1. Optimization for Operation of Power Systems with Performance Guarantee
by Gokturk Poyrazoglu
University at Buffalo, SUNY
Department of Electrical Engineering

2. Electricity Supply Chain
Consumer
Distribution
Transmission
Generation

3. Optimal Power Flow Problem
Dispatch Generating Units while minimizing Total Cost
In a single time period ( i.e. 1 hour)
Standard form of NLP
min 𝑥 𝑓 𝑥 subject to 𝑔 𝑥 = ℎ 𝑥 ≤ 𝑥 𝑚𝑖𝑛 ≤𝑥≤ 𝑥 𝑚𝑎𝑥
Non-convex Feasible Region of OPF
Courtesy of EDF Energy

4. Unit Commitment t=5 t=4 t=3 t=2 Time t=1
Select and Dispatch Generating Units while minimizing Total Cost
In a multi time period ( i.e. 1 day or 1 week)
t=5
t=4
t=3
t=2
Time
t=1

5. Unit Commitment with DC Model
Select and Dispatch Generating Units while minimizing Total Cost
In a multi time period ( i.e. 1 day or 1 week)
t=5
t=4
t=3
t=2
Time
t=1

6. Security Constrained Unit Commitment
Secure transaction between two consecutive periods
𝑥 3
𝑥 1 ∗
𝑥 5 ∗
𝑥 2 ∗
𝑥 4 ∗
𝑥 3 ∗
t=5
t=4
t=3
t=2
Time
t=1

7. Power Systems Modeling
Quadratic Cost function
Min and Max Capacity Limits
Bus2
Bus1
Bus3 G1 G2
Load
Max Power Flow Limits
Power Balance Equalities
Min and Max Voltage Limits
Bus G
Load

8. Transmission Switching Simulation VS Reality
Bus2
Bus1
Bus3 G1 G2
Load

9. University at Buffalo, SUNY Department of Electrical Engineering
PART I Optimal Topology Control with Physical Power Flow Constraints and N-1 Contingency Criterion
This study was partially published in the following journal and conferences.
University at Buffalo, SUNY
Department of Electrical Engineering

10. Problem Definition AC OPF with optimal topology control:
AC Optimal Power Flow:
Nonlinear and nonconvex programming (NLP or QCQP)
AC OPF with optimal topology control:
Mixed integer nonlinear and nonconvex programming (MINLP)
Same N-1 reliability consideration as the original topology
Line contingencies
Large-scale problem

11. Challenges Comparison of OPF solutions among multiple topologies
NLP solvers seek for a local optimum
Guarantee a better topology?
NLP solutions may not reflect the quality of the global solution
Therefore, comparison between two NLP solutions does not guarantee optimal topology

12. Range for the global optimizer
Approach
Semi-Definite Programming (SDP) relaxes nonconvex feasible range of OPF
SDP may find a feasible solution = the global solution
SDP solution provides a lower bound of the global optimizer
 Global solution is sandwiched between an SDP and a local solution
Upper Bound by an NLP Solver : possibly a local optimizer
Range for the global optimizer
Lower Bound by an SDP Solver:
possibly infeasible solution

13. Optimal Power Flow (OPF) Problem
Quadratic Cost function
Definition of Real and
Reactive Power Flows
(Bi-directional flow)
Voltage Magnitude
Reliability Limits
Real and Reactive
Power Balance
Maximum
Power Flow Limits
(bi-directional flow)
Real and Reactive
Generation Capacity
Limits
The vector x is lifted up to a matrix W
= Convex feasible region
OPF problem is given in the form of Non-convex Quadratically Constrained Quadratic Programming (QCQP)

14. SDP Relaxation Real and Reactive Real and Reactive Generation Capacity
Power Balance
Real and Reactive
Generation Capacity
Limits
&
Voltage Magnitude Reliability Limits
Maximum Power Flow Limits(bi-directional flow)
Schur Compliment of Quadratic Cost function
Positive Semi-definite
Matrix &
Reference Angle

15. Topology Χ and Υ (original topology)
SDP Relaxation
Topology Χ and Υ (original topology)
Utilize the sandwich structure of the global solution
NLP solution of Χ < SDP solution of Υ ≤ Global solution of Υ
A local optimizer of Χ is better than the global optimizer of Υ
Topology Χ Topology Υ
Aim to find such a topology Χ in this study

16. Search for a Better Topology
NLP solution to Υ
SDP solution to Υ
NLP solution to Χ
 Candidate for Χ
Step I: Candidate for Χ: Guaranteed a better topology than Υ
Step II: Check the same N-1 reliability criterion as ones for Υ

