Abstract- Load shedding during contingency conditions is an efficient solution to alleviate transmission lines over loadings. Minimization of total load shedding considering loads importance has great significance in these situations. This problem requires simultaneous optimization of two or more conflicting objectives, such as minimization of the total load shedding, minimization of transmission lines over loadings and voltage violations minimization. The objectives are in conflict since the improvement of one of them leads to the deterioration of another.
A modified version of Non-Dominated Sorting Genetic Algorithm (NSGA-II) is used as an effective optimization tools for solving the minimum weighted load shedding problem during contingency conditions. Also relation between transmission lines overloading and amount of load shedding is surveyed. IEEE 14 bus test system is used as a study case and results are discussed.
Keywords-: Contingency, energy not supplied, generation rescheduling, load shedding, NSGA-II.
Nomenclature
number of buses,
number of generators,
normal state indicator (superscript),
contingency state indicator (superscript),
bus index (subscript),
importance factor of load in ith bus,
load active power demand in normal state,
load active power demand in contingency state,
active power generation in normal state,
active power generation in contingency state,
active power losses in contingency state,
load reactive power demand in normal state,
load reactive power demand in contingency state,
reactive power generation in normal state,
reactive power generation in contingency state,
reactive power losses in contingency state,
minimum active power generation,
maximum active power generation,
minimum reactive power generation,
maximum reactive power generation,
minimum amount of load which must be supplied,
apparent power flow from bus i to j,
apparent power limit of line,
voltage low limit,
ith bus voltage,
voltage up limit.
Introduction
The emergency states may occur in a power system as a consequence of a sudden increase of system load, the unexpected outage of a transmission line, a generator or failure in any of the system components. This state may result in some problems such as lines overloading, under frequency, voltage collapse and angle instability [1].
Generation rescheduling and/or load shedding can be used to overcome the mentioned problems, effectively. Load shedding schemes have become quite important in present day systems, where there is a lack of adequate spinning reserve margin or a shortage of tie line capacity to make up the lost generation [2]. Optimal load shedding has been taken into consideration from various aspects and by means of various techniques in many papers.
It is a common practice for utility companies to perform load shedding by using under frequency relays to disconnect the predetermined load when the frequency drops below set values [3]. A load shedding method which considers the frequency decay rate is also applied for utilities in [4].
An adaptive under frequency load shedding to protect electric power systems from dynamic instability and frequency collapse is presented in [5]. A methodology for setting the under frequency load shedding relays is described in [6] which is based on the combination of artificial neural networks and the Monte Carlo simulation. In [7] an adaptive ANN based load shedding algorithm is proposed which considers system stability. Coordination of load shedding scheme and superconducting magnetic energy storage (SMES) unit to enhance the transient stability of a large industry cogeneration facility has developed in [8]. In [9] a non-linear programming methodology for evaluating load shedding as a corrective action to improve the dynamic security of power systems is proposed. A scheme to load shedding from the voltage stability point of view is presented in [10].
In many other researches the load shedding problem is studied from steady state point of view. In [2] and [11], a steady state load shedding scheme based on Genetic Algorithm (GA) method is proposed which considers operational constraints. [12] proposes an interior point based optimization model to minimize the load curtailments necessary to restore the equilibrium of operating point with relaxation of restrictions. The developed methodology minimizes the load shedding by considering their importance.
Most of research efforts have been essentially addressed to local load shedding (LLS) schemes [9] which may result in increasing of overall power system load shedding. Also using of some conventional techniques such as linear programming requires some approximations and linearization in the model of system which tends to non-optimal solutions [13].
None of the existing papers considers the exact relation between the amount of load shedding and lines overloading or voltages violations. In this paper a modified version of non-dominated sorting genetic algorithm is used to determine the location and amount of load shedding in power system as well as generator rescheduling in post contingency conditions. Also simultaneous minimizing of total load shedding with respect to loads importance, transmission lines overloading and voltage violations are taken into consideration. Severity of transmission lines over loading in terms of load shedding is studied.
