Chapter 8
Modeling Preventive and Corrective Actions Using Linear Formulation
Tom Van Acker and Dirk Van Hertem
Research Group Electa, Department of Electrical Engineering, University of Leuven, Belgium
8.1 Introduction
Power system reliability management can be defined as a sequence of decisions which are taken under uncertainty in order to meet a certain reliability criterion (see also Chapter 2). A reliability criterion is a principle imposing a basis to determine whether or not the reliability level of a given operating state of the power system is acceptable [1]. This can be expressed as a set of constraints that must be satisfied by the decisions taken by the system operator (SO). An example of such a reliability criterion is the N-1 criterion, which states that the considered power system must be able to withstand any credible single contingency, for example, the loss of a generator, in such a way that a new operational set point can be reached without violating the security constraints of that power system. The SO makes use of flexibility options that are inherent to a power system to reach a new, secure operating point after a contingency. These flexibility options are also referred to as actions and the distinction can be made between preventive and corrective or curative actions.
Preventive actions are those flexibility options preventively activated to adapt the system state. Through preventive actions, a set-point is chosen which is inherently safe and anticipates contingencies by building in reserves. As such, there is no need for corrective operator intervention after a certain contingency occurs.
Corrective actions are those flexibility options available to adapt the system state immediately after a contingency has occurred, sufficiently quickly to avoid cascading of the event. Typically, corrective actions are (partly) automatic actions.
It is the responsibility of the SO to select the actions, both preventive and corrective, that result in optimal reliability, which ideally leads to a minimal global socio-economic cost. In order to obtain such optimality, the system operator needs to implement all considered actions in an Optimal Power Flow (OPF) [2], which is adapted to take the possibility of contingencies into account. This type of OPF is generally referred to as a Security Constrained OPF (SCOPF) [3].
This chapter presents the various reliability management actions and a possible formulation to integrate these actions into a security constrained OPF environment. The different actions are represented using a linearized formulation. Linearization allows to keep the computational costs low while retainting a sufficiently accurate approximation of the behavior of the system [4]. Many other formulations can be found in the literature. The main objective is to show how each of the different control actions can be implemented in a consistent manner and to illustrate the impact of such control actions.
8.2 Security Constrained OPF
The general formulation of a SCOPF is given by (8.1)–(8.5):
In this formulation, 
, 
and 
represent the control variables, state variables and parameters respectively. The index 0 and 
indicate the preventive base case and the corrective contingency cases. Equations (8.2), (8.3) respectively ensure that the power flow equations and the operating limits are respected for the preventive base case while (8.4), (8.5) does this for the considered corrective contingency cases. Equations (8.2) and (8.4) contain the power flow equations and other equality constraints of the OPF. The inequality constraints, (8.3) and (8.5), are for instance used to represent the limits of the power system components. For the purpose of this chapter, a DC SCOPF [5] is considered and all actions are formulated towards their implementation in a DC SCOPF. In a DC SCOPF, the power flow equations are written in their approximated linear version, the DC power flow equations. However, the techniques presented in this chapter can be extended toward other SCOPF formulations, taking into account the assumptions related to the alternative SCOPF formulation.
8.3 Available Control Actions in AC Power Systems
In this section, the most common available actions in an AC power system are listed and described. There are four main type of actions:
- 1. Actions that affect the active power injections in the system: generator redispatch, demand side management and load curtailment [6].
- 2. Topological actions influencing power flows through the system: phase shifting transformer control [7], switching actions [8].
- 3. Reactive power and voltage management: generator voltage set point changes, on-load tap changer transformer control.
- 4. Special protection schemes: Specific actions triggered by specific grid related events.
This chapter focuses on the first two types of actions.
