5th Southeast Symposium on Contemporary Engineering Topics (SSCET), 2014
A Parallel Solution to Stochastic Power
System Operation with Renewable Energy
Yong Fu, Ph.D.
Associate Professor
Electrical and Computer Engineering
Mississippi State University
New Orleans, LA
September 19th, 2014
Parallel Computing
o With development of high performance computing technique, parallel
computing technique can significantly improve computational efficiency of
optimization problem with utilization of multi-processors and multi-threads.
o These improvements cannot be achieved by the architectures of the
machines alone, it is equally important to develop suitable mathematical
algorithms and proper decomposition & coordination technique in order to
effectively utilize parallel architectures
A Typical Power System Operation Problem
– Security Constrained Unit Commitment
 Objective Function – Minimize
 Generation and startup/shutdown costs
 Generating Unit Constraints





Load (MW)
300
Unit 3
250
Generation capacity
Minimum ON/OFF time limits
Ramping UP/DOWN limits
Must-on and area protection constraints
Forbidden operating region of generating units
Unit 2
200
Unit 1
150
100
1
4
7
10
13
16
19
22
Hours
 System Operation Constraints






Power balance
System reserve requirements
Power flow equations
Transmission flow and bus voltage limits
Limits on control variables
Limits on corrective controls for contingencies
Large Scale, Non-Convex,
Mixed Integer Nonlinear
Problem
Who Use SCUC and How?
 ISOs: PJM, MISO, ISO New England, California ISO, New York ISO and ERCOT

GENCOs
TRANSCOs
DISTCOs

ISO

Security-Constrained
Unit Commitment

ISO (SCUC) and Market Participants
Day Ahead Market (DAM) determines the 24-hourly
status of the generating units for the following day
based on financial bidding information such as
generation offers and demand bids.
Day Ahead UC for Reliability (RUC), which focuses
on physical system security based on forecasted system
load, is implemented daily to ensure sufficient hourly
generation capacity at the proper locations.
Look-Ahead UC (LAUC), as a bridge between dayahead and real-time scheduling, constantly adjusts the
hourly status of fast start generating units to be ready to
meet the system changes usually within the coming 3-6
hours.
Real-Time Market (RTM) further recommits the very
fast start generating units based on actual system
operating conditions usually within the coming two
hours in 15-minute intervals.
Stochastic SCUC

In stochastic programming, the decision on certain variables has to be made before the stochastic
solution is disclosed, whereas others could be made after.

The set of decisions is then divided into two groups:
 A number of decisions are made before performing experiments. Such decisions are called firststage decisions and the period when these decisions are made is called the first stage.
 A number of second-stage decisions are made after the experiments in the second stage.

Stochastic models containing above two groups of variables, first-stage and second-stage decision
variables, are called two-stage stochastic programming.
Min
x , y1 ,, yk
K
c x  Min  p k d T y k
T
k 1
s.t. Ax  b, x is binary,
Hx  Fy k  ek , k  1,  , K
Stochastic SCUC --- Example
G1
13 $/MWh
40MW~80MW
G2
42 $/MWh
15MW~ 40MW
1
2
L1 75MW
L2 75MW
G3
16 $/MWh
10MW~40MW
Load

W

System
Base Case
50 MW
Wind
(WM)
Load
(MW)
Base case
-
20
100
Scenario 1
G3
15
105
Scenario 2
L2
23
95
G3 can adjust dispatches by 5 MW
G2 is quick-start unit with 30 MW QSC
Scenario 1
Scenario 2
0 MW
? MW
52.5 MW
0 MW
15
MW
0
MW
50 MW
52.5 MW
Base Case
105 MW
95 MW
23 MW
Scenario 1
Scenario 2
60 MW
42.5 MW
15 MW
52 MW
52.5 MW
0 MW
30
MW
0
MW
42.5 MW
20 MW
75 MW
15 MW
65 MW
0 MW
? MW
100 MW
20 MW
Solution 2
Equipment
Outage
75 MW
80 MW
Solution 1
Cases
20 MW
75 MW
0
MW
52.5 MW
100 MW
15 MW
105 MW
95 MW
23 MW
Current Work
Start
Unit=1:NG
Unit
Commitment
calculation
Unit
Commitment
calculation
Unit
Commitment
calculation
Unit
Commitment
calculation
Optimal Power
Flow
Calculation
Optimal Power
Flow
Calculation
Time=1:NG
Optimal Power
Flow
Calculation
Optimal Power
Flow
Calculation
End
o Amdahl’s law: an upper
bound on the relative
speedup achieved on a
system with multi-processors
is decided by the execution
time of the application
operating sequentially.
Proposed Approach
Start
Time=1:NG
Unit=1:NG
Unit
Commitment
calculation
Unit
Commitment
calculation
Unit
Commitment
calculation
Unit
Commitment
calculation
Optimal Power
Flow
Calculation
Optimal Power
Flow
Calculation
Optimal Power
Flow
Calculation
Optimal Power
Flow
Calculation
End
o
o
Structure of Algorithm: Scenario-based stochastic model is adopted to analyze the uncertainties
of load and wind energy in this paper. Instead of master-and-slave structure, UC and OPF
subproblems are solved simultaneously in the proposed parallel calculation method.
Convergence performance: In an iterative solution process, the number of iterations affects the
overall computational time. Several convergence acceleration options, including initialization
and update of penalty multipliers, truncated auxiliary problem principle and trust region
technique, are used to improve the convergence performance and efficiency in a scenario-based
study.
Decomposition Strategy
 Mathematically, the stochastic SCUC can be formulated as a mixed integer
programming (MIP) problem as shown in
NS
  s Fs ( x, y s )
Min
s 0
s.t. Ax  bys  d , Eys  h
 Variable Duplication Technique
NS
  s Fs ( x, y s )
Min
s.t.
s 0
Ax  bys  d ,
Eyˆ s  h,
y s  yˆ s
 Augmented Lagrangian Method
NS
Min
s.t.
T
  s Fs ( x, y s )   ( yˆ s  y s ) 
s 0
Ax  bys  d ,
Eyˆ s  h
c
yˆ s  y s
2
2
Algorithms for Parallel Solutions
 Auxiliary Problem Principle (APP) Method
 Diagonal Quadratic Approximation (DQA) Method
 Alternating Direction Method of Multipliers (ADMM)
 Analytical Target Cascading (ATC) Method
Iterative Solution Procedure
Decomposition structure:
Original Augmented
Lagrangian Problem
Optimal Power Flow
Subproblem
Unit Commitment
Subproblem
UC
Subproblem
Unit 2
UC
Subproblem
Unit 1
UC
Subproblem
Unit NG
OPF Subproblem
Period 1
OPF Subproblem
Period NT-1,
Scenario 0
OPF Subproblem
Period NT-1
OPF Subproblem
Period NT-1,
Scenario 1
Two separated auxiliary problem:
Decision variables for
the current iteration
NS
NT NG
s 0
t 1 i 1
OPF Subproblem
Period NT
OPF Subproblem
Period NT-1,
Scenario NS
Given values
from the previous
iteration
1)
1)
Luc  Min   s   [ F ( Pg ,its , I it )  SUDit  c g ,its ( Pg ,its ) 2  (its  c g ,its ( Pˆg( k,its
 Pg( k,its
)) Pg ,its ]
NS
NT NG
s 0
t 1 i 1
1)
1) ˆ
Lopf  Min   s   [c g ,its ( Pˆg ,its ) 2  (its  c g ,its ( Pˆg( k,its
 Pg( k,its
)) Pg ,its ]
Case Study – IEEE 118-bus Testing System
o Case 1: Deterministic case
o Case 2: Stochastic case with 3 scenarios
Zone 1
Zone 2
7
2
13
1
33




