WO2011112365A2 - Efficient security-constrained optimal power flow (sc opf) analysis using convexification of continuos variable constraints within a bi-level decomposition scheme - Google Patents

Abstract

A scheme is presented that utilizes the convexification of continuous variables in the modeling of the Security-Constrained Optimal Power Flow (SC-OPF) problem to create discrete variables that allow for column-generation tools to be used in the solution of the SC OPF problem. One such relation is the voltage law relation for AC branch control flows, and the convexification utilizes a complex plane representation to create a convex solution of discrete values that can be used to perform a feasibility analysis of the various contingency cases. As a result of the convexification, analysis tools, such as column-generation decomposition associated with discrete variables are available for use in solving the SC OPF problem and increase the efficiency and accuracy of the solution.

Description

EFFICIENT SECURITY-CONSTRAINED OPTIMAL POWER FLOW (SC OPF) ANALYSIS USING CO VEXIFICATION OF CONTINUOS VARIABLE CONSTRAINTS WITHIN A BI-LEVEL DECOMPOSITION SCHEME

Cross-Reference to Related Applications

This application claims the benefit of US Provisional Application No. 61/311,803, entitled "Bi-Level Decomposition Scheme for Security-Constrained Optimal Power Flow (SCOPF) Problem Supporting Both Preventive and Corrective Control Modes", and US Provisional Application No. 61/311 ,804, entitled "Discrete Optimal Power Flow (DOPF) Model Formulation and Solution Method Involving Mixed AC -DC Power Flow

Constraints", both filed March 9, 2010 and herein incorporated by reference.

Technical Field

The present invention relates to the creation of a scheme that utilizes the convexification of continuous variables in the modeling of the Security-Constrained Optimal Power Flow (SC-OPF) problem to allow for column-generation tools to be used in the solution of the SC OPF problem. Background of the Invention

The global electric industry is facing a number of challenges: an aging

infrastructure, growing demand, and rapidly changing markets, all of which threaten to reduce the reliability of the electricity supply. Deregulation of the electricity supply industry continues, and the drive to increase efficiencies in power systems have been particularly relevant in the attempt to develop new processes for intelligent observation and management of the grid.

Increasing demand due to economic and demographic variations, without additional generation investments, has led transmission and distribution systems worldwide to their limits of reliable operation. According to the North American Electric Reliability Council (NERC), transmission congestion is expected to continue over the next decade. Growth in demand and the increasing number of energy transactions continue to outstrip the proposed expansion of transmission systems.

The primary objective of operation and security management in the electric industry is to maximize the infrastructure use while concurrently reducing the risk of system instability and blackouts. The optimal power flow (OPF) problem is utilized to minimize a certain objective over certain power network variables under a set of given constraints. The variables may include real and reactive power outputs, bus voltages and angles. The objective may be the minimization of generation cost or maximization of user utilities. The constraints may be bounds on voltages or power levels, or that the line loading not exceed thermal or stability limits. OPF algorithms based on successive linearization techniques are widely used to solve different problems in power system planning, operation and control.

Security-constrained (SC) OPF problems are a special class of OPF problems that consider constraints derived from a normal system state (hereinafter referred to as the "base case"). Contingencies are abnormal events that adversely impact the normal operation of the electric power network. These abnormal events may include events such as a line that is out of service due to a lightning hit, a tree hitting a power line, a bus taken out of service, or the like.

Contingency analysis is performed in an effort to identify events that may cause cascading outages or unrecoverable situations and determine the necessary actions required to prevent these situations from occurring in the first place. The contingencies themselves are defined by the user and each contingency may involve one or more branches in the power network, bus or equipment outages, or a shift or change in generation and/or load. In practice, there may be thousands of contingency events defined for a particular system.

The definition of system security in actual power system operation varies throughout the power industry. Different operation policies and rules are applied to define security requirements. A widely accepted system security is referred to as "N-1 security", where the main objective is to keep the system in a normal state during both the normal system operation (the above-described "base case") and in the presence of "one" major contingency in a predefined list of contingencies. In order to satisfy the N-1 security criteria, the power system should be secure (i.e., no violations) after the occurrence of any single contingency in the system. This is also referred to as the implementation of "preventive" control actions in the system (i.e., "preventive mode" of SC OPF) as will be discussed in detail below.