8.3.1 Generator Redispatch
Generator redispatch uses the available upward and downward power reserves in the power system to change the power injection and consequently change the power flows within the grid [9]. Generator redispatch can be used to alleviate congestion or to compensate for a generator outage. The implementation of generator redispatch is very case specific and depends on the market structure as well as the SO. These power reserves are stipulated in contracts or purchased on the market in a liberalized system. The SO can ask to shut down, start up or change the power output of a generator. These actions need to be financially compensated by either the SO or the generator, depending on the type of redispatch. In a vertically integrated system, these reserves are made available to the SO.
8.3.2 Load Shedding and Demand Side Management
As an alternative to generator redispatch, the demand can also be altered in order to keep the power balance within the system. Two methods exist that facilitate a change of the power demand: load shedding and demand side management [10].
The unannounced interruption of supply is called load shedding. The cost of load shedding corresponds to the value of lost load, which is the estimated amount a customer is willing to pay to avoid a disruption in their electricity service. The value of the loss of electricity can be expressed as a customer damage function (CDF), which is a non-linear function of load type, time and duration of the interruption. Demand side management is the modification of consumer demand in exchange for financial compensation, based on a prearranged agreement. Demand side management allows shifting a part of a consumer's energy requirements to another point in time. The cost of demand side management is directly linked to either the electricity price at a specific point in time or a predefined price.
8.3.3 Phase Shifting Transformer
A phase shifting transformer (PST) is able to control the active power flow through a certain transmission line, thus influencing the power flows throughout the entire grid [11].
Figure 8.1 Representation of transmission line with a PST.
A PST can be represented as a reactance 
in series with a phase shift as depicted in Figure 8.1 [11]. The power flow through the line is altered by inserting an angle 
, which changes the phase angle over the line from 
to 
, and consequently the voltage drop over the line impedance 
and the power flow through it. As the flow through the line with the phase shifter is changed, the flows throughout the meshed system will change following Kirchhoff's laws. The angle 
of the phase shifting transformer is controllable within certain limits, 
and 
. A PST operated by the SO can significantly influence the power flows in the system while the cost associated with its control is very low.
8.3.4 Switching Actions
Reconfiguration of the grid topology using switching actions allows altering the power flow in the system, alleviating possible congestion. Switching as a means of control in a power system is described in [12]. This can enable cheaper redispatch or avoid the need for load shedding when balancing the system. Two main methods exist to alter the topology of a transmission system. The first method, in literature often referred to as transmission switching, allows for the connection or disconnection of a branch within a transmission grid. A second method, named bus bar switching, allows adapting the topology of a transmission grid at breaker level, allowing the substation topology to be altered. Bus bar switching also includes transmission switching. When including switching actions in calculations, each switching action adds a possible topology to the solution set, increasing the complexity of the problem.
8.3.5 Reactive Power Management
Reactive power management is possible through five main control actions: generator excitation controls, switching capacitors or inductors, the use of FACTS controllers, the use of under load tap-changing transformers or the adjustment of the set point of inverter connected generation.
In transmission systems, which are largely inductive in nature, the effect of reactive power compensation has a direct effect on the voltage near the compensation. From a reliability point of view, reactive power compensation is crucial to maintain the voltage within acceptable limits and to avoid cascading of events or voltage instability.
8.3.6 Special Protection Schemes
Special protection schemes form a special type of power system protection components. They are designed to detect a particular system condition that is known to cause unusual stress to the power system and to take a specific and predefined action to counteract the observed condition in a controlled manner [13]. Special protection schemes are linked to a particular (set of) event(s) and control actions. These control actions can include line switching, generator tripping, load shedding and fast HVDC redispatch.
8.4 Linear Implementation of Control Actions in a SCOPF Environment
In this section, the linear implementation of different control actions in an SCOPF environment is discussed. As voltage or reactive power cannot be represented by a DC power flow, reactive power management is not covered. Also, special protection schemes are not shown as they form particular solutions for which no general formulation is possible. The other control actions of 8.3 are treated. The presented formulation generates a finite set of linear constraints that produces a search space in the form of a convex polytope. Such a formulation results in a minimal computational burden. Implementation of the preventive and corrective formulation for the same action are fairly similar in notation. In each subsection, the formulation of the preventive implementation of an action is discussed, after which the changes for the corrective implementation are highlighted. The index 0 for the preventive variables introduced in Section 8.2 in (8.1)–(8.3) is omitted as it is redundant.
