Pre-processing of the optimal coordination of overcurrent relays

Electric Power Systems Research 75 (2005) 134–141

Pre-processing of the optimal coordination of overcurrent relays Hossein Kazemi Karegar a , Hossein Askarian Abyaneh b,∗ , Vivian Ohis c , Matin Meshkin d a

Department of Electrical Engineering, Zanjan University and the Center for Renewable Energy of Research and Application, Zanjan University, Zanjan P.O. Box 313, Iran b Department of Electrical Engineering, Amir Kabir University of Technology and Zanjan University, Hafez Avenue No. 424, P.O. Box 15875-4413, Tehran, Iran c Department of Electrical and Computer Systems Engineering, Monash University, Clayton, Melbourne, Vic. 3800, Australia d Department of Electrical Engineering, Amir Kabir University of Technology, Tehran, Iran Received 10 August 2004; accepted 1 February 2005 Available online 9 June 2005

Abstract This paper presents a method to reduce the number of constraints and detect those that make the optimization of overcurrent relay settings to be infeasible. By reducing the number of constraints, the feasibility of the optimization is increased and the run-time of the program is decreased. It should be noted that, in the optimal coordination methods, the operating times of the relays are minimized in the frame of the objective function subjects to the so-called coordination constraints. However, a new method for optimum coordination of protective relays based on only constraints has been developed by authors of the paper. The pre-processing reduction constraints method introduced in this paper can be used for any optimal coordination of overcurrent relays. The new method is supported by results obtained from a typical test network and a real power system network. © 2005 Elsevier B.V. All rights reserved. Keywords: Optimal coordination; Power system protection; Relay coordination

1. Introduction Setting and coordinating of protective devices in a power system is a tedious and time-consuming job. A well-designed coordination computer program can perform relieve to relay engineer of this routine task. There are two approaches to coordinate protective devices in power systems: non-optimal and optimal [1,2]. It has been shown that the linear optimal programming can be applied to optimize relay settings in interconnected power systems [3]. The optimal coordination of overcurrent (OC) relays is a linear-programming problem for minimizing the operating times of OC relays; this is referred to as an objective function subject to the coordination constraints. The ∗

Corresponding author. Tel.: +98 21 6466009; fax: +98 21 6406469. E-mail addresses: h kazemi [email protected] (H.K. Karegar), [email protected] (H.A. Abyaneh), [email protected] (V. Ohis), matin [email protected] (M. Meshkin). 0378-7796/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2005.02.005

constraints define coordination criteria based on primary and backup (P/B) pair relays [2,4]. The critical and the worst-case scenario for a power system should be considered in the optimal relay coordination process. In addition, consideration of future modifications to the power system and dynamic changes will produce more constraints [3–6]. Recently, a new optimal coordination method has been developed by the author in which there is no need to compose an objective function [7]. In other word, solving constraints equations can give optimal results directly. This leads to simplify the optimal coordination problem. In this method, all constraints must be fulfilled, however, if some of the constraints conflict, there will be no solution and the optimal problem becomes infeasible. The aim of this paper is to: first recognize the constraints that make the optimization problem infeasible, and then decrease the complexity of the optimal coordination problem of OC relays by reducing the number of constraints. Of course,

H.K. Karegar et al. / Electric Power Systems Research 75 (2005) 134–141

the efficient constraints are kept and the useless ones are removed. In this paper, the method to designate the constraints is provided. An area called possible solution area (PSA) is introduced for each relay pair, i.e. primary and backup relays. The PSA is a square, which is bounded by the maximum and minimum values of the time multiplier setting (TMS) or time dial setting (TDS) for each primary and backup relays. A line on the two dimensional plane, i.e. PSA, is defined. It describes the relationship between the backup and primary relays operating times, this is referred to as a constraint line. Based on the intersection of constraint lines and PSA, the constraint lines are divided into four categories among them only one is valid. Improvement of the feasibility is made by comparing the crossing points of valid constraint lines with the PSA. The method is fully described in the paper. The application of the proposed method on an 8-Bus network and England Norweb power system will be shown in the paper. A solution for infeasible problems is referred as well as processing time reduction. The detail approach is described in the next scetions.