SC OPF in preventive mode is conservative, because it does not consider the system's post-contingency (i.e., corrective) control capabilities. By introducing corrective rescheduling to the N-1 security concept, three different modes of control adjustments that affect the SC OPF are defined: (1) preventive mode; (2) corrective mode; and (3) preventive/corrective mode.

In the preventive mode, all control variables are optimized such that no post-contingency adjustments are necessary in order to avoid violation of the base case and post- contingency constraints. This is the most secure solution mode, since no operator intervention is required following an anticipated contingency. The consequences of such a solution are a higher pre-contingency objective function, and a generally more difficult problem to solve. In some cases the preventive mode solution may not even exist, especially for more severe contingencies.

In the corrective mode, the control variables are permitted to adjust after the contingency occurs. This is a less secure mode of operation since operator action is required soon after the occurrence of a contingency to reach an acceptable operating state. Such a problem is generally easier to solve, since there are more degrees of freedom in the control adjustments. The corrective mode is solved as a sequence of independent optimization problems, one per contingency.

Execution of the SC OPF function in either mode is time consuming. Historically, the performance problems are dealt with by introducing in the model a relatively small number of critical contingencies. This approximation presents an unresolved modeling problem for all known SC OPF formulations. That is, by fixing just a small subset of most critical contingencies, there is no guarantee that other contingencies labeled as non-critical will not become critical after a new SC OPF solution. The only practical solution to this problem is to directly involve a large number of critical contingencies in the SC OPF formulation, resulting in a very large optimization problem to be solved.

Adding to the difficulty in arriving at a SC OPF solution is that a number of the variables in the problem are continuous quantities with non-convex solutions.

Thus, a need remains for addressing the enormity of the overall SC-OPF problem that allows for a reasonable solution to be created in an efficient and expedient manner.

Summary of the Invention

The needs remaining in the art are addressed by the present invention, which relates to the creation of a scheme that utilizes the convexification of continuous variables in the modeling of the Security-Constrained Optimal Power Flow (SC-OPF) problem to create discrete variables that allow for column-generation tools to be used in the solution of the SC OPF problem.

In accordance with the present invention, non-convex relations are submitted to a convexification process that creates a discrete solution. One such relation is the voltage law relation for AC branch control flows, and utilizes a complex plane representation to create a convex solution of discrete values that can be used to perform a feasibility analysis of the various contingency cases. In one embodiment, the complex plane representation results in the formation of a ring sector characterized by a plurality of rectangles (or other convex shapes) that allows for the variables to be defined in a manner that allows for discretization of the complete SC OPF problem.

As a result of the convexification, analysis tools associated with discrete variables are available for use in solving the SC OPF problem. In accordance with the present invention, a bi-level inner/outer loop process is used to look for infeasible contingencies in the inner loop, add their signatures to the overall problem and then re-solve the OPF in the outer loop. A selected column-generation technique can be used in either the inner loop or outer loop (or both) to remove the need to process irrelevant constraints and provide an efficient overall solution to the SC OPF analysis.

Both preventive and corrective mode solutions are then modeled in an inner loop/outer loop arrangement by adding various contingencies in a controlled manner.

Inasmuch as all of the variables are discrete, column-generation techniques can be used in both the inner loop and the output loop to arrive at a feasible solution in an expedient manner.

While the "voltage law" relation is first subjected to the convexification process, other non-convex variables in the general OPF problem may also be subjected to this process, if desired, to create a more robust solution.

Other and further aspects and features of the present invention will become apparent during the course of the following discussion and by reference to the

accompanying drawings.

Brief Description of the Drawings

Referring now to the drawings, FIG. 1 is a network graph of an electric utility arrangement that can be analyzed used the process of the present invention;

FIG. 2 is a complex plan representation of the feasibility set for both the cross- and self-admittance terms of the voltage law;

FIGs. 3 - 5 illustrate an exemplary process of applying convexification to the ring sector representation of the voltage law feasibility sets in FIG. 2;

FIGs. 6 and 7 illustrates alternative, sub-optimal convexification arrangements for the ring sector shown in FIG. 2;

FIG. 8 is a flowchart associated with the convexification process of the present invention;

FIG. 9 is a flowchart of the bi-level, dual-loop process of solving the SC OPF problem; and

FIG. 10 is a block diagram illustrating the applicability of the convexification process to the SC OP problem.