Proposed algorithm is applied to IEEE 14 bus test system during two critical contingencies and results are discussed. This paper is organized as follows: The concept of multi objective optimization is explained in section 3. Problem formulation is shown in section 4. Non-dominated sorting genetic algorithm and applying this algorithm to the mentioned problem is brought in section 5. Section 6 introduces the study case and the next sections include the results/discussions and conclusion, respectively.
Multi Objective Optimization Problem
Mathematically, multi objective optimization problems can be written as equation 1.
(1)
In the presence of constraints as equations (2) and (3):
(2)
(3)
is the vector of control variables. The solution of (1) is usually not unique. The concept of pareto optimality may be explained in terms of a dominance relation [14]. For a multi objective problem having objective functions to be simultaneously minimized, a solution is said to dominate the other solution if is better than for at least one objective and is not worse for any other , where and as equation (4).
(4)
Here, the symbol denotes domination operator. The above concept is used to find a set of non-dominated solutions in search space. The solutions that are non-dominated regarding the entire search space are called pareto optimal solutions [15].
Problem Formulation
Alleviating or elimination of transmission lines over loadings and voltage profile improvement in contingency conditions by means of load shedding and generators rescheduling is formulated as an optimization problem with non-linear constraints as follows:
Objective Function
This problem has three objective functions which must be simultaneously minimized. These three components are as equations (5) to (7).
(5)
Equation (5) represents the sum of weighted difference between pre contingency and post contingency active power consumption of all buses and equals to total weighted load shedding. Equation (6) represents the total amount of over loadings in all over loaded transmission lines.
(6)
Equation (7) represents the sum of voltages violations in all buses.
(7)
Constraints
The equality and inequality constraints are described in equations (8) to (15). Active and reactive power balance equations are expressed as equations (8) and (9), respectively.
(8)
(9)
Control variables constraints are the real power of generators and load demand of buses which are shown by equations (10) and (11), respectively.
(10)
(11)
In equation (11), we have restricted load shedding of buses between pre contingency value and a minimum amount i.e. . In other word, it has assumed that the load shedding in bus i cannot be greater than . Operating constraints are as follows:
(12)
(13)
(14)
This object is achieved by optimal determination of control variables. Control variables are shown in Fig. 1 schematically.
…
…
Schematic diagram of control variables
Solution Method
Multi-objective optimization (MO) is an appropriate tool for handling different incommensurable objectives with conflicting relations or not having any mathematical relation with each other [16]. NSGA is an evolutionary based multi objective optimization tools which has great advantages in comparison to other evolutionary algorithms.
The steps of NSGA, developed to find a set of the best tradeoffs between the three objectives in the load shedding problem, are as follows:
Each individual is a string of numbers which represents control variables. Each individual is a feasible solution in search space. The population is initialized as usual. Once the population is initialized the population is sorted based on non-domination into each front. The first front being completely non-dominant set in the current population and the second front being dominated by the individuals in the first front only and the front goes so on. Each individual in the each front are assigned rank (fitness) values based on front in which it belong to.
In addition to fitness value a new parameter called crowding distance is calculated for each individual. The crowding distance is a measure of how close an individual is to its neighbors. Large average crowding distance will result in better diversity in the population.
Parents are selected from the population based on the rank and crowding distance. The selected population generates offsprings from crossover and mutation operators. The population with the current population and current offsprings is sorted again based on non-domination and only the best N individuals are selected, where N is the population size. Detailed Description of NSGA-II is represented in [17]. Fig. 2 summarizes the basic steps of the NSGA-II applied to load shedding minimization problem.
Basic steps of the NSGA-II applied to weighted load shedding minimization problem
Case Study
The IEEE 14-bus test system which is shown in Fig. 3 has been selected as case study in this paper [18]. Initial operating state of the system is shown in table I. Generator in 1st bus is considered as reference generator. Variables with dark background are control variables.
IEEE 14 bus test system
It is supposed, that the increasing or decreasing the active power generation of all generators is limited to 20 percent of their current generation. In other word we have: and . Also, it is supposed that for all of load buses. This equation means that, load shedding in bus i, can not be greater than 50 percent of load demand in this bus. Also apparent power limits of the transmission lines have been given in the appendix.
Initial operating state of system
Generation Power
Consumption power
Voltage magnitude
Bus No.