2. Review of the optimal coordination problem This section is devoted to the review of the notation and concepts presented in Ref. [7] to give a better coherency to this paper. In the coordination program, two types of tap settings, current and time dial settings must be calculated. The current setting for each relay is determined by two parameters: the minimum fault current and the maximum load current [2,8]. However, the variables of interest in the optimal coordination problem are the TDS/TMS [4]. To find the TDS/TMS using optimal programs, the objective functions and constraints are given by in Refs. [2,3] Minimize:
Obj = (Wi f (Ipi , Ii )xi ) (1)

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within the program after each xi calculation [7]. A×X≥B

(6)

Matrix A is composed of two parts, upper and lower, as shown in (7). The upper part is a unity matrix because the constraints relating to the minimum of TDS are located there as can be seen in (7). The lower part of matrix A relates to the other constraints. For example, the first constraint of the optimal problem is related to relays i and j, where relay j is the backup for relay i. Then the first constraint parameters of the backup and primary coefficients, i.e. f (Ipj , Ij ) and −f (Ipi , Ii ) are placed in the first row of the lower part of matrix A. The second constraint located in the second row. This process continues for every constraint. Each row represents a fault either close to the relay or at the far end bus of related line for a specific line outage. This eventually completes matrix A.       TDSmin X1 1 0 0 0......0        X2   TDSmin   0 1 0 0......0          .  0 0 1 0......0  .           .   ......  .....................         (7)  0 0 0 0 . . . . . . 1   Xm  ≥  TDSmin         - - -   - −  - - - - - - - - - - -       Xm+1   CTI            .  .    Aij CTI Xn As mentioned in Section 1, in Ref. [7], solving this equation gives optimal results without any need of composing an objective function. Although the pre-processing method, which will be described in the next section can be performed either with or without considering the objective function. However, the results, which will be shown in Section 5, are obtained by applying the method of this paper to the optimal method without an objective function.

i

3. Problem statement

Subject to: tj − ti ≥ CTI

(2)

ti = f (Ipi , Ifi )xi

(3)

f (Ipj , Ifj )xj − f (Ipi , Ifi )xi ≥ CTI

(4)

li ≤ xi ≤ ui ,

(5)

lj ≤ xj ≤ uj

Expression (2) shows the difference between the operation times of the relays i and j must be greater than CTI. To express (2) in term of TDS/TMS (3) is substituted in (2) and (4) is obtained. The constraints constructed from (4) and (5) can be expressed in the form of (6). It should be noted that to ensure the calculated xi in each case does not exceed ui , i.e. the maximum of TDS/TMS, a check is automatically made

As explained in the previous section, the optimization process is feasible if the obtained solutions satisfy all constraints. If any constraint is in conflict, then the optimal problem becomes infeasible. It is necessary to recognize these conflicting constraints before performing the optimal programming process. Fig. 1 illustrates how constraints can have no effect on the optimal solution. In Fig. 1, for a given fault at point F, the original coordination constraint between the primary relay i and the backup relay j, without considering any configuration change, is given by (4). If the constraint variables, i.e. xi and xj , are beyond their maximum and minimum limits, then the solution of problem is impossible.

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Fig. 1. A typical primary and backup relays on the part of a power system.

If the far end circuit breaker (CB), i.e. CB1, is open, then the relays see different network topology and faults current seen by the relays i and j will be different from the original case. The new constraint, which must be added to the constraints set in (7) is given by (8).   f (Ipj , Ifj )xj − f (Ipi , Ifi )xi ≥ CTI

(8)

The normal situation of the network is explained by expression (7). To explain how some constraints make the solution of (7) impossible, consider Fig. 2. In this figure, the constraint line A is described by (4) in the plane xi and xj . The slope and crossing points of the line with xi and xj axes depend on the coefficients of its variables and the CTI. This line divides the xi and xj plane into two sections, upper and lower. The upper section contains any possible optimal settings of a P/B pair satisfying (4), while expression (5) defines the possible solution area in the xi and xj plane. This square area is bounded by the maximum and minimum values of the variables xi and xj , shown in Fig. 2. As can be seen from this figure, although all points on the upper section of the line satisfy the equation, only those that lie within the maximum and minimum of TDS/TMS are chosen. This is the common area between the upper section and PSA. This area is defined as the feasible solution area (FSA). Consider the situation where any of the constraint lines are above the PSA. This will make the optimization impossible. In the traditional optimization coordination process there is no way to detect this and the process does not simply converge [7]. However, by a pre-analysis method non-valid constraints can be detected and removed, then the problem can be solved. The full analysis on how to detect useless constraints and the method to have a feasible solution are given in the next section.