8.4.1 Generator Redispatch
The implementation of preventive generation redispatch within an OPF requires the addition of a cost term (8.6) to the objective function (8.1) and the addition of constraints (8.7) to (8.9).
The generator redispatch cost for a generator 
is equal to the marginal price of preventive generator redispatch 
multiplied by the redispatched generation 
. The total preventive redispatch cost is the summation of the cost of preventive generation redispatch for all generators 
. The marginal price parameter 
changes depending on the type of generator and on the nature of the redispatch. Piece-wise linear formulation [14] of the objective function allows integration of such a non-constant parameter 
. This formulation is convex as long as the marginal price parameter 
for upward preventive redispatch of a generator 
exceeds that of downward preventive redispatch 
. An example of such a piece-wise linear representation of the preventive redispatch cost is depicted Figure 8.2.
Figure 8.2 Piece-wise linear representation of the preventive generator redispatch cost.
Equation (8.7) ensures that the output limits of each generator 
are respected. Equation (8.8) determines the preventive generator redispatch 
, which is equal to the difference between the generator set point after preventive generator redispatch 
and the set point provided by the market after the closure of the day-ahead market 
in the case of a liberalized market structure. In the case of a vertically integrated system, the 
set point reflects the generator outputs made available to the SO. Equation (8.9) ensures that the preventive generator redispatch 
is within the downward 
and the upwards reserves 
.
The implementation of corrective generation redispatch within an OPF requires the addition of a cost (8.10) to the objective function (8.1) and the addition of constraints (8.11) to (8.13).
The corrective redispatch cost for a generator 
is equal to the marginal price of corrective generator redispatch 
multiplied by the corrective redispatched generation 
, taking into account the probability of the occurrence of a certain contingency 
. If a contingency 
occurs on a certain generator 
, that generator can be excluded from the total cost. This can be done by multiplying the cost by a parameter 
. The constant 
is equal to zero when the generator 
is unavailable during a contingency 
, and one in all other instances of 
. The integration of different marginal price parameters for upward and downward redispatch can be done using piece-wise linear constraints, similar to preventive redispatch. The total cost is the summation of the cost of redispatch for all generators 
and contingencies 
.
Equation (8.11) ensures that the output limitations of the generators are respected. The addition of 
on both the left and the right hand side of the equation ensures that during a contingency involving the generator 
, the power output 
of the generator is zero. Equation (8.12) determines the corrective redispatch 
, which is equal to the differences between the corrective and preventive generator set points. Equation (8.13) ensures that the corrective redispatch 
is within the contracted downwards 
and the upwards reserves 
, which takes the ramp rates of the generators into account. The downward redispatch limit is disabled for a contingent generator using the term 
, where 
is a large number.
8.4.2 Load Shedding and Demand Side Management
In order to implement load shedding as linear constraints, each load can be subdivided into smaller loads, each with a constant load shedding cost. The implementation of preventive load shedding within an OPF requires the addition of a cost term (8.14) to the objective function (8.1) and the addition of constraints (8.15) to (8.16).
The load shedding cost for a load 
is equal the marginal price of load shedding 
multiplied by the amount of shed load 
, and the total cost is the summation of the cost of load shedding for all loads 
.
Equation (8.15) ensures that the active power consumed by the load 
is limited between 
and 
. Equation (8.16) ensures that the shed load 
is equal to the difference between the initial load set point 
and the actual preventive load set point 
. The implementation of corrective load shedding uses a similar formulation. The cost of corrective load shedding (8.17) needs to be calculated for each contingency 
taking into account the probability of each contingency 
.
The constraints are again similar to the preventive actions, but need to be generated for all contingencies 
. Constraint (8.16) needs to be adapted into (8.18) for corrective load shedding in order to reflect the difference between the preventive set point 
and the corrective set point 
instead of the difference between the initial set point and the preventive set point.
