. det(HTR-1H)=0.
Therefore, the state estimation cannot be solved and the system is unobservable.
8