7.2.3.1 Obtaining the Optimality Conditions In IPM

The abstract OPF formulation (7.1)–(7.6) can be compactly written as follows:

7.7

7.8

where , , and are twice continuously differentiable functions, is a vector grouping both decision variables and state variables, the vectors , , and having sizes , , and , respectively.

The IPM requires four steps to obtain optimality conditions as follows.

First step: slack variables are added to inequality constraints, transforming them into equality constraints and non-negativity conditions on slacks:

7.9

7.10

where the vectors and are called primal variables.

Second step: the non-negativity conditions are added to the objective function as logarithmic barrier terms, leading to an equality constrained optimization problem:

7.11

7.12

where is a positive scalar called the barrier parameter, whose key role in IPM will be explained later on.

Third step: the equality constrained optimization problem is transformed into an unconstrained one, by building the Lagrangian:

where the vectors of Lagrange multipliers and are called dual variables and the vector gathers all variables, that is, .

Fourth step: the perturbed Karush–Kuhn–Tucker (KKT) first order necessary optimality conditions of the problem are obtained by setting to zero the derivatives of the Lagrangian with respect to all variables [18]:

7.13

where is a diagonal matrix of slack variables, is the gradient of , is the Jacobian of , and is the Jacobian of .

Central to the IPM is the theorem [18], which proves that as tends to zero the solution converges to a local optimum of the original problem (7.7), (7.8). IPM solution algorithms (e.g., pure primal dual [13–15], predictor-corrector [13–15], multiple centrality corrections [16, 17]) exploit this finding, solving the perturbed KKT optimality conditions (7.13) while gradually decreasing to zero as iterations progress. To facilitate understanding, the basic primal dual IPM algorithm is outlined hereafter.

7.2.3.2 The Basic Primal Dual Algorithm

1. 1. Set iteration count . Chose . Initialize such that slack variables and their corresponding dual variables are strictly positive .
2. 2. Solve the linearized KKT conditions (7.13) for the Newton direction :
   7.14
3. where is the second derivatives Hessian matrix.
4. 3.

   Compute the maximum step length along the Newton direction such that :

   7.15

   where is a safety factor²

5. aiming to maintain the strict positiveness of slack variables and their corresponding dual variables.

6. Update the solution:

   

7. 4. Convergence test. A locally optimal solution is found and the optimization process terminates when change in primal feasibility, scaled dual feasibility, scaled complementarity gap and objective function from one iteration to the next meet some tolerances (typically ) [13–15]:
   7.16
   7.17
   7.18
   7.19
8. where is the complementarity gap.
9. 5.

   Update the barrier parameter:

   

   where the centering parameter is usually set as .

   Set and go back to step 2.

   As with any approach, the IPM has also some drawbacks: (i) the heuristic to decrease the barrier parameter, (ii) the positivity condition on slack variables and their dual variables, which may shorten the Newton step length, slowing down convergence, and (iii) lack of efficient warm/hot start, which should generally further speed-up OPF solution methodologies (see Figure 7.3).

7.2.4.1 Description of the Test System

The main characteristics of the abstract OPF formulation are best unveiled for educational purposes on a small system. Figure 7.1 sketches a 5-bus system, inspired from [19], whose bus and line data are provided in Tables 7.1 and 7.2, respectively.

In Table 7.1, all notations are self-explanatory, except of which denotes the ramping ability of a generator (both up and down) as a corrective control action for the assumed time horizon for removing post-contingency violated limits.

Bus	MW	MVar	MW	MVar	pu	pu	pu	MW	MW	MVar	MVar	MW
1	1100	400	−	−	0.954	0.92	1.05	−	−	−	−	−
2	500	200	−	−	0.950	0.92	1.05	−	−	−	−	−
3	−	−	700.0	69.5	1.0	0.92	1.05	150	1500	−500	750	200
4	−	−	600.0	304.9	1.0	0.92	1.05	150	1500	−500	750	200
5	−	−	333.8	146.9	1.0	0.92	1.05	150	1500	−500	750	200

In Table 7.2, , , and are the resistance, the reactance, and the susceptance, respectively, of the line linking bus and bus . Note that the line current limit (in A) corresponds to 1100 MVA limit and hence to 11 pu current limit in 100 MVA base.