Detailed Description

While the general subject of Security-Constrained Optimal Power Flow (SC OPF) has been the subject of study for decades, a continuing stumbling block has been the need to evaluate the solution as a mixed integer problem, requiring the use of different types of analyses at different times, including viewing the non-convex variables (such as "voltage law") as a quadratic. By virtue of performing a convexification on these variables in accordance with the present invention, it is now possible to reformulate the OPF problem with discrete parameters, opening up the possibility to utilize solution approaches heretofore unavailable. For example and as discussed in detail below, various column- generation techniques (Benders cuts, Dantzig- Wolfe) can be used in a bi-level loop analysis system to arrive at a reasonable solution.

As mentioned above, the base case OPF problem seeks to produce an optimum power flow solution that balances generation and consumption for all buses within a power distribution system - across all connected branches - such that the overall system state (as measured by the underlying physical parameters such as voltages and phase angles) is feasible and safe. The base case model utilizes the following set of parameters: (1) AC and DC energy dispatch control (i.e., bus generation flow, bus load flow, branch power flow); (2) AC bus voltage and phase angle regulation; (3) AC transformer and phasor tap and shunt switch selection; (4) AC/DC converter control.

In studying the electric power network, the following components and their control states are considered, where these components are shown in the network graph of FIG. 1. An AC bus 10 represents a node in the network graph, where the bus is defined by the following controls and states:

• Load 12: utilizes/consumes both active power (measured in MW) and reactive power (measured in MVar)

• Generator 14: creates/generates both active power (MW) and reactive power

(MVar)

• Voltage and phase angle of bus 10: restricted to a certain range around a nominal value

• Shunt capacitor 16: (optional) can be fixed or "switched", has a positive value and moderates the active and reactive "net" power produced at the bus

• Shunt inductor 18: (also optional) can be fixed or "switched", has a negative value and moderates the active and reactive "net" power produced at the bus

In studying AC bus 10 in terms of developing an OPF solution, the main parameters of concern are the bounds for both load 12 and generator 14, as well as the bounds for the voltage and phase angle.

An AC branch (or line) 20, as shown in FIG. 1, represents an arc in the network graph and transmits both active and reactive power. AC branch 20 is best represented by the following controls and states:

• Power flow: defined as active and reactive power injected into branch 20 by both of its connected AC buses 10 (located above and below AC branch 20 on the network graph of FIG. 1)

• Adjusted impedance: (optional) and represents the branch impedance adjusted by an attached transformer 22

In studying AC branch 20 in terms of developing an OPF solution, the main parameters of concern are the admittance matrix (as discussed below and based on the branch impedance and branch fixed shut admittance) and the flow capacity ratings (that is, the minimum and maximum total flow as constrained by thermal ratings).

AC transformer 22 is an extra add-on to AC branch 20 that may either transform the voltages at the connected buses (traditional "transformer" mode), or shift the difference between the phase angles of the connected buses ("phasor" mode). AC transform 22 itself is modeled by the following controls and states:

• Tap choice

• Tap turns ratio (transformer mode)

· Tap phase shift (phasor mode)

DC bus 24 represents a node in the network graph and is represented by "DC voltage" and "current injection" controls, where the main parameters for DC bus 24 are the bounds for the voltage and current. DC branch (or line) 26 is represented as an arc in the network graph of FIG. 1 and transmits a constant current in the network. DC branch 26 is modeled by its DC current, where the main DC bus parameter is its resistance.

A converter is shown as an arc in the network graph of FIG. 1 and is used to connect AC and DC buses. Converters can be of two types: a rectifier 28 (AC to DC converter) and an inverter 30 (DC to AC converter). For the purposes of OPF, a converter is defined by the following controls and states:

· Firing and overlap angles, AC side voltage (a transformer being associated with a converter)

• Power factor, active power injection

The main converter parameters used in OPF analysis are: bounds for all converter controls and states, as well as the commutating impedance. A voltage source converter 32 is a special type of converter that does not involve firing/overlap angles and directly controls voltage, power factor and active power injection. FACTS (flexible AC transmission system) devices 34 are not modeled as distinct components for the OPF analysis, but rather as tight bounds on certain controls/states of other network components that are regulated by FACTS. Finally, the OPF is generally performed for an "area", which is defined as a grouping of buses used for defining area power exchange and inter- area power transfer constraints. An "area" generally involves several interfaces that correspond to a group of branches that connect one area to another.