Demand side management can be implemented in the same way as load shedding, taking into account that both an increase as a decrease of the load power is possible.
8.4.3 Phase Shifting Transformer
A transmission line equipped with a PST can be modeled as depicted in Figure 8.3. The effect of the phase shifting angle 
can be modeled as two additional active power injections with opposite signs on each side of the PST, represented by the impedance 
. The power injections 
and 
alter the power flow through the transmission line. The sum of the injections of both fictive generators needs to be zero, as the PST only shifts power, but does not generate or consume power. Their injections are also be limited, taking into account the technical limits 
and 
of the PST.
Figure 8.3 Model of a PST.
The implementation of a PST within an OPF requires the addition of a cost term (8.19) to the objective function (8.1) and the addition of constraints (8.20) to (8.22).
The cost for a PST 
is equal to the product of the marginal price of preventive usage of the phase shifting transformer 
, accounting for the wear of the transformer, and the difference of the phase shifting transformer setting 
. The total cost is the summation of the cost of phase shifting for all phase shifting transformers 
.
Equation (8.20) ensures that the fictive power 
injected at node 
is equal to the power 
extracted at node 
(Fig. 8.3). Equation (8.21) limits the fictive power injection 
, and consequently also 
, taking into account the limits on 
. Equation (8.22) determines 
, which is equal to the absolute difference between the initial set point 
and the set point 
of the phase shifting transformer in the preventive grid state. As it is the objective of the optimization to minimize the cost, the absolute value 
can be implemented in a linear way by replacing equations (8.21) and (8.22) with equations (8.23), (8.24) and (8.25). Either 
or 
deviates from zero if the set point of the PST is changed, representing the absolute value 
. The other variable is then equal to zero.
The implementation of corrective actions with a PST is similar to the preventive formulation. The cost for corrective PST usage (8.26) needs to be calculated for each contingency 
, taking into account the probability of each contingency 
.
The constraints are similar to the preventive formulation but need to be generated for all contingencies 
individually. Constraint (8.22) needs to be adapted into (8.27) for the corrective usage of the PST in order to reflect the difference between the preventive set point 
and the corrective set point 
instead of the difference between the initial set point and the preventive set point.
8.4.4 Switching
Breakers can be modeled in an OPF as lossless elements using on/off constraints [15]. An on/off constraint is a constraint that is activated when the corresponding binary variable 
of the considered breaker 
is equal to one. The implementation of preventive switching of a breaker within an OPF requires the addition of a cost (8.28) to the objective function (8.1) and the addition of constraints (8.29) to (8.31).
The cost for operating a breaker 
is equal to the product of the marginal price of preventive usage of the breaker 
, accounting for the wear of the breaker, and the change of the state of the breaker 
. The total cost is the summation of the cost of breaker usage for all breakers 
.
Equation (8.29) ensures that the power flow 
through a breaker 
is equal to zero if the breaker is open (
). If the breaker is closed (
), (8.29) ensures that the active power limits of the breaker are respected. The actual power flow 
through the breaker is determined by the power balance equations included in the standard OPF formulation. Equation (8.30) sets the voltage angles 
of both end nodes of the breaker equal to each other when the breaker 
is closed. If the breaker is open, (8.30) ensures that the voltage angles are independent from each other. Equation (8.31) determines 
which is the absolute difference between the status of the breakers for their initial state 
and their preventive state 
. The absolute value in (8.31) can be linearized using the same technique as described for the phase shifting transformers.
The implementation of corrective switching of a breaker is again similar to the preventive formulation. The cost for breaker usage (8.32) needs to be calculated for each contingency 
taking into account the probability of each contingency 
.
Also the constraints are similar, but need to be generated for all contingencies 
. Constraint (8.31) becomes (8.33) for the corrective usage of the breakers in order to reflect the difference between the preventive set point 
and the corrective set point 
instead of the difference between the initial set point and the preventive set point.