	Bus	Bus
Line			kV			S	A
L1	1	2	400	3.2	16	160	1587.7
L2	1	3	400	6.4	32	320	1587.7
L3	1	4	400	3.2	16	160	1587.7
L4	2	5	400	6.4	32	320	1587.7
L5	3	4	400	6.4	32	320	1587.7
L6	4	5	400	6.4	32	320	1587.7

The following generator cost curves, in (US) , are assumed:

,

,

.

For the sake of simplicity, current and voltage limits are considered the same in both pre-contingency and post-contingency states. Furthermore, only N-1 lines outages are considered in the contingencies set, hence . The contingencies are named according to the disconnected line (e.g., L1, L2, L3, L4, L5, and L6).

7.2.4.2 Detailed Formulation of the OPF Problem

The compact OPF formulation (7.1)–(7.6) is specifically detailed hereafter for the 5-bus system, adopting a rectangular coordinates model for complex bus voltages:

where is the voltage magnitude at bus .

The formulation models the most comprehensive operating mode, that is, “also corrective” (see Table 7.3); the simplifying assumptions behind the other modes are explained later on.

The objective (7.1) is economic dispatch (i.e., minimum generation cost):

7.20

The decision variables of this problem are generator active powers and generator voltages, which are limited within some ranges (7.26) and (7.27), respectively.

Equality constraints (7.2) and (7.4) comprise mainly the active and reactive power flow equations at each bus () and for each system state ():

7.21

7.22

where is the set of adjacent buses to bus in state , other notations being self-explanatory.

The reference³ voltage angle is set at bus 5:

Inequality constraints (7.3) and (7.5) include operational limits on (longitudinal) branch current, voltage magnitude, and generator reactive power:

7.23

7.24

7.25

and limits on decision variables:

7.26

7.27

Finally, the coupling constraints (7.6) take on the form:

7.28

7.29

Note that, for the sake of formulation simplicity, the branch current is approximated by the longitudinal current in (7.23).

7.2.4.3 Analysis of Various Operating Modes

Table 7.3 provides, for various operating modes, the values of the objective, the six decision variables, and the critical (i.e., violated, binding, or near-binding) operation constraints in post-contingency states.

	Objective	Decision variables						Critical constraints
Operating	cost
modes		MW	MW	MW	pu	pu	pu	pu	pu	pu	pu
unoptimized	65439	700.0	600.0	333.8	1.000	1.000	1.000	0.88	0.90	12.7	11.8
no contingencies	61042	847.0	637.0	150.0	1.050	1.050	1.047	0.94	−	12.5	10.6
preventive	76709	617.0	319.5	695.5	1.035	1.030	1.050	−	−	11.0	11.5
preventive	76842	693.4	246.4	693.2	1.050	1.029	1.050	−	−	11.0	11.0
also corrective	70550	668.2	442.7	519.1	1.050	1.050	1.050	−	−	10.8	11.0

Assume first that the system operates in the unoptimized mode (this mode does not consider contingencies). One can observe that by simulating each contingency using a classical power flow tool ( is assumed as slack generator), severe violations of voltage and line current limits occur for some contingencies. Specifically, the lowest voltage corresponds to the third contingency (loss of line L3) and is recorded at bus 1 ( pu), while line L3 is overloaded by 1.7 pu and 0.8 pu for contingencies two and four, respectively. Note incidentally that the use of a DC model would miss identifying such low voltage problems.

Assume next that the system operates in the “no contingencies” mode where the pre-contingency state is optimized but contingencies are still disregarded. This mode corresponds to solving the OPF formulation (7.20), (7.21), (7.22), (7.23), (7.24), (7.25), (7.26), (7.27) while keeping only the variables and constraints corresponding to the pre-contingency state (i.e., ). One can remark that the optimization leads to a good reduction of the generation cost (i.e., from 65 439 to 61 042 ) due to a more economical dispatch of active power in absence of network congestions. Indeed, the most expensive generator is dispatched at its technical minimum while the cheapest generator has the largest production increase. This active power re-dispatch has an even stronger negative effect on losses, which slightly increase by 0.2 MW, than the positive impact of reactive power redispatch via an increase in the voltage profile (e.g., almost all voltages at generators reach their upper bound). Also, although not in the optimization objective, constraints violations due to contingencies are alleviated to a certain extent.