17. Parallel Algorithm AC Transmission Topology Control with N-1 Reliability Criterion
SOLVE NLP
SEPERATE
CHECK CONDITIONS
SOLVE SDP

18. Computation Time

19. Simulation Environment
Modified IEEE 30 bus system
6 generators
39 lines + 2 tie lines
Generation capacity: 360 MW
Peak load: 320 MW
Area 2: load pocket
High cost for generators Gen 5 and Gen6
Computation with a Linux server with GHz

20. Operating Cost
Up to 10% cost reduction

21. Real Power Losses
Reduced losses
Increased losses

22. Conclusion
Efficient algorithm to implement topology control in the restructured electricity market experiments
Identified topology
Guaranteed a better one
Satisfy the same reliability criterion
Topology control finds a solution with
Decreased system cost but
Losses may increase

23. University at Buffalo, SUNY Department of Electrical Engineering
PART II Scheduling Maintenance for Reliable Transmission System (The SMaRTS Model)
This study will be partially published in the following journal and conferences.
University at Buffalo, SUNY
Department of Electrical Engineering

24. What is the SMaRTS Model?
Scheduling Maintenance for Reliable Transmission System
Combination of studies on power economics
Security Constrained Unit Commitment
Transmission Topology Control
Outage Coordination
N-1 Reliability Criterion
TTC
N-1
SCUC
SMaRTS

25. Security Constrained Unit Commitment (SCUC)
Selection of generator units
Minimize total operating cost
Security on ramp up and ramp down of units
Mixed Integer Linear Programming (MILP)
Being used widely in business

26. Transmission Topology Control (TTC)
Selection of branches to keep in service
Minimize total operating cost
Security on ramp up and ramp down of units
Mixed Integer Linear Programming (MILP)
No reported use in business

27. Challenges of Topology Control
Transient Stability Concerns
Switching effect
Cost of Switching
Who will pay?
Financial Transmission Rights (FTR) Market
Unwillingness of market participants

28. The Proposed Model : SMaRTS
Transmission lines require
Preventive Maintenance
Schedule maintenance
Optimal time
Follow same procedure
No stability concern
No extra cost for switching
No impact on FTR market

29. Feasible Region of SMaRTS
No outage request
SMaRTS = Unit Commitment
Outage requests on all lines
SMaRTS = Topology Control
SMaRTS
SCUC
TTC

30. Business-As-Usual VS The SMaRTS

31. Problem Formulation Objective Function: No Load Cost of Generators ($)
Generation ($)
No Load Cost of Generators ($)
min 𝜃, 𝑃 𝐺 , 𝑃 𝑘 ,𝑢 𝑠,ℎ,𝑧,𝑚,𝑎 𝑡 𝑔∈Γ 𝐶 𝑔 𝑃 𝑔,𝑡 𝐺 𝛽 + 𝑔∈Γ 𝑁𝐿 𝑔 𝑢 𝑔,𝑡 + 𝑔∈Γ 𝑆𝑈 𝑔 𝑠 𝑔,𝑡 + 𝑘∈Ψ 𝑃𝑀 𝑘 𝑡 𝑚 𝑘,𝑡 − 𝑎 𝑘
In per unit
Start-Up Cost of Generators ($)
Partial Maintenance
Cost ($)

32. Power Balance Constraints & Power Flow Definitions
𝑃 𝑘𝑖𝑗,𝑡 ≤ 𝐵 𝑘 𝜃 𝑖,𝑡 − 𝜃 𝑗,𝑡 + 1− 𝑧 𝑘,𝑡 𝑀 , ∀𝑡, 𝑘∈Ψ
𝑃 𝑘𝑖𝑗,𝑡 ≥ 𝐵 𝑘 𝜃 𝑖,𝑡 − 𝜃 𝑗,𝑡 − 1− 𝑧 𝑘,𝑡 𝑀 , ∀𝑡, 𝑘∈Ψ
𝑃 𝑘𝑖𝑗,𝑡 = 𝐵 𝑘 𝜃 𝑖,𝑡 − 𝜃 𝑗,𝑡 , ∀𝑡, 𝑘∈Ψ
θ 𝑟𝑒𝑓,𝑡 = , ∀𝑡
𝑔∈Γ 𝑃 𝑔 𝑖 ,𝑡 𝐺 − 𝑃 𝑖,𝑡 𝐷 = 𝑘∈ 𝑖,∗ 𝑘∈Ω 𝑃 𝑘𝑖𝑗,𝑡 − 𝑘∈ ∗,𝑖 𝑘∈Ω 𝑃 𝑘𝑖𝑗,𝑡 , ∀𝑡, 𝑖∈ Φ 𝐺
− 𝑃 𝑖,𝑡 𝐷 = 𝑘∈ 𝑖,∗ 𝑘∈Ω 𝑃 𝑘𝑖𝑗,𝑡 − 𝑘∈ ∗,𝑖 𝑘∈Ω 𝑃 𝑘𝑖𝑗,𝑡 , ∀𝑡, 𝑖∈ Φ/Φ 𝐺
Power flow definitions
- Maintenance requested Lines
- Other lines
Reference Voltage Angle
Real Power Balance Equalities