8.5 Case Study of Preventive and Corrective Actions
To show the effect of different preventive and corrective actions, they are implemented in a DC SCOPF and tested on the Roy Billinton Test System (RBTS) (Figure 8.4). The RBTS consists of five substations. A load (Table 8.2) is connected to each substation, except for substation S1. The nodes of the RBTS are interconnected by seven transmission lines (Table 8.3). Eleven generators (Table 8.4) are located in the grid, of which four are connected to substation S1 and seven to substation S2.
Figure 8.4 Roy Billinton Test System [16].
In this section, three case studies are conducted. Each case study is a one-hour Day Ahead Congestion Forecast (DACF) where all possible N-1 contingencies are considered. The Day Ahead Congestion Forecast (DACF) corresponds to the instance in the grid scheduling where the system operator receives all planned market positions from the different stakeholders and from the neighboring systems. The DACF is used to assess whether the expected operating points are within the security boundaries. If not, the system operator will take preventive actions, or prepare corrective actions. In order to keep the results clear, it is assumed that load and renewable generation remain constant during the considered hour. During each case study, the SO has different preventive and corrective actions at his disposal. The available actions during each case study are depicted in Table 8.1.
Table 8.1 Available actions for each case study
| Preventive Actions | Corrective Actions | ||||||
| Redisp. | PST | Switch | Redisp. | Shed | PST | Switch | |
| 1 |  |
– | – |  |
 |
– | – |
| 2 |  |
 |
– |  |
 |
 |
– |
| 3 |  |
– |  |
 |
 |
– |  |
The preliminary set points of all generators (
) are submitted to the SO. Based on that data, a DACF analysis is conducted by the SO in order to determine the preventive and corrective actions it has to take in order to have a stable system at all times. During this analysis, the considered contingencies are failure of one generator or one transmission line, with the probability of failure 
denoted in their respective tables.
Table 8.2 Load data
| Load | Sub. |  |
 |
 |
| [MW] | [MW] | [€/MWh] | ||
| L1 | S2 | 10 | 20 | 10206 |
| L2 | S3 | 75 | 85 | 10206 |
| L3 | S4 | 30 | 40 | 10206 |
| L4 | S5 | 10 | 20 | 10206 |
Table 8.3 Transmission line data
| Line | Length | Admittance |  |
 |
| [km] | [p.u.] | [MW] | [-] | |
| T1, T6 | 48 | 5.555 | 85 | 0.001713 |
| T2 | 160 | 1.666 | 50 | 0.005710 |
| T3 | 128 | 2.083 | 50 | 0.004568 |
| T4, T5, T7 | 32 | 8.333 | 71 | 0.001142 |
Table 8.4 Generator data
8.5.1 Case Study 1: Generator Redispatch and Load Shedding (CS1)
In the first case study, preventive generation redispatch, corrective generation redispatch and corrective load shedding are the only available actions. The preventive redispatch allows adapting the generation set point compared to those set by the market 
. As preventive generation redispatch deviates from the ideal market scheduling, it comes at a cost. It is only activated if the market set points cause congestion in the system or if preventive redispatch is cheaper then corrective redispatch taking into account the probability of the contingencies. The market set points 
provided in Table 8.4 cause congestion on the transmission lines T2 and T3 as a result of an excessive generation output on substation S2. Preventive generation redispatch alleviates this congestion by reducing the generation of generator G5 and G10 of substation S2 and increasing the generation of generators G1 and G2 of substation S1 (Fig. 8.5).
Unbalances in generation and load can be caused by a contingency, either by the failure of a generator or because of congestion caused by an outage of a transmission line. In this first case study, these unbalances can only be corrected by corrective generation redispatch or by shedding load. The corrective generation redispatch and shed load are depicted in Figure 8.6 for contingency C1 to C18.
