This document discusses steady-state security regions in power systems. It presents methods to construct steady-state security regions, including an optimization method and expanding method. Numerical examples are provided to compare the size of security regions for the IEEE 6-bus and 30-bus test systems using different methods. The appendix further discusses the standard form of linear programming problems.

1. University of technical education HCMC
Faculty of Electrical and Electronics
Engineering
Power System Optimization
www.hcmute.edu.vn
Chapter 9
STEADY-STATE
SECURITY REGIONS
Nguyễn Anh Toàn

2. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction
Objectives
2.Model Presenting the concept and definition of security
region
3. Numerical Example
Explaint various methods to contruct steady state
4. Appendix
security regions of power system
5. Conclusion

3. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction
Agenda
1 Introduction
2.Model
2 Model
3 Numerical Example
3. Numerical Example
4 Appendix
5
4. Appendix
Conclusion
5. Conclusion

4. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction Introduction
Used to find the best or optimal solution
2.Model
Requires that all the mathematical functions
in the model be linear functions.
3. Numerical Example
The linear model consists of the following
components:
4. Appendix • A set of decision variables.
• An objective function.
5. Conclusion • A set of constraints.

5. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction Models
Objective Function:
n 1
M m
2.Model MaxZ Wi ( PGi PGi ) (9.72)
i nd 1
Subject to constrains
3. Numerical Example
n 1
M m
(P Gi P ) Gi ( PGi max PGi min ) (9.90) ij min ( Aik Ajk ) PGk ij max (9.95)
i nd 1
m
PGi PGi PGi min i=n d +1,..,n-1 (9.91) m
PGk when Aik -A jk 0
PGi M
PGi PGi max i=n d +1,..,n-1 (9.92) PGk ={ M (9.96)
4. Appendix PGk when Aik -A jk 0
nd n 1
M
( Pi - PGi )=PGnm (9.93) PM
G P0 +
G P0
G (9.97)
i 1 i nd 1
nd n 1
m Pm
G P0 +
G P0
G (9.98)
( Pi - P )=PGnM
Gi (9.94)
i 1 i nd 1
5. Conclusion

6. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction Models
X2
1000 Let’s take a closer look at
2.Model the optimal point
800 Infeasible
3. Numerical Example 600
4. Appendix
Feasible
Feasible
region
region
X1
5. Conclusion
400 600 800

7. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction Numerical Examples
To assess or compare the size of Ωp for
2.Model different means, the following performance
index is introduced:
n 1
(PM
Gi Pm )
Gi
i nd 1
3. Numerical Example PI= n 1
(9.99)
( P max
Gi P min )
Gi
i nd 1
4. Appendix
or
M m
PGi PGi
PI= i=n d +1,.....,n-1 (9.100)
PGi max PGi min
5. Conclusion

8. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction Numerical Examples
Table 9.14 Comparison of security region results for IEEE 6-bus system.
2.Model
Security Generator Generator Total
Method
regions PG4 PG5 PI%
PM
Gi 4.200 2.200
3. Numerical Example m
PGi
Method 1 0.184 1.378 71%
PI i % 96% 31%
PM
Gi 3.750 2.649
4. Appendix m
Method 2 PGi 2,449 1.400 37%
PI i % 31% 47%
Method 1: optimization method.
5. Conclusion Method 2: the expanding method.

9. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction Numerical Examples
Table 9.15 Comparison of security region results for IEEE 30-bus system
Gen Gen
Security Gen Gen Gen Total
method PG PG
regions PG2 PG5 PG8 PI%
2.Model 11 13
PM
Gi 0.800 0.500 0.350 0.300 0.384
Method m
PGi 0.439 0.150 0.100 0.100 0.120 85%
1
3. Numerical Example
PI i % 80% 100% 100% 100% 94%
PM
Gi 0.712 0.402 0.350 0.300 0.400
4. Appendix
Method m
PGi 0.428 0.150 0.148 0.100 0.177 70%
2
PI i % 47% 72% 81% 100% 80%
5. Conclusion Method 1: optimization method.
Method 2: the expanding method.

10. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction Numerical Examples
2.Model Table 9.16 Results for security regions on IEEE 6-bus system (p.u.)
Security
-cut PG4 PG5 PG6
regions
M
3. Numerical Example PGi 4.200 2.2240 3.8990
1
P m
Gi 0.1840 1.3700 0.0000
M
PGi 4.0050 202245 4.5480
0.6 m
PGi 0.1701 1.1500 0.0000
m
PGi 4.0050 2.2245 4.7100
0.5 m
PGi
4. Appendix 0.1620 1.0693 0.0000
M
PGi 3.9755 2.4245 5.1849
0 m
PGi 0.1215 1.000 0.0000
5. Conclusion

11. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction Appendix
Standard Form of LP
2.Model
Max/min z = c1x1 + c2x2 + ... + cnxn
subject to:
a11x1 + a12x2 + ... + a1nxn (≤, =, ≥) b1
3. Numerical Example a21x1 + a22x2 + ... + a2nxn (≤, =, ≥) b2
:
am1x1 + am2x2 + ... + amnxn (≤, =, ≥) bm
4. Appendix
xj = decision variables
bi = constraint levels
cj = objective function coefficients
5. Conclusion aij = constraint coefficients

12. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction Infeasibility
No point, simultaneously,
2.Model lies both above line 1 and
below lines 2 and 3.
3. Numerical Example
2
4. Appendix
5. Conclusion 1
3

13. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction Unbounded solution
2.Model
3. Numerical Example
4. Appendix
5. Conclusion

14. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction Unbounded solution
Primal Dual
2.Model Objective max cTx min bTy
Variables x1, …, xn y1,…, ym
Constraint matrix A AT
3. Numerical Example Right-hand vector b c
Constraints ith constraint: · yi ¸ 0
versus ith constraint: ¸ yi · 0
Variables ith constraint: = yi unrestricted
4. Appendix
,
xj ¸ 0 jth constraint: ¸
xj · 0 jth constraint: ·
xj unrestricted jth constraint: =
5. Conclusion

15. FEEE
University of technical education HCMC
POWER SYSTEM OPTIMIZATION
Faculty of Electrical and Electronics Engineering
Ensuring Enhanced Education
Group No 5
www.hcmute.edu.vn/feee/ Chapter 9: STEADY-STATE SECURITY REGIONS
1. Introduction
2.Model
3. Numerical Example
4. Appendix
GROUP 5
5. Conclusion