33. Box Constraints Generation be in between capacity limits
𝑃 𝑔 𝐺,𝑚𝑖𝑛 𝑢 𝑔,𝑡 ≤ 𝑃 𝑔,𝑡 𝐺 ≤ 𝑢 𝑔,𝑡 𝑃 𝑔 𝐺,𝑚𝑎𝑥 , ∀𝑡, 𝑔∈Γ −𝑃 𝑘 𝑚𝑎𝑥 𝑧 𝑘,𝑡 ≤ 𝑃 𝑘𝑖𝑗,𝑡 ≤ 𝑧 𝑘,𝑡 𝑃 𝑘 𝑚𝑎𝑥 , ∀𝑡, 𝑘∈Ψ −𝑃 𝑘 𝑚𝑎𝑥 ≤ 𝑃 𝑘𝑖𝑗,𝑡 ≤ 𝑃 𝑘 𝑚𝑎𝑥 , ∀𝑡, 𝑘∈ Ω Ψ −𝑅 𝑔 ≤ 𝑃 𝑔,𝑡 𝐺 − 𝑃 𝑔, 𝑡−1 𝐺 ≤ 𝑅 𝑔 , ∀𝑡, 𝑔∈Γ
Generation be in between capacity limits
Max. Power flow limits are enforced
The change in generation should be applicable.

34. Maintenance Scheduling Constraints
Definition of maintenance start variable
Definition of approval status & Limitation on partial maintenance (optional)
Limit on maintenance duration
Complete maintenance in planning horizon

35. Unit Commitment Constraints
𝑠 𝑔,𝑡 − ℎ 𝑔,𝑡 = 𝑢 𝑔,𝑡 − 𝑢 𝑔, 𝑡−1 , ∀𝑡, 𝑔∈Γ 𝜏=𝑡− 𝑝 𝑔 +1 𝑇 𝑠 𝑔,𝜏 ≤ 𝑢 𝑔,𝑡 , ∀𝑡, 𝑔∈Γ 𝜏=𝑡− 𝑤 𝑔 +1 𝑇 ℎ 𝑔,𝜏 ≤1− 𝑢 𝑔,𝑡 , ∀𝑡, 𝑔∈Γ
Definition of unit commitment variables
Limit on minimum up-time
Limit on minimum down-time

36. Numerical Example Modified 30-bus system
IEEE Reliability Test Case Data
Low power transfer between Areas
13 MW
Hourly Peak Data is from IEEE Reliability Test Case - A summer weekday

37. Requested Maintenance Starting Period
Outage Requests
Line ID
From Bus
To Bus 15 4 12 10 7 6 18 8 9 38 28 3 31 24 25
Business-as-Usual Model
Transmission Owner submit TIME
Line ID
Requested Maintenance Starting Period 15
Hour 8 7
Hour 6 18
Hour 12 9
Hour 15 38
Hour 20 31
Hour 9

38. Total Operating Cost ($) Total Operating Cost ($)
Results
The Optimal Solution
The SMaRTS Model
Business-as-Usual Model
Total Operating Cost ($)
Approved Line ID(s)
53,051.83
--- 1
52,807.70 7 2
52,725.15
7, 18 3
52,722.56
7, 18, 9 4
52,722.47
7, 18, 9, 31 5
52,723.99
7, 18, 9, 31, 38 6
52,841.17
All
Total Operating Cost ($)
Approved Line ID(s)
53,051.83
---- 1
53,054.75 38 2
53,062.72
38, 9 3
53,094.97
38, 9, 7 4
53,204.15
38, 9, 7, 18 5
54,264.13
38, 9, 7, 18, 31 6
55,030.72
All