With these definitions in place, it is possible to first understand the base case OPF and then address the advances associated with the present invention in terms of convexification of non-convex variables, allowing for the utilization of column-generation decomposition techniques to simplify the iterations performed by the inner and outer loops in the process of addressing contingencies to the base case. The power network is modeled via standard graph N = { V(N), E(N) } where V(N) is the set of buses and E(N) is the set of branches. Within V(N) the following subsets are distinguished: AC bus set V(AC) and DC bus set V(DC). Within E(N) , the branches subsets include the set of AC branches E(AC), the set of DC branches E(DC) and the set of converters V(C). Inter-area interface branch groupings E(I) are separately defined.

In performing the analysis, it is understood that every control and state variable has a "box" constraint (i.e., lower and upper bound constraint values), where not all of these box constraints are specifically mentioned in the following analysis. An example of box constraint is associated with the active power generation control variable P_ig , which is subject to "generation" bound box constraints:

P ig mm < P ig < P ig max

For the purposes of the present discussion, a "choice" variable is defined as a component where alternative values are available, for example, with a transformed/phasor top or with each shunt switch. It is to be noted that that all "choice" variables, denoted w_k in the inventive model, are not only are subject to be within the [0; 1] box, but also should satisfy the Exclusive Choice constraint:

The set of AC bus constraints consists of the fundamental set of network constraints called the "power flow conservation" set (PFC) and are defined as follows for both the active power P and reactive power Q:

P i.g - P il + P i., shunt = V P e + V p c

eeS⁺ (i) ceS⁺ (i)

Qig ^~ Qil + Qi, shunt ⁼ c

eeS⁺ (i) + Q

ce ΣS⁺ (i)

The left-hand side of these constraints reflect the net power injection generation; the right- hand side reflects the net power flow (coming from both V(AC) and V(C)). The shunt term in the PFC constraint is given by: p i., sh,unt = V / > w i.k . p ik,

k=l

Qi,shunt ^ ' ^ ik Qik

k=l

The set of AC branch constraints includes: (1) the "voltage law" (VL), defined as follows:

I C' ^■ ' ·; (G_v * cos(<¾ ) + 5, * sin(^ ))

Qi_j = ^~V_aB_a + V_tj {G_tj * sin(^ ) - cos(^ ))

It is to be noted that the variables V_tj are given by the voltage tensor product represented by bilinear constraints: v ^Y ij. = v ^Y i.^■ V ^Y i ' where it is to be noted that the variables 9_tj are given by the angle difference relation represented by the following relation:

where∑w _kA _k is the phase shift due to a phasor (if any).

k=l

The branch flow cone (BFC) constraint consists of the following second-order (Lorentz) cone constraint:

p ² ₊ n ² < and the branch flow bilinear (BFB) constraints are defined as follows:

P: Q > /·^" : .

The BFC constraints are convex and easy to handle by convex solvers, while BFB constraints are non-convex and difficult for any optimization solver, where this non- convex problem will be addressed below.

Note also that for the sake of convenience, every physical undirected AC branch in the following model has two logical directed AC branches: and These branches are not coupled, but their parameters are calculated based on the same physical branch characteristics (e.g. impedance).

The set of transformer constraints includes: (1) joint mixed-integer-linear tap voltage box constraints:

(2) link with bus voltage:

(3) link with branch (tapped) voltage:

k=l

Also note that in the model there are two logical transformers corresponding to one physical transformer: one for (I ) line and the other for (j,i) line. These two logical transformers not only have their parameters generated by the same underlying set of taps of the physical transformer, but they also are explicitly coupled through an artificial component referred to as "coupling". Coupling holds any pair of component IDs and couples all of their choice variables one-to-one with each other such as: for (t, , t₂ )≡ coupling : w_≠ = w_%i

Phasors are similarly coupled. As noted above, the tap choice variables w_tk are restricted to be in [0; 1] box. These tap choice variables can also be restricted to be binary by either of the following two methods:

a) Mixed-Integer integrability constraint:

" {0;i} b) Non-linear integrability constraint:

The first of two methods implies using a "mixed integer programming" (MIP) solver and is more expensive computationally while providing more responsive transformer control. For example, the solver adjusts transfer taps rather than bus voltages if the transformer control is responsive (and vice versa). The second method introduces another set of non-linear non-convex constraints that can be solved by NLP or SLP methods and can be less expensive (than MIP) computationally, but provides less responsive transformer control. Impedance adjustment constraints produce a choice of the admittance matrix (G, B) that depends on transformer/phasor tap position. For each

"impedance adjustment" record, a choice variable is introduced that is one-to-one (i.e. does not require extra integrability constraints) linked to transformer/phasor tap choice through the following linear constraints:

W_am > W_tk - /(t_am < t_tk≤ tam )

The chosen admittance matrix is then given by the sum:

(G, B) =∑w_ak - (G, B)_ak

k=l

The set of "area" constraints involve aggregation of branch power flows into area interface flows:

Q_f = ∑G.

ee/(/)

and then adding BFC and/or BFB flow capacity constraints for the resulting interface flows.