7.2.4.4 Iterative OPF Methodology

In order to avoid building unmanageable OPF problems for large real-life applications, practical OPF solution methodologies (see Figure 7.3) include progressively only some contingency constraints in the OPF [2, 10, 20–22]. These constraints should be carefully chosen (e.g., by decreasing order of the overall number of violated constraints). This requires (in its simplest OPF methodology form) iterating between the core optimizer and the security analysis (SA) to check potential constraint violations for contingencies not included in the OPF.

Such a methodology is illustrated in its simplest form, assuming starting “from scratch,” for system operation in the (conservative) preventive mode. The latter corresponds to solving the OPF formulation (7.20), (7.21), (7.22), (7.23), (7.24), (7.25), (7.26), (7.27), (7.28), (7.29) while imposing that decision variables have the same values in both pre-contingency and post-contingency states. This can be conceptually expressed⁴ with two specific settings in the coupling constraints: (i) in (7.28), which means than only generator takes care of post-contingency active power imbalance (this ensures coherency with SA, where is the slack generator), and (ii) in (7.29).

In this methodology one first optimizes the pre-contingency state, which corresponds to the “no contingencies” mode. Then, SA is performed for this state. The results of the SA in terms of post-contingency line currents and bus voltages for the six postulated contingencies are gathered in Table 7.4. Note that only one constraint is violated, specifically line L3 becomes overloaded due to contingency L2. Next, the OPF is solved in preventive mode but including only the harmful contingency L2. The results at this intermediate step of the OPF procedure are presented in Table 7.3 under the label “preventive.” Note that performing SA again on this solution reveals that contingency L4 leads to the overload of line L3. The OPF is run again including both contingencies L2 and L4 and the results are provided in Table 7.3 under the label “preventive.” Because no other contingency leads to constraint violation, this is also the optimal solution of the preventive mode.

Obviously the cost to ensure power system security with respect to the contingencies leads to an increase in the generation cost (e.g., from 61 042 to 76 709 ).

Concerning the physical interpretation of the generation dispatch pattern, comparing the solutions in “no contingencies” and “preventive” modes, one can remark that in order to diminish the loading of line L3 due to contingency L2, a significant amount of active power is redirected via the L4 (see Figure 7.1). As a consequence, generators and reduce their output while increases its production by 545.5 MW. Further comparing the solutions “preventive” and “preventive” modes, one can observe that in order to reduce the loading of line L3 due to any of “conflicting” contingencies L2 and L4, the compromise solution consists in further decreasing the production of generator .

Contingency	Longitudinal current ()						Voltage () at bus
loss of	L1	L2	L3	L4	L5	L6	1	2	3	4	5
line	pu						pu
L1	0.0	5.5	6.2	5.4	2.8	3.5	1.009	0.987	1.05	1.05	1.047
L2	0.9	0.0	12.5	4.7	8.1	2.3	0.985	0.987	1.05	1.05	1.047
L3	1.9	10.6	0.0	7.3	1.6	4.5	0.938	0.953	1.05	1.05	1.047
L4	5.6	6.7	10.6	0.0	1.8	1.6	0.985	0.953	1.05	1.05	1.047
L5	2.3	8.2	6.1	3.2	0.0	1.1	1.005	1.001	1.05	1.05	1.047
L6	3.5	6.2	8.7	2.5	2.1	0.0	1.005	1.001	1.05	1.05	1.047

Finally, applying the same methodology for the “also corrective” mode, one sees that only contingency L2 is binding at the optimum. Due to the larger number of degrees of freedom⁵, this operating mode leads to a reduction in the generation cost compared with the preventive mode (e.g., 70 550 vs. 76 842 ), but naturally the cost still remains larger than in the “no contingencies” mode, the difference reflecting the cost needed to cover the contingencies. The optimization process in this operating mode takes advantage of the post-contingency generators' active power available and may end up with opposite generation re-dispatch, for example:

• contingency L2: MW, MW, MW;
• contingency L4: MW, MW, MW.