Fig. 3. Four categories of constraint lines.

4. Theoretical approach 4.1. Constraints recognition This section is based on the intersection of constraints lines of the P/B pairs with the PSA. In this phase, the validity of each constraint will be examined. As mentioned in the previous section, the optimal solution for a P/B relays pair exists if the upper section and the PSA have a common area. Based on this, constraints lines shown in Fig. 3 are classified into four categories as follows: -

non-valid; pre-obtained; redundant; valid.

If a constraint line is sited above the PSA and does not have any common area with the PSA, then it is classified as a non-valid constraint. If this constraint is not removed from the constraints set, then the optimal coordination program will be disturbed, i.e. the optimization will become impossible. The criterion to designate a constraint as a non-valid one is stated by (9) because if the right hand side of a constraint for the point A is smaller than the CTI, then it means that there is no intersection between that constraint and the PSA. In this expression the characteristics of point A are Max(xj ) and Min(xi ). fj (Ipj , Ifj ) Max(xj ) − fi (Ipi , Ifi ) Min(xi ) < CTI

If a constraint line crosses the PSA at only one point, i.e. the point A, then the only optimal solutions for the variable xi and xj are Min(xi ) and Max(xj ). This type of constraints is known as the pre-obtained, which can be recognized by (10). fj (Ipj , Ifj ) Max(xj ) − fi (Ipi , Ifi ) Min(xi ) = CTI

Fig. 2. PSA and FSA.

(9)

(10)

The redundant constraint is a constraint in which its line does not intersect with the PSA and is sited below it. In this case, any arbitrary point within the PSA is a possible solution. Therefore, this constraint can be excluded from the constraints set because it does not inhibit any point within PSA.

H.K. Karegar et al. / Electric Power Systems Research 75 (2005) 134–141

Fig. 4. Comparing two valid constraints.

Fig. 5. Crossing point of two valid constraints inside PSA.

Eq. (11) describes how to find a redundant constraint. The point B is specified by (Max(xi ), Min(xj )). fj (Ipj , Ifj ) Min(xj ) − fi (Ipi , Ifi ) Max(xi ) ≥ CTI

(11)

Finally, the valid constraint is a constraint whose line passes through the PSA. The upper-bounded area between the valid line and the PSA is the FSA where the optimal solutions of xi and xj exist. Therefore, the valid constraints are kept in the constraint set while others are removed. As a consequence, the constraint set includes fewer constraints and the solution area is reduced.

xjB =

(CTI + fi (Ipi , Ifi ) Max(xi )) fj (Ipj , Ifj )

In the previous section, the valid constraints are identified. Now, if more than one valid constraint for each relay pair exists, then identification of whether their lines intersect inside the PSA or not, different procedure should be considered. Fig. 4 illustrates how the efficient constraint line is selected. If two valid constraint lines (1 and 2) cross each other outside PSA, then one of them can be removed from the constraints set because the obtained solutions of one of them satisfy the other one. In this figure, FSA1 and FSA2 overlap, therefore, all solutions of the constraint 1 are valid for constraint 2. Hence, constraint 2 can be removed from the constraints set. To find the position of a constraint in comparing with another one, it is sufficient to obtain the crossing points of each valid constraint with the lines Min(xi ) and Max(xi ), as shown in Fig. 4. The points A and B define the intersections of constraint line 1, and D and C for constraint line 2. If the values of the points A and B associated with the xj axis, i.e. xjA and xjB are greater than the values of xj axis of the points D and C, then the constraint line 1 is above the other. Therefore, the constraint line 2 can be removed from the constraints set and the FSA becomes smaller. The values of the xj axis of points A–D are obtained by substituting the value of Min(xi ) and Max(xi ) in a corresponding constraint line. For example, the values of xjA and xjB are given in Eqs. (12) and (13). (CTI + fi (Ipi , Ifi ) Min(xi )) fj (Ipj , Ifj )

(13)

If two valid constraint lines meet each other inside the PSA, then both constraints are kept in the constraints set. This is because none of the FSAs can totally overlap another. This is shown in Fig. 5. The flowchart of the proposed methodology is shown in Fig. 6. The pre-processing block is indicated by blocks (3–6) of Fig. 6. If non-valid constraints are detected in block (3), then some changes on current settings or relays types may be required.