8.5.2 Case Study 2: Generator Redispatch, Load Shedding and PST (CS2)
The implementation of a PST can help to alleviate congestion during certain contingencies and consequently enable cheaper generation redispatch. Case study 2 introduces a PST P1 to transmission line T4 (Figure 8.4). For the purpose of this case study, the angle limits are set to 
0.4 (rad). The usage of the PST causes wear of the transformer tap changer and consequently comes at a cost. The cost of using the PST is set equal to an arbitrary value of 400 €/(rad). The implementation of the PST P1 reduces congestion and realizes less or cheaper generator redispatch during contingencies C12 and C17 (Figure 8.7).
Figure 8.5 Case Study 1: Preventive generator redispatch.
Figure 8.6 Case Study 1: Corrective generator redispatch. Each square of the first eleven rows depicts the corrective generator set point 
relative to the maximum set point 
of that generator for a contingency. The different contingencies are shown in columns. The corrective generator set point is a combination of the preventive generator set point 
, depicted in gray, adjusted with possible upwards or downwards corrective generation redispatch, respectively depicted in green and orange. Each square in the last row depicts the part of the total load that is shed during a contingency.
Contingencies C12 and C17 exclude either transmission line T1 or T6, which are equivalent, resulting in the same grid state. This causes congestion on both T6/T1 and T2. The total power that can be transported by T6/T1 (85 MW) and T2 (50 MW) is 135 MW, which is insufficient to supply the loads connected to substations S3, S4 and S5 with a total demand of 145 MW. In order to correct this unbalance, 10 MW of load L2 is shed. To supply the remaining load, the flow through the grid needs to be adapted. In case study 1, this was accomplished by downwards redispatch of the generators G2, G3 and G4 of substation S1 and upwards generator redispatch of G5 and G11 of substation S2. The downwards redispatch of G3 and G4 comes at a high cost as these are renewable generators and their downwards redispatch cost significantly exceeds that of a conventional generator.
The redispatch can be avoided by adapting the power flow in the power system using a PST. In case study 2, the set point of the PST for contingencies C12 and C17 is adapted to change the power flow through T4 from 0 to 21 MW. This influences the power flow in the system in such a way that the power flow through T3 is reduced to 20 MW compared to 30 MW in the case without a PST, negating the need for redispatch between generators of substations S1 and S2. Load shedding of 10 MW (L2) still takes place as the lines T6/T1 and T2 are incapable of supplying the entire load of substation S3, S4 and S5. The corresponding downward redispatch is provided by generator G2 of substation S1.
Figure 8.7 Case study 2: The difference in corrective generation redispatch and load shedding between CS1 without PST in the grid and CS2 with a PST connected to transmission line T4 for contingencies C12 (T1) and C17 (T7).
8.5.3 Case Study 3: Generator Redispatch, Load Shedding and Switching (CS3)
An alternative approach to adapting the power flow in a transmission system is changing its topology. Case study 3 introduces transmission line switching, which allows eliminating transmission lines by opening their breakers. This causes wear of the breakers and consequently comes at a cost. For the purpose of this case study, the cost of breaker operation is set at an arbitrary value of 400 €. Elimination of transmission lines T4 and T5 influences the power flow in the system is such a way that the power flow through T3 is reduced to 10 MW (Fig. 8.8). This allows the negative redispatch needed to compensate for the load shedding of 10 MW (L4) to be done by generator G5.
A note must be made of the fact that these switching actions reduce the RBTS system to a radial system, which in reality may not be an acceptable strategy.
Figure 8.8 Case study 3: The difference in corrective generation redispatch and load shedding between CS1 without transmission line switching in the grid and CS3 with transmission line switching for contingencies C12 (T1) and C17 (T7).
8.6 Conclusions
This chapter presents the main reliability control actions. These actions allow the system operator to influence the operating point. They allow the operator to either work with a sufficient margin through preventive control, or to quickly return to the safe operating area using corrective control actions. In order to calculate the optimal use of these control actions, an optimal power flow program can be used. Each of the control actions add a number of constraints to the optimization program, and alter the objective function. In this chapter, the linear implementation of control actions in an OPF is given. An extensive case study is used to show how different actions influence both system reliability and system cost on a small test system.