7.3.3.1 Detailed Formulation of the RB-OPF Problem

The compact RB-OPF formulation (7.30)–(7.39) is detailed hereafter for the 5-bus system for preventive mode operation.

The objective function (7.30) is the economic dispatch (7.20).

The set of constraints specific to RB-OPF (7.36)–(7.39) take on the form:

7.40

7.41

7.42

7.43

Active and reactive power flow equations (7.21), (7.22), at each bus () and each system state (), are slightly altered to take into account the new load curtailment control variables :

7.44

7.45

In terms of inequality constraints, the RB-OPF includes the set of constraints (7.23)–(7.27) as the deterministic OPF.

Finally, the constraints specific to the preventive mode take on the form:

7.46

7.47

7.48

7.3.3.2 Numerical Results

Two scenarios, called relaxed and constrained, are studied. In both scenarios load shedding is performed under a constant power factor. The relaxed scenario studies the impact of the risk constraint (7.43) only and relaxes constraints (7.41) and (7.42). The constrained scenario considers additionally that load shedding is limited to 10 of the initial power in (7.41) and the maximum allowed overall amount of load shed per contingency is set to 200 MW in (7.42). Unless otherwise specified it is assumed that each contingency has probability .

Figure 7.2 displays (dotted line) the operation cost obtained with the proposed RB-OPF approach, without short-term constraints, for increasing values of system risk between two extremes: (i) conservative N-1 secure operation which corresponds to and hence to conventional OPF, as no load shedding is allowed; and (ii) risky N-0 secure operation, where the risk constraint is practically relaxed (e.g., for roughly ). The figure also provides the corresponding overall amount of load shed corresponding to each risk level. One can observe that as the maximum allowed risk increases, the overall load shed increases and the operation cost decreases. This curve could constitute an important basis for a transmission system operator (TSO) to trade-off the cost savings incurred by choosing a more risky operation.

The potential consequences of the operation risk assumed can be evaluated in terms of post-contingency overall and individual load shedding. These are provided in Table 7.5 for contingency L2, the only contingency that is binding in all cases, as it leads to an active current constraint on line L3. Because the curtailment of load in bus 1 has a larger impact on the loading of critical line L3 (and thereby on the cost reduction), as a consequence, it is fully exploited in all cases, as shown in the table.

	Relaxed			Constrained
	Overall	Bus 1	Bus 2	Overall	Bus 1	Bus 2
0	0.0	0.0	0.0	0.0	0.0	0.0
0.001	10.0	10.0	0.0	10.0	10.0	0.0
0.002	20.0	20.0	0.0	20.0	20.0	0.0
0.005	50.0	50.0	0.0	50.0	50.0	0.0
0.010	100.0	100.0	0.0	100.0	100.0	0.0
0.015	150.0	150.0	0.0	150.0	110.0	40.0
0.016	160.0	160.0	0.0	160.0	110.0	50.0
0.017	169.4	169.4	0.0	160.0	110.0	50.0
0.020	169.4	169.4	0.0	160.0	110.0	50.0

7.4.4.1 Detailed Formulation of the Worst Uncertainty Pattern Computation With Respect to a Contingency

For simplicity, the worst case scenarios are computed solely with respect to overload and further assume that only one line (known beforehand) is overloaded by the worst uncertainty pattern. The reader is referred to [30] for a detailed comprehensive algorithm to reveal the overall set of overloaded lines whatever the uncertainty pattern and the processing of several overloads in the worst case scenario.

The compact formulation (7.56)–(7.60) is detailed hereafter for contingency .