4.2. Efficient constraints selection

xjA =

137

(12) Fig. 6. Flowchart of the optimal program with pre-processing.

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H.K. Karegar et al. / Electric Power Systems Research 75 (2005) 134–141 Table 4 Load data Bus

P (MW)

Q (MVAR)

2 3 4 5

40 60 70 70

20 40 40 50

Table 5 Primary/backup pairs

Fig. 7. The 8-Bus network.

5. Case study The performance of the proposed method was evaluated by its application to an 8-Bus network and a real power system network.

P/B no.

Primary relay

Backup relay

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 7 12 2 8 6 12 13 6 7 14 8 5 2 4 6 10 2 14 11

6 13 13 7 7 14 14 8 5 5 9 9 4 10 3 2 11 1 1 12

5.1. Test system The 8-Bus network shown in Fig. 7 consists of 8 buses, 7 lines, 2 generator-transformers and 14 OC relays. The information data of the network are shown in Tables 1–6. Table 1 Line data Bus–Bus

R (km)

X (km)

Y (km)

Length (km)

1–2 1–3 3–4 4–5 5–6 2–6 1–6

0.0040 0.0057 0.0050 0.0050 0.0045 0.0044 0.0040

0.050 0.0714 0.0563 0.045 0.0409 0.05 0.05

0 0 0 0 0 0 0

100 70 80 100 110 90 100

All OC relays are assumed directional and have standard normally inverse characteristic time curves based on (14). t=

If Ip

0.14 0.02

−1

x

(14)

To evaluate the proposed method, the constraints related to the two types of faults have been taken into account. They are close-in and line-end faults. The close-in fault is assumed to be at the beginning of a transmission line and the line-end Table 6 Relay data

Table 2 Generator data Bus

Sn (MVA)

Vn (kV)

X bus (%)

7 8

150 150

10 10

15 15

Table 3 Transformer data Bus–Bus

Sn (MVA)

Vp (kV)

Vs (kV)

X (%)

1–7 8–6

150 150

10 10

150 150

4 4

Relay number

Type

CT ratio

Plug setting (%In)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

TDS TMS TDS TDS TDS TMS TMS TMS TDS TDS TDS TMS TDS TMS

2000 2500 1500 2000 2500 2500 1500 2000 1500 2000 2000 2000 2000 1500

100 180 120 130 120 140 130 130 130 80 160 60 100 120

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fault is considered to be at the end of the line. For each fault type, three below cases are considered: - Case I: no topological changes are considered. - Case II: the far end line circuit breaker is opened, but the CBs’ of all the lines to the near bus are closed. - Case III: both the far end CB and the lines connected to the near bus except the line on which the backup relay exists is opened. Based on these cases, each P/B pair in Table 5 poses six constraints in the optimal coordination program. Three of them are related to the close-in faults while the others are associated with line-end faults. Therefore, the total number of constraints is 120, out of which 79 were evaluated in this investigation. For the other 41 constraints, the direction of the fault currents in primary or backup relays was not matched with the direction of the relays. Furthermore, 28 constraints, derived from expression (4), must be added to 79, making the total number of the constraints to 107. If the coordination is done without considering the proposed method, the optimization process will not converge because investigation using the new method shows that four of the constraints lines lay outside the PSA. These constraints are related to the primary relay 11 and its corresponding backup relay 12. The constraints are shown in expressions (15–18) and their locations to the corresponding PSA are sketched in Fig. 8. Only the required part of the PSA is shown in this figure. 4.6003x12 − 13.4403x11 ≥ 0.4

(15)

4.0727x12 − 9.8028x11 ≥ 0.4

(16)

3.4079x12 − 6.7121x11 ≥ 0.4

(17)

3.1667x12 − 5.8548x11 ≥ 0.4

(18)