The set of DC bus constraints consists of the fundamental set of network constraints defined as the "current flow conservation" (CFC) set: = ∑/. - ∑/.

It is to be noted that /. reflects external current injection from AC/DC converters.

Therefore, if there are no converters are attached to a particular DC node /, the box constraint for /. is simply: 0≤ ≤ 0 . The set of DC branch constraints consists of the DC version of the Voltage Law as described above, in this case: '^· '^· ·

where the set of converter constraints consists of: (1) the AC/DC Voltage relation:

V_DC = - (cos(a) + cos( ))

V_DC = V_AC ^■ cos(a + μ) - Ι_ΙΧ · R_com

(2) the power factor relation: cos(i?) =— (cos(a) + cos( ))

2 , and

Q = P - tan(R) (3) the active power injection from the AC side:

P ¹ = V ' DC I ¹ DC

It is to be noted that the sign of active power injection for a rectifier is opposite of the sign of active power injection for the inverter.

With all of these definitional relations, the base case OPF objective function is the generic convex quadratic cost of all buses' generation and load and can be expressed as follows: min ∑ c- ( )² + C_a' ( )² + (a_g J + C_u ^r (β, )² + 2¾ + B_a'P_a + B_i ^r _gQ_ig + B_a'Q_a ieV(AC) '- where each active/reactive load or generation term reflects a linear transformation (i.e. centering and scaling with respect to a given reference value) of the corresponding model variables.

As mentioned above, one problem with arriving at solutions for the base case OPF is that certain relations (such as, for example, the "voltage law") are non-convex, in the form of a quadratic and are difficult to solve, particularly when involved with the multiple variables in play for the OPF problem. Thus, in accordance with the present invention, a methodology is presented to perform a discretization of these non-convex relations. In particular, a convexification of the original non-convex relation has been created that yields discrete variable results. These discrete values, when used with the remaining portions of the model, then enable the use of different tools to address limitations in the remaining problem and arrive at a credible, useful solution.

The discretization methodology of the present invention will be explained below as applied to the "voltage law" relation associated with the AC branch constraints, inasmuch as this is a significant aspect of the overall SC OPF analysis. In particular, the process begins with introducing a complex branch flow (Fy) including only two terms, defined as follows:

F.. = P.. + i ^■ Q..

Since the original voltage law included only two terms (a first term defining the self-admittance that depends on a single bus voltage and a second term defining the cross-admittance term that depends on phase angles and the cross-product of voltages), the relation for the complex branch flow can also be represented as:

77 _ p - self _,_ p - cross

ij ^~ ij ij

The Euler representation of complex numbers in polar coordinates for both self- and cross- admittance terms is then used to arrive at the following:

rr¹ cxp(/- : ^;; · / · ( r^: ))

The modules, r™^ss and r. ¹ , are driven by both voltages and admittances: r = ln^. - lnt_{1 +} lnr₇. - lnt₂ + lni?,

rf^f = In V_t - In t_x + In V_t - In t . + In R.. where R_Y =

The phase shifts, - ^mss and - ^elf , are driven by admittances only: a^™ = arcsin a-,- = arcsin

The following relations switch to using the logs of bus voltages (equivalent to measure them in logarithmic scale). This transformation does not introduce new non-linearity into the model since the voltages only have box constraints. Note that the voltage tensor product is now a linear sum in the expression for the modules r?^oss and r^^lf .

Therefore, in accordance with the present invention, it has been found that this logarithmic transformation introduces a straightforward way to preserve the "rank 1" property of the voltage tensor product (where this property is one of the above-mentioned drawbacks of the SLP method). The chosen transformer taps (in log scale) also appear linearly in the new expression for the modules. As a result, the methodology of the present invention creates a new way to introduce binary transformer tap choice in a way similar to the switching of shunts and phasor shifts. As a result, this type of binary transformer tap choice eliminates the need for the inflated set of transformer constraints in the OPF model formulation and MIP enforcement of the binary choice performs much faster for this reformulation.