The objective function is the maximization of the current in the line linking buses and in the state post-contingency :

7.66

The decision variables of this optimization problem are the uncertain power injections ( and ), which are allowed to vary in certain ranges:

7.67

7.68

and limits on the active power of the generator responsible for frequency regulation:

7.69

The active power and imposed voltage of other generators are fixed, in both states, at the values provided by the conventional OPF ( and ):

7.70

7.71

The power flow equations at each bus () including uncertain injections for both the base case and the state post-contingency (i.e., ) are:

7.72

7.73

The reference voltage angle constraint is set as:

Operational limits on (longitudinal) branch current, voltage magnitude, and generator reactive power can be written as:

7.74

7.75

7.76

Note that, because the violated line currents in the state post-contingency are assumed known, other branch current constraints for this state are dropped; hence, constraint (7.74) includes only the set of branch current constraints in the base case.

7.4.4.2 Detailed Formulation of the OPF to Check Feasibility in the Presence of Corrective Actions

The compact formulation (7.61)–(7.65) is detailed hereafter for the 5-bus system.

The objective function is the minimization of the amount of generation redispatch, with respect to the values provided by the conventional OPF, to remove violated constraints in the state post-contingency :

7.77

The decision variables of this optimization problem are the generators active power and terminal voltage:

7.78

7.79

The power flow equations at each bus () for the state post-contingency are:

7.80

7.81

The reference voltage angle constraint is set as:

Operational limits on (longitudinal) branch current, voltage magnitude, and generator reactive power can be written as:

7.82

7.83

7.84

Finally, the coupling constraints take on the form:

7.85

7.86

7.4.4.3 Detailed Formulation of the R-OPF Relaxation

A detailed specific formulation of the R-OPF relaxation problem including a finite number of worst case scenarios (), where corresponds to the nominal scenario, is provided hereafter.

The objective function is economic dispatch (i.e., minimum generation cost):

7.87

The decision variables of this problem are the generators active power and terminal voltage, which are limited within some ranges:

7.88

7.89

The active and reactive power flow equations at each bus () and for each uncertainty scenario () and system state () are:

7.90

7.91

The reference voltage angle constraint is:

Operational limits on (longitudinal) branch current, voltage magnitude, and generator reactive power can be written as:

7.92

7.93

7.94

Finally, the coupling constraints (7.55) are:

7.95

7.96

7.4.4.4 Numerical Results

The main characteristics of the abstract OPF-UU formulation are best shown for educational purposes using the 5-bus system in Figure 7.1.

Uncertainty can be assumed to stem from renewable generation present in lower voltage grids, which have not been explicitly modeled, the renewable generation being simply aggregated to load buses 1 and 2. Uncertainty is modeled as variable active and reactive power injections at the two load buses in the range of −5 to +5 of the nominal active/reactive load.

Following the proposed methodology for solving the R-OPF problem, one first computes a reference schedule for the nominal scenario via classical OPF formulation in “also corrective” mode including the six postulated contingencies (see Table 7.6).

The solution provided by this classical OPF computes, via the formulation (7.66), (7.67), (7.68), (7.69), (7.70), (7.71), (7.72), (7.73), (7.74), (7.75), (7.76), the worst uncertainty pattern for each contingency with respect to thermal overload. Note that, due to the network simplicity, only one worst uncertainty pattern, common for all contingencies, is revealed. The worst uncertainty pattern corresponds to a case where the output of renewable generation at buses 1 and 2 is zero. Accordingly, the pre-contingency state corresponding to the worst uncertainty pattern differs, with respect to the classical OPF solution, by additional active/reactive power drawn from the network (i.e., 55 MW/20 MVAr at bus 1 and 25 MW/10 MVAr at bus 2, respectively), the active power balance being ensured by an increase of 83.3 MW at G5, while the generators' reactive power production change so as to maintain their imposed voltages in the prevailing operation conditions.

Further, only two out of six contingencies lead to overloads for their worst uncertainty pattern. Specifically, contingency L2 overloads line L3 with 0.95 pu and contingency L4 also overloads line L3 with 0.9 pu. In both cases, upon solving the formulation (7.77), (7.78), (7.79), (7.80), (7.81), (7.82), (7.83), (7.84), (7.85), (7.86), one observes that the overload cannot be fully removed despite the best use of corrective actions. Specifically, for contingency L2, the best redispatch scheme consists in reducing the active power of G4 and increasing the active power of G5, which creates a counterflow in the overloaded line L3. The latter line current cannot be reduced to more than 11.53 pu. For contingency L4, despite the best redispatch between generators G3 and G4 (where the former increases its production), which creates a counterflow in the overloaded line L3, its latter current is reduced only to 11.12 pu.

Next, in order to compute the intermediate robust optimal solution, one solves the R-OPF formulation (7.87), (7.88), (7.89), (7.90), (7.91), (7.92), (7.93), (7.94), (7.95), (7.96) which includes the reference scenario and the sole worst case scenario found (common to both contingencies).

Table 7.6 provides, for various operating modes, the values of the objective, the six decision variables, and the binding operation constraints. Note that the robust optimal solution encompasses both the active power and the terminal voltage of each generator.

In the second loop of the algorithm, one notices that the worst uncertainty pattern does not change. Because the latter scenario is already covered by the R-OPF solution, this means that the algorithm reaches a fixed point and hence this schedule is the robust optimal solution. The operation cost of robustness exceeds roughly 10% the cost of the most likely scenario (see Table 7.6).

	Objective	Decision variables						Critical constraints
Operating	cost
modes		MW	MW	MW	pu	pu	pu	pu	pu
also corrective	70550	668.2	442.7	519.1	1.050	1.050	1.050	10.8	11.0
robust solution	77235	700.5	228.7	703.2	1.050	1.047	1.050	10.5	10.8

7.5.1.1 Overall OPF Solution Methodology

In order to facilitate the understanding of the major basic concepts of an OPF problem, the latter has been assumed as a clean mathematical programming problem (e.g., NLP) which can be solved directly by an appropriate optimization method (e.g., IPM). However, real-world OPF problems are far more complex. For instance, the major challenges to today's deterministic OPF problems are [2–6]: (i) the huge size, (ii) the management of discrete control variables, and (iii) the incorporation of several problem peculiarities that reflect TSO practical needs (e.g., usable solution in terms of number and sequence of corrective actions, etc.) and operating rules (e.g., conditional corrective actions, constraints and control variables priorities, etc.). Regarding item (iii), it must be emphasized that some practical aspects of the OPF problem may not be formulated analytically in an advantageous form exploitable by an optimizer, requiring one to resort to heuristic techniques during the OPF solution methodologies [2]. However, since benchmarks of practical real-world OPF problems are not available,⁸ the academic realm typically only addresses items (i) and (ii), interpreting, in the most ambitious vision, a real-life OPF problem as a large scale non-convex mixed-integer nonlinear programming (MINLP) problem [22].

The huge size of real-life OPF problems stems from the large number of contingencies that have to be secured (e.g., at least all N-1 events) and the large size of the power system. Because taking a direct approach to such a problem is computationally inefficient if even feasible, methodologies for solving practical OPF problems require decomposition into (and iterations over) a certain number of modules (see Figure 7.3) including in the most generic also corrective mode:

• Core NLP optimizers for the master problem and slave (i.e., contingency) problems: The master problem includes constraints for the base case and some potentially binding contingencies while each slave problem relates to a post-contingency state and checks whether it is feasible thanks to corrective actions.
• Manager of discrete variables module: sets discrete control variables according to some approach (e.g., round-off [36], progressive round-off [22, 37], sensitivity-based [38]) or dedicated techniques for topology control (transmission switching) [39]. Note that this module may interact several times during the solution process with optimizers for both master and slave problems.
• Security analysis module: performs classical post-contingency power flow calculations to check whether constraints violations occur.
• Contingency selection (or filtering) module: selects from the set of contingencies that violate constraints some promising potentially binding contingencies to be added to the master optimizer [20, 21, 40, 41].

An acceptable solution of the OPF problem is found (and iterations stop) when no contingency leads to constraint violation in the presence of corrective actions.