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Table 7 Reduced constraints results of 8-Bus network Total constraints Reduced valid constraints Reduced redundant constraints Total number of reduced constraint in phase-II Percentage of reduced constraints (%)

107 44 20 64 60

However, in Fig. 6, its variation is only shown up to 2 because the rest of PSA has no effect on the process. To illustrate how the proposed method reduces the number of constraints, the plug setting of relay 11 has changed from 60 to 120%. In this case, 64 constraints were reduced in which 20 of them were designated, as redundant constraints and the rest of them, i.e. 44, were valid constraints. Table 7 shows the obtained results. As indicated in this table, 60% of the constraints were removed. 5.2. Real power system The next example is a modified part of Norweb England power system [9]. The network consists of 11 buses, 8 lines, 1 generator, 9 transformers and 39 directional OC relays shown in Fig. 9. The reactance of the generator is equal to 0.1278 p.u. based on Vb = 33 kV and Sb = 100 MVA. Other information including lines and transformers data of the network is given in Table 8. The plug settings (PS) percentage and the current transformer ratios are given in Table 9. All time settings of relays are according to the TDS with the minimum of 0.5 and the maximum of 11.

Four constraints related to the relays 11 and 12 are recognized as non-valid. The expressions (15) and (16) are related to the close-in faults’ situations, and the other inequalities are associated to the line-end faults. In Table 6, the time setting of relay 11 is based on the TDS, and varies from 0.5 to 11.

Fig. 8. Non-valid constraints lines.

Fig. 9. Part of Norweb power system of England.

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Table 8 Transformers and line data of Norweb power system

Table 10 Reduced constraints results of Norweb power system

Item no.

Start bus

End bus

Positive reactance (X1 ), p.u.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6

2 3 4 5 4 10 10 9 9 6 5 7 7 8 8 6 11

0.0143 0.0517 0.0187 0.056 0.0217 0.98 0.98 1 1 0.0269 0.0269 1 1 1 1 0.0322 0.5

Total constraints Reduced valid constraints Reduced redundant constraints Total number of reduced constraint in phase-II Percentage of reduced constraints (%)

287 144 1 145 50

In this case, the constraints related to the close-in and far end faults with the same condition as the previous case in this section, are evaluated. The total numbers of constraints are 287. By applying the new method, 145 constraints were removed from the constraints set. One constraint was recognized as a redundant constraint by the method given in subsection “constraints recognition” and 144 constraints were identified by the method in section “efficient constraints selection”. Therefore, 50% of the constraints were removed from the constraints set. The obtained results are shown in Table 10.

Table 9 Relay data of Norweb power system

6. Conclusion

Relay no.

Bus no.

CT ratio

PS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 2 3 4 5 4 10 10 9 9 6 5 7 7 8 8 6 11 9 8 7 10 11

800 800 800 800 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 400 600 400 400 400 400 600 600 400 400 400 400 600 400 400 400 400 400 400

125 125 125 125 100 80 125 100 80 100 100 100 100 125 125 100 80 80 100 80 80 125 150 100 100 80 125 125 150 100 100 100 125 125 100 150 80 80 125

This paper has shown that by using the pre-processing method the number of constraints can be reduced. This increases the possibility of a solution and decreases the processing time. The optimization methods for coordination of overcurrent relays with objective function have been described in the paper first. Then the new coordination approach based on the constraints without objective function was explained. After that the way of reducing the constraints and its application to two different networks were given. The obtained results from an 8-Bus and an industrial network showed that the number of constraints was reduced by 60 and 50%, respectively. As a result, the processing time and the required memory decreased.

Appendix A. List of symbols

li , ui xi , xj Ifi , Ifj Ipi , Ipj fi , fj ti , tj t If Ip x Wi TDSi TDSmin TDSmax TMS

lower and upper limits exist on the relay TDS/TMS of the primary and backup relays i and j fault current through relays i and j current multiplier setting of relays i and j characteristic curve function of relays i and j operation time of relays i and j operation time fault current through the relay pick-up current TDS/TMS of the relay operation time weighted coefficient relay i time dial setting minimum time dial setting maximum time dial setting time multiplier setting

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Ipi CTI: Ii f A X, B

relay i pick up current value co-ordination time interval relay i measured fault current overcurrent relay i characteristics curve TDS coefficients matrix TDS and constraint vectors

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