With this understanding, it is now possible to view the feasibility sets of F_y ^elf and f^^ross on the complex plane. As mentioned above, the log- voltages and the phase angles are only box-constrained. Therefore, for a fixed transformer tap choice, the feasibility set for either of the two modules is itself a box (i.e., a closed interval). The same constraint holds true for the phase angle difference 6>. - 0. - a^^oss .

Therefore, since the phase angle is fixed to be equal to { - 0 y ^lf ) for the self-admittance term F_y ^elf , the feasibility set can be presented as a line, as shown on the plot of FIG. 2. The feasibility set for the cross-admittance term F^^ross is defined as a sector of a ring, as shown in

FIG. 2, defined by box intervals for its radius and angle. Different transformer tap choices will result in the selection of different sections, weighing them with exclusive binary choice variables.

In light of this discussion and by reference to FIG. 2, it is concluded that the feasibility set of F_y and - as a result of the power flow conservation constraint - the net power injection P_ig - Pa + (Qi_g - Qii) become a weighted sum of different ring sectors on the complex plane. A ring sector itself is not a complex set. Therefore, a sum of ring sectors is also not complex. Moreover, such a sum may take various geometric forms on the complex plane, consisting of pieces of ring sectors; this is a very unsystematic and highly non-convex result and is not conducive for use in the OPF problem in hand.

Continuing, however, it is possible to convexify a ring sector if its angle span is small. This process is illustrated in FIG. 3, defining a "small span" of the sector of angle γ, starting with a first edge of the sector. A midpoint M of this small span is determined. Midpoint M is then covered by a rectangle R (shaded region) formed by the ring sector's tangent lines T at midpoint M. The process moves to the next "small span" , as shown in FIG. 4, and the next midpoint M is selected and the associated rectangle created. For the exemplary beginning ring sector as shown in FIG. 2, a series of three "convexification" routines is required, resulting in the arrangement as shown in FIG. 5. As a result of this process, the complete ring sector is now defined by a combination of several disjoint small sectors, with each small sector covered by a separate rectangle. This result in shown in FIG. 5 as a set of three rectangles Ri, R₂ and R₃ covering the original ring sector. The rectangles are themselves convex representations and therefore, convexify the ring sector with high accuracy, as a function of the combination of properly centered, scaled and oriented rectangles. The midpoints Mi, M₂ and M₃ become anchor points for the following analysis.

The degree of coverage accuracy in this convexification process can be controlled by increasing the number of rectangles (smaller span angles) and by careful location of their placement. Theoretically speaking, a ring sector can be convexified by any other geometric shape (since it exhibits the necessary convex property), but it has been found that the "natural" coverage of other shapes (even triangles) is not as good as rectangles. However, it can also be shown that merely increasing the number of rectangles does not often lead to higher accuracy. FIGs. 6 and 7 show alternative placement and locations of rectangles along the same ring sector; clearly the arrangement of FIG. 5 is preferable. Indeed, by virtue of the proper placement of just a few anchor points (midpoints Mi, M₂ and M₃ in FIG. 5), an accurate coverage of the ring sector is provided.

As will be discussed in detail below, it is also possible that the complete coverage may not be necessary at the beginning of the OPF solution process. Additionally, it will remain possible to add in other anchor points as the solution progresses in real-time if the individual performing the analysis determines that coverage in an additional space by another rectangle is useful. Indeed, this concept is the core idea of the inventive mixed-integer programming (MIP) heuristic for placing new anchor points.

With this representation in mind, if each sector is covered by rectangles, and if there is a selection process for the rectangles within a sector such that only one of the rectangles is chosen at a time, then the feasibility set of complex branch flow Fy and net power injection P_ig - Pa + (Qi_g - Qii) (by virtue of the power flow conservation constraint) becomes a sum of rectangles; that is, a convex set (the sum of convex shapes is itself convex). Thus, in accordance with the present invention, the overall OPF problem has now become convexified and, as a result, is capable of creating viable solutions in an expedient manner, allowing for tools associated with discrete variables to be utilized.