39. Total Operating Cost ($) Total Operating Cost ($)
Decision Support
BUSINESS-AS-USUAL MODEL
Total Operating Cost ($)
Approved Line ID(s)
53,051.83
---- 1
53,054.75 38 2
53,062.72
38, 9 3
53,094.97
38, 9, 7 4
53,204.15
38, 9, 7, 18 5
54,264.13
38, 9, 7, 18, 31 6
55,030.72
All
SMaRTS
THE SMaRTS MODEL
Total Operating Cost ($)
Approved Line ID(s)
53,051.83
--- 1
52,807.70 7 2
52,725.15
7, 18 3
52,722.56
7, 18, 9 4
52,722.47
7, 18, 9, 31 5
52,723.99
7, 18, 9, 31, 38 6
52,841.17
All

40. The SMaRTS Model with N-1 Reliability
University at Buffalo, SUNY
Department of Electrical Engineering

41. N-1 Reliability Criterion3 2 G D
Loss of Transmission Lines
Loss of Generators B C 1 3 2 G D A C 1 3 2 G D A B 1 3 2 G D A B C 1 3 2 G D A B C 1 3 2 G D
Status of Transmission Line at time t
Status of Transmission Line at contingency c

42. Proposed N-1 Reliability The 3rd indexC 1 3 2 G D
Time t & Contingency c
zk, t, c : status of Line k
Pkij, t, c : power flow on Line k
Pg, t, c : power generation at Gen g
Contingency Elements
Status of
Transmission Lines
Time

43. Discarding 3rd index by Physical Property1 3 2 G D
Time t & Contingency c
zk, t, c : status of Line k
Pkij, t, c : power flow on Line k
Pg, t, c : power generation at Gen g
Loss of Transmission Lines B C 1 3 2 G D A C 1 3 2 G D
zk, t, A = zk, t, 0 ,∀k – A
zk, t, A = zk, t, 0 , k = A
Pkij, t, A ≠ Pkij, t, 0 ,∀k
Pg, t, A ≠ Pg, t, 0 ,∀k
zk, t, B = zk, t, 0 ,∀k – B
zk, t, B = zk, t, 0 , k = B
Pkij, t, B ≠ Pkij, t, 0 ,∀k
Pg, t, B ≠ Pg, t, 0 ,∀k

44. Utilizing Physical Meaning
Discard 3rd index &
a variable
Transmission Line Contingency
Enforce the flow to be ZERO
Generation Contingency
Enforce the generation to be ZERO

45. Numerical Example Modified 30-bus system
IEEE Reliability Test Case Data
Low power transfer between Areas
Hourly Peak Data is from IEEE Reliability Test Case - A summer weekday
Tie lines
Radial Lines

46. Requested Maintenance Starting Period
Outage Requests
Line ID
From Bus
To Bus 15 4 12 10 7 6 18 8 9 38 28 3 31 24 25
Business-as-Usual Model
Transmission Owner submit TIME
Line ID
Requested Maintenance Starting Period 15
Hour 8 7
Hour 6 18
Hour 12 9
Hour 15 38
Hour 20 31
Hour 9

47. Results Business-as-Usual Model The SMaRTS Model
Total Operating Cost ($)
Approved Line ID(s)
53,781.71
--- 1
53,756.47 7 2
53,753.67
7, 18 3
53,736.75
7, 18, 31 4
7, 18, 31, 38 5
Infeasible - 6
infeasible
Total Operating Cost ($)
Approved Line ID(s)
53,781.71
---- 1
53,783.72 38 2
53,795.89
38, 7 3
53,842.87
38, 7, 18 4
54,211.41
38, 7, 18, 31 5
Infeasible - 6
The SMaRTS Model w/ no Partial Maintenance
Total Operating Cost ($)
Approved Line ID(s)
53,781.71
--- 1
53,772.01 7 2
53,773.44
7, 38 3
53,790.29
7, 18, 38 4
53,833.21
7, 18, 31, 38 5
Infeasible - 6
infeasible

48. Results

49. Impact of N-1 Reliability

50. PART I - SDP Based Transmission Topology Control
Conclusion
PART I - SDP Based Transmission Topology Control
AC Model with physical flow limits
Up to 9% cost reduction
Guaranteed to find a better topology
PART II - The SMaRTS Model
A new method to utilize Transmission Switching in Outage Coordination
Method to discard additional variable for N-1
Up to 4% daily cost reduction
NYISO - daily operating cost is $15 Million.

51. Gokturk Poyrazoglu gokturkp@buffalo.edu
Thank you.
Gokturk Poyrazoglu
University at Buffalo, SUNY
Department of Electrical Engineering