55
11
56
34
53
14
46
17
4
54
45
12
6
54 thermal units
3 wind farms
118 buses
186 branches
44
117
15
3
43
36
35
37
18
57
52
47
42
39
16
58
51
19
41
5
59
48
40
49
50
60
38
8
9
30
10
20
113
31
29
73
32
66
21
62
69
67
64
65
28
114
71
26
22
75
118
76 77
115
25
27
68
23
80
116
24
98
70
78
79
97
86
85
90
63
72
74
87
61
81
88
89
96
84
83
82
95
93
91
112
94
107
92
106
100
102
101
106
109
105
103
108
Zone 3
111
104
110
99
Deterministic Case Study
The converged result is obtained after 39 iterations.
150
Unit 36 at Hour 5
150
Power(MW)
Power(MW)
200
100
P op f,36,5
50
Unit 45 at Hour 5
100
50
P op f,45,5
P uc,36,5
0
10
20
Iterations
30
0
40
300
300
250
250
200
Unit 36 at Hour 21
150
100
P op f,36,21
50
0
P uc,36,21
0
10
20
Iterations
30
40
Power(MW)
Power(MW)
0
P uc,45,5
0
10
200
20
Iterations
40
Unit 45 at Hour 21
150
100
P op f,45,21
50
0
30
P uc,45,21
0
10
20
Iterations
30
40
Deterministic Case Study
Items
Centralized
SCUC
Parallel
SCUC
Changes
Total Cost ($)
1,583,700
1,584,997
+0.08%
Time (Seconds)
19
8
-58%
Stochastic Case Study (3 scenarios)
Items
Centralized
SCUC
Parallel
SCUC
Changes
Cost ($)
1,582,840
1,583,565
+0.046%
Time (Seconds)
1,083
20
-96%
Case Study – A 1168-bus Power System
1200
1000
Power Output (MW)
o A practical 1168-bus power
system with 169 thermal units,
10 wind farms, 1474 branches,
and 568 demand sides.
o It could be nearly impossible to
get a near-optimal stochastic
SCUC solution for this system
by applying a traditional
centralized SCUC algorithm.
o However, the proposed parallel
stochastic SCUC algorithm
provides solutions.
800
600
400
Popf,8,1,1
200
Puc,8,1,1
0
0
50
100
150
Iterations
Unit 8 at Hour 1
200
250
Case Study – A 1168-bus Power System
# of Scenarios
# of Iteration
Total Time (sec.)
0
1
2
3
4
5
6
7
8
9
10
315
330
299
327
277
278
248
243
242
237
231
109.55
146.06
139.31
163.47
142.28
140.28
139.52
130.94
133.25
148.33
131.95
Conclusions
o The proposed stochastic SCUC approach minimizes the operation cost of
system by possibility expectation of each scenarios, which can adaptively
and robustly adjust generation dispatch in response to constraints in
different scenarios.
o In comparison with traditional stochastic SCUC, optimal power flow
problem does not have to wait for unit commitment decision, both
problems can be solved simultaneously, which is more computational
efficient in both day-head and real-time power markets.
o The ideas can be applied to various power system applications: state
estimation, economic dispatch, and planning.
Thanks !
fu@ece.msstate.edu