Indeed, in the simple case where the rectangles are generated by tangent lines and polar coordinate intervals (as shown in FIGs. 3 - 5), the voltage law convexification becomes equivalent to a first-order approximation of the Euler representation's exponent for the complex power flow F_tj in the neighborhood of the given anchor point ( ^,⁰ , Qy ) :

_\ _ p0,self _\ self p0, cross i cross _ 0,cross β

ij ij ij ij ij - , ,

\ Q _ Q0,self _r ^seV , Q0, cross i cross , pO, cross β

ij ij ij ij

While this linearization bears some similarities to Successive Linear Programming (SLP) linearization, it is considered to be a superior solution since it uses log-scale voltages and transformer taps and, as a result, preserves the voltage tensor product "rank 1" property.

In the simplest case where only one anchor point is used, this approximation is similar to the DC representation of the OPF problem. The convexification with a single anchor point following the procedure outlined above is considered a preferred method for performing the DC approximation since it consider voltages as variables (i.e. not fixed as a conventional DC approximation assumes).

In the general case evaluation in accordance with the present invention, an MIP heuristic will manage the solution process through placing anchor points and generating an OPF model convexification within rectangles according to the following workflow:

1. Choose starting anchor points

2. Pre-solve a few-iterations of the dual of the convexified OPF relaxing constraints due to the OPF objective function (i.e. solving for any feasible solution without optimization) 3. Produce updated dual variables and infeasibility reduction directions

4. Generate a linear cut for the chosen anchor points 5. Choose a step size towards a chosen direction

6. Update anchor points set through branching by making the chosen step

7. Go back to the step 2 until either a feasible solution is found or the infeasibility check is true

8. Pre-solve a few-iterations of the primal of the convexified OPF

9. Produce updated primal variables and reduced cost directions

10. Go back to the step 4 unless either optimal solution is found or iteration limit is reached FIG. 8 contains a flowchart outlining these steps as they are used in the convexification process.

This process resembles the SLP process, however a branch-and-cut mechanism is used instead of penalties. Branching gives a considerable advantage so that the previously checked anchor points are not discarded but rather are traversed in a tree-like recursive way. Cutting provides a way to discard anchor points that are checked to be infeasible or inferior.

It is believed that in accordance with the utilization of convexification in accordance with the present invention that the MIP heuristic which dynamically places anchor points and handles the bearing anchor points choice process is more robust in its performance than SLP and much faster than generic MIP. In the simplest case, the process of the SC OPF solution should be as fast as solving the DC approximation.

As mentioned above, there other non-convex relations in the OPF model. In particular, the branch flow bilinear (BFB) constraints, as well as the converter equations, are non-convex. BFB is not convex due to its branch capacity lower bound. In most cases, it is possible to ignore BFB unless it is imperative to enforce (e.g. for some inter- area transfers). In such cases, it is true that the combination of BFC and BFB together form a full ring in the complex plane. Therefore, the same rectangle coverage idea applies so that BFC and BFB can be convexified and managed by the same MIP process as outlined above for use with the voltage law.

The converter relations are a bit harder to analyze, but a preliminary analysis confirms that they can also be convexified and managed by the same MIP process, although the quality of the approximation is lower (as a result of some simplifying assumptions necessary to be made). However, it is noted that in realistic cases, the fraction of converter components is negligibly small compared to the AC and DC nodes, so approximation accuracy is not crucial (i.e. the main reason to consider converters is network coupling). As mentioned above, contingencies are abnormal events that adversely impact the normal operation of the electric power network. Contingency analysis is performed in an effort to identify events that may cause cascading outages or unrecoverable situations and determine the actions that can be taken to prevent them from occurring in the first place. The contingencies themselves are pre-defined by the user and each contingency may involve one or more branch, bus or equipment outage, or a shift or change in generation and/or load. Indeed, in a standard electric power network there may be thousands of predefined contingency events.

For the purposes of the present invention, the SC OPF needs to adhere to a set of pre-defined emergency limits for each constraint (as defined above) for every given system contingency. The SC OPF also needs to adjust the given controls, in either preventive or corrective mode, for the affected transformers, phase shifters, switched shunts (capacitors or inductors) and load adjustments. As discussed above, "preventive mode" implies that the network control (generator scheduling, load shedding, transformer tapping, etc.) must be chosen for a base case such that the network remains secure under any contingency in a given set. That is, if any of the contingencies occurs, the network must remain feasible without any control modification. Corrective mode implies that the network control is chosen to adapt to the given contingency case such that the network thereafter becomes secure. That is, if the contingency happens, then the network adapts to it with control modification from the base case control to the contingency control. While adhering to the pre-defined emergency limits and adjusting affected controls (in either preventive or corrective mode), the SC OPF needs to always minimize the total number of controls that need to modified.