References
- 1 Kundur, P., Paserba, J., Ajjarapu, V., Andersson, G., Bose, A., Van Cutsem, T., Canizares, C., Hatziargyriou, N., Hill, D., Vittal, V., Stankovic, A., and Taylor, C. (2004) Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions. IEEE Transactions on Power Systems, 19 (3), 1387–1401, doi: 10.1109/TPWRS.2004.825981.
- 2 Wood, A.J. and Wollenberg, B.F. (1996) Power generation, operation and control, 2nd Edition, John Wiley & Sons, Ltd.
- 3 Capitanescu, F., Martinez Ramos, J.L., Panciatici, P., Kirschen, D., Marano Marcolini, A., Platbrood, L., and Wehenkel, L. (2011) State-of-the-art, challenges, and future trends in security constrained optimal power flow. Electric Power Systems Research, 81 (8), 1731–1741, doi: 10.1016/j.epsr.2011.04.003.
- 4 Conejo, A.J., Castillo, E., Minguez, R., and Garcia-Bertrand, R. (2006) Decomposition techniques in mathematical programming: engineering and science applications, Springer Science & Business Media.
- 5 Stott, B., Jardim, J., and Alsaç, O. (2009) Dc power flow revisited. IEEE Transactions on Power Systems, 24 (3),1290–1300.
- 6 Christie, R.D., Wollenberg, B.F., and Wangensteen, I. (2000) Transmission management in the deregulated environment. Proceedings of the IEEE, 88 (2), 170–195, doi: 10.1109/5.823997.
- 7 Han, Z.X. (1982) Phase shifter and power flow control. IEEE Transactions on Power Apparatus and Systems, PAS-101 (10), 3790–3795, doi: 10.1109/TPAS.1982.317064.
- 8 Hedman, K.W., Neill, R.P.O., Fisher, E.B., and Oren, S.S. (2009) Optimal transmission switching with contingency analysis. IEEE Transactions on power systems, 24 (3), 1577–1586.
- 9 Van den Bergh, K., Couckuyt, D., Delarue, E., and D'haeseleer, W. (2015) Redispatching in an interconnected electricity system with high renewables penetration. Electric Power Systems Research, 127 (January), 64–72, doi: 10.1016/j.epsr.2015.05.022.
- 10 Capozza, A., D'Adamo, C., Mauri, G., and Pievatolo, A. (2005) Load shedding and demand side management enhancements to improve the security of a national electrical system. 2005 IEEE Russia Power Tech, PowerTech, pp. 1–7, doi: 10.1109/PTC.2005.4524568.
- 11 Verboomen, J., Van Hertem, D., Schavemaker, P., Kling, W., and Belmans, R. (2005) Phase shifting transformers: principles and applications. 2005 International Conference on Future Power Systems, (February 2016), p. 6, doi: 10.1109/FPS.2005.204302.
- 12 Glavitsch, H. (1985) Switching as means of control in the power system. International Journal of Electrical Power & Energy Systems, 7 (2), 92–100, doi: 10.1016/0142-0615(85)90014-6.
- 13 Anderson, P.M. and LeReverend, B.K. (1996) Industry experience with special protection schemes. IEEE Transactions on Power Systems, 11 (3), 1166–1179, doi: 10.1109/59.535588.
- 14 Somuah, C.B. and Khunaizi, N. (1990) Application of linear programming redispatch technique to dynamic generation allocation. IEEE Transactions on Power Systems, 5 (1), 20–26, doi: 10.1109/59.49081.
- 15 Henneaux, P. and Kirschen, D.S. (2016) Probabilistic security analysis of optimal transmission switching. IEEE Transactions on Power Systems, 31 (1), 508–517.
- 16 Billinton, R. and Allan, R.N. (1996) Reliability evaluation of power systems, Plenum Press.