Significant progress in the OPF state-of-the-art has been achieved recently in the following methodologies for problem decomposition: all potentially binding contingencies together (possibly with post-contingency network compression) [20–22]; adaptive Benders decomposition [42] (which improves solution quality but requires larger computational effort compared to the classical Benders decomposition-based OPF methodology [9, 21]); and alternating direction method of multipliers [42]. The core optimizer in these methodologies utilizes powerful general-purpose local solvers (mostly based on the IPM) for NLP (e.g KNITRO and IPOPT). The use of high performance computations renders these computationally demanding OPF methodologies practical for solving realistic large-scale OPF problems (e.g., for large systems up to roughly 10 000 buses and 12 000 contingencies [22]), even starting “from scratch.” Because the core optimizer in these methodologies is the IPM, further improvement can be obtained by decomposing the structure of the linear system of equations to be solved in the interior point method [40]. Anyway, to cope with stringent OPF running time industry needs, powerful computing infrastructure is deployed enabling distributed (parallel) computations and hence fostering OPF decomposition methods.

7.5.1.2 Core Optimizers: Classical Methods Versus Convex Relaxations

In most today OPF commercial software used by the industry, the core optimizer relies on SLP, SQP, Newton's method, and IPM. Although solvers for LP and QP are reliable and fast, SLP and SQP methodologies may provide approximate solutions of the NLP core problem, especially for highly nonlinear OPF problems (e.g., reactive power dispatch). Newton's method and IPM can provide exact (at least) local optimum for the underlying NLP core problem, but while Newton's method may have difficulties in handling inequality constraints, IPM (both general-purpose and produced by academia, tailored specifically for the OPF problem) is not fully reliable [17] and does not warm/hot start well. The warm/hot start feature of an algorithm (via the latest solution at hand or an intermediate approximated solution at a given iteration) is supposed to generally accelerate OPF solution methodologies, especially if successive master problem solutions are close to each other, or their active set of constraints do not change significantly.

A research framework that has attracted much interest recently is devoted to convex relaxations⁹ of the non-convex NLP problems intervening in the core optimizers [44–46]. The major goal of these approaches is to provide, in certain conditions, the global optimum of the non-convex NLP continuous relaxation of the OPF problem. Other notable advantages of these approaches are: provision of a lower bound of the NLP problem (which allows evaluation, provided that the relaxation is tight, of the quality of the local optimizer solution), problem infeasibility guarantee, reliability, and polynomial solution time. Most of these approaches rely on either semi-definite programming (SDP) relaxation [44] or the more general high-order moment-based relaxation [45] (in fact SDP relaxation corresponds to the first order moment).

Various research with convex relaxations report very useful empirical evidence that, generally, the solution provided by local optimizers is the global optimum (or a solution of very high quality) [42, 45, 46]. Next, extensive experiments conducted using a diverse data set point out that [46]: (i) in most cases SDP relaxation does not recover the global solution (since the duality gap is greater than 0%) and (ii) convex relaxations methods are generally slower and scale less well than local optimizers on large OPF problems. Further research with convex relaxations is therefore needed in order to reduce the computational burden, improve the convergence precision of SDP [45], solve real-life OPF problems (which, inter alia, may present negative Lagrange multipliers), and so on.

“It is important to emphasize that, due to today's limitations of the MINLP state-of-the-art, when solving real-life large-scale AC OPF problems under stringent time requirements, the aspect of whether the solution of the OPF continuous relaxation dealt with by the core optimizer is the global optimum or not becomes definitely of secondary importance compared to the primary aim which is to provide, within the allowed computational budget, a (possibly sub-optimal) solution which satisfies the constraints.” [6]. Furthermore, it is worth mentioning that the core algorithm classification (approximate, at least local, or global) relates to the ideal case of a perfect system state forecast, which is less and less the case in today's increasingly uncertain operational environment (especially in day-ahead, which has been the main OPF application timeframe).

Although there is definitely room to improve convex relaxations in the coming years and even make them usable by industry as a complement for classical well-established non-convex local optimizers (in terms of solution quality and certificate of problem infeasibility), it is unlikely that these relaxations will eventually supersede local optimizers.

Finally, in order to efficiently solve increasingly complex practical OPF problems, there will be no single winner in terms of method adopted by the core optimizer, but rather modern optimizers will embed different solution methods (like the powerful solver KNITRO), being able to seamlessly switch between them in order to adapt on-the-fly to the features of the optimization problem at hand.