Typically, in the preventive mode, only the state vector x = [V, θ\ is adapting to the contingency case; the rest of the network control variables u = [P_g, Pi, φ, r, V_g, Q_s] remain fixed. Thus, the control vector u couples together the base case and all of the contingency case constraints. Alternatively, in the corrective mode, there is no such coupling and the SC OPF problem is decomposable into stand-alone, case-by-case contingency OPF analysis.

As a result, the preventive mode SC OPF is much more difficult to solve, since each contingency case network constraint must be present in the model - the model size grows quickly beyond the limits of even the most powerful OPF solvers. This problem is addressed by the present invention as discussed below by using a common two-tier decomposition framework of contingency column-generation, where it is possible to utilize this approach by virtue of first applying convexification to the non- convex relations (such as the "voltage law" relations) to create discretized variables. The following discussion will first describe the contingency case OPF model in general, then describe the specific improvement associated with the use of column-generation.

It is first presumed that any contingency C can be parametrically represented by a set of four indicator vectors:

• Line Elimination indicators, as defined by the set:

{z^eij e {0,l} , V(/, ) _e E(N)}

• Line Modification indicators, as defined by the set:

{z , e {0,l} , V(/, 7) e E(N)}

• Bus Elimination indicators, as defined by the set: {z^ei e {0,1} , Vz e V(N)}

• Bus Modification indicators, as defined by the set: {z"\- e {0,1} , Vz e V(N)} For the "base case" model, all of these indicator vectors have a zero value. Thus, depending on the physical parameters of a given contingency, one or more of these vectors will have a nonzero value. Any contingency case defined within the contingency family C can be obtained from the same, single base case without adding new topological elements, only a parameter modification is required (such as emergency limits). This property is exploited in the SC OPF framework by introducing the methodology of "capacity relaxation", as described below.

For each of these vectors, the term "elimination" changes the network topology by completely remove the selected line or bus from the system graph (defined as N) so that all constraints associated with that removed element disappear from the OPF model. In the case of "modification", there is no change in the network topology and, therefore, no removal of any constraints. Instead, the parameters associated with the "modified" element change their parameter values. In particular, for any model parameter p, its modification from the base case value po to the contingency case value pi can be represented in a continuous fashion by a weighted sum:

p(Z^m) = (\ - Z^m)p₀ + Z^m _Pl .

Accordingly, any contingency case problem can be represented by the base case OPF model and the binary vectors couple (z^e,z^m ) by performing the following steps: (1) eliminate the OPF model constraints for removed elements as given by Z²; (2) modify the OPF parameters affected by the contingency as given by Z"¹ (3) add constraints for all OPF model elements, fixing preventive controls at the given base case level; (4) modify the capacity box constraints for all OPF model elements by "controlled parametric relaxation"; and (5) substitute the OPF model objective function with the step of "maximizing the capacity relaxation tightness".

In accordance with the present invention, the last two steps in the process function to make the contingency OPF model as well-defined as the base case model, ensuring that it will always converge (if the base case OPF converges in the first instance). For elements unaffected by the contingency, the capacity relaxation should always be looser than the base case capacity bounds, and give exact base case bounds only at the maximum possible relaxation tightness. For elements removed by the contingency, the capacity relaxation should start with wider bounds than the base case capacity bounds, but effectively shut down the capacity to zero at the maximum possible relaxation tightness. For those parameters modified at step (2) of the process, Z"¹ is treated as variables defined within [0,1]. The modification starts with the zero value (base case), with the solver trying to maximum Z"¹ from zero to the maximum value of one, and include these modified parameters as part of the parametric relaxation into the objective function.

Summarizing the above, it is found that the OPF model has the following

characteristics: (a) some constraints are eliminated; (b) all capacity bounds are initially looser than defined in the base case; (c) all of the model parameters are initially equal to the base case values; and (d) preventive control functions to fix the constraints at base case levels that are known to be feasible. As a result, the contingency OPF model contains the base case situation and, therefore, is guaranteed to have one feasible solution (that is, it is always converging).

Additionally, the contingency OPF has the following characteristics: (e) parametric capacity relaxation providing the exact contingency case situation at the maximum possible relaxation tightness; and