MVAr
MW
MVAr
MW
-16.9
232.32
0.00
0.00
1.06
1
42.4
40.0
12.7
21.7
1.04
2
23.4
0.00
19.0
94.2
1.01
3
0.00
0.00
-3.9
47.8
1.01
4
0.00
0.00
1.6
7.6
1.02
5
12.2
0.00
7.5
11.2
1.07
6
0.00
0.00
0.00
0.00
1.06
7
17.4
0.00
0.00
0.00
1.09
8
0.00
0.00
16.6
29.5
1.05
9
0.00
0.00
5.8
9.0
1.05
10
0.00
0.00
1.8
3.5
1.05
11
0.00
0.00
1.6
6.1
1.05
12
0.00
0.00
5.8
13.5
1.05
13
0.00
0.00
5.0
14.9
1.03
14
Results and Discussion
In this section the relation between the transmission lines over loadings and the amount of load shedding in power system is studied. Algorithm is applied to case study during two contingencies. The first contingency is a single contingency and the second one is a double contingency. Also, effect of loads importance factor on obtained solutions is surveyed.
First Contingency: Line 1-2 Outage
According to power flow results at the pre contingency operating condition, maximum transmitted power flows in line 1-2 equals to 157 MVA. Therefore, it seems that the removal of this line would be a critical contingency. Removing line 1-2 causes to over loadings of lines 1-5 and 4-5.
Table II shows the new system operating condition after this contingency by means of proposed algorithm. Over loadings of mentioned lines are eliminated because of load shedding and generators rescheduling. Symbol ↑ indicates increasing the generators generation and symbol ↓ indicates decreasing the generators generation.
Table II shows that the generator at bus 1 decreases its generation to 110 MW. In fact, 110 MW is the maximum power, that generator 1 can generate in this contingency. If the generation of generator 1 is more than 110 MW then the line 1-5 will over load, it is as a result of power sending by means of generator at bus 1 to the rest of the system only through the line 1-5.
By decreasing the generation of generator 1, generator 2 increases its generation to provide total power demand of system. The percentage of load shedding for each bus is shown at the bottom of table II. The total load shed applied to customers is shown in the last row of this table.
New operating state after first contingency by means of proposed algorithm (All buses have equal importance factors)
Control Variables
Bus No.
110.00↓
1
Active power generation (MW)
48.00↑
2
36.83↑
3
Reactive power of condensers (MVAr)
17.21↑
6
1.73↓
8
0.00
1
The percentage of load shedding (%)
31.67
2
48.64
3
45.59
4
41.29
5
49.63
6
0.00
7
0.00
8
43.99
9
32.88
10
42.63
11
27.15
12
34.06
13
13.88
14
108.93
Total shed power (MW)
Fig. 4 shows the relation between the amount of load shedding and the transmission lines over loading. As shown in fig. 4 the over loadings of the transmission lines will decrease as the amount of load shedding increases. Finally the over loadings of lines reach to zero in the particular amount of load shedding. This minimum amount of load shedding is inevitable during mentioned contingency to eliminate transmission lines over loadings.
If the amount of load shedding is less than this minimum amount, the lines over loadings will not be eliminated and if the amount of load shedding is greater than this minimum amount, the power system will experience unnecessary load shedding.
Sum of transmission lines over loadings in terms of total load shedding
In order to survey the effect of importance factors on obtained solutions, the mentioned results are re-obtained by considering the following assumptions:
(16)
The results in this case are shown in table III. It can be deduced from table III that, by increasing importance of loads in buses number 12, 13 and 14 to three times of theirs previous value, load shedding in these buses is decreased considerably.
New operating state after first contingency by means of proposed algorithm (Buses have different importance factors)
Control Variables
Bus No.
110.00↓
1
Active power generation (MW)
48.00↑
2
25.81↑
3
Reactive power of condensers (MVAr)
14.66↑
6
4.38↓
8
0.00
1
The percentage of load shedding (%)
46.42
2
49.93
3
50.00
4
28.73
5
18.69
6
0.00
7
0.00
8
48.73
9
33.05
10
28.45
11
8.47
12
26.70
13
17.62
14
110.37
Total shed power (MW)
Fig. 5 shows the impact of increasing loads importance factor on the amount of load shedding in three buses while keeping other buses importance factor equal to 1. As shown in fig. 5 by increasing the importance factor of buses the amount of load shedding decreases.
Effect of loads importance factors on the amount of load shedding in buses 12, 13 and 14
Second Contingency: Line 1-5 and Generator 2 Outage
In order to reveal the performance of mentioned algorithms, results are obtained by applying it into another contingency that is simultaneous outage of line 1-5 and generator at bus 2. In this case, line 1-2 is over loaded. Table IV shows the new operating condition in this contingency by means of proposed algorithm.
New operating state after second contingency by means of proposed algorithm (All buses have equal importance factors)
Control Variables
Bus No.
220↓
1
Active power generation (MW)
0.00↓
2
11.34↓
3
Reactive power of condensers (MVAr)
3.62↓
6
19.82↑
8
0.00
1
The percentage of load shedding (%)
0.12
2
6.43
3
24.70
4
25.94
5
46.35
6
0.00
7
0.00
8
31.71
9
49.05
10
23.41
11
41.19
12
40.29
13
49.48
14
54.97
Total shed power (MW)
Fig. 6 shows the relation between the amount of load shedding and the transmission lines over loading. As shown in fig. 6 the over loadings of the transmission lines will decrease as the amount of load shedding increases. The minimum amount of load shedding to eliminate over loadings of transmission lines is equal to 54.97 MW.
Sum of transmission lines over loadings in terms of total load shedding
In addition, in order to survey the effect of loads importance factors on obtained solutions in second contingency, results are re-obtained by considering the following assumptions:
(17)
Results have been shown in table V. It is clear from table V that, by increasing importance factors of loads in buses number 12, 13 and 14 to ten times of theirs previous value, load shedding in these buses is decreased considerably.
Fig. 7 shows the impact of increasing loads importance factor on the amount of shed load in three buses while keeping other buses importance factor equal to 1. As shown in fig. 7 by increasing the importance factor of buses the amount of load shedding decreases.
Conclusion
In this paper, minimizing of total load shedding by considering operating constraints such as transmission lines maximum permissible power flow and maximum permissible fluctuations of buses voltage is solved. Also relation between load shedding and sum of power system transmission lines overloading were discussed.
In order to give priority to loads of different buses, importance factor was allocated to each bus and effect of importance factors on load shedding in each bus was considered. In order to reveal effectiveness of the methodology, the proposed method was applied to power system during two critical contingencies.
Simulation results showed that there is a specific amount of load shedding corresponding to each contingency in which the lines over loadings will reach to zero. This amount of load shedding is minimum load which must be shed. If the amount of load shedding is less than this minimum amount, the lines over loadings will not be eliminated and if the amount of load shedding is greater than this minimum amount, the power system will experience unnecessary load shedding.
Also, increasing the importance factor of buses, decreases the amount of load shedding in those buses but the total amount of load shedding remains constant.
New operating state after second contingency by means of three algorithms (Buses have different importance factors)
Control Variables
Bus No.
220↓
1
Active power generation (MW)
0.00↓
2
15.37↓
3
Reactive power of condensers (MVAr)
20.41↑
6
14.99↓
8
0.00
1
The percentage of load shedding (%)
6.239
2
24.999
3
11.019
4
15.841
5
43.880
6
0.00
7
0.00
8
43.929
9
41.401
10
10.519
11
0.000
12
0.077
13
6.848
14
54.37
Total shed power (MW)
Effect of loads importance factors on the amount of load shedding in buses 12, 13 and 14
appendix
In this paper it is supposed that the transmission lines apparent power limits are as table VI.
Apparent power limit for 14 bus test system transmission lines
Line No.
From Bus
To Bus
Apparent Power Limit
(MVA)
1
1
2
220
2
1
5
110
3
2
3
110
4
2
4
110
5
2
5
110
6
3
4
110
7
4
5
110
8
4
7
70
9
4
9
70
10
5
6
55
11
6
11
40
12
6
12
70
13
6
13
55
14
7
8
55
15
7
9
55
16
9
10
55
17
9
14
55
18
10
11
55
19
12
13
55
20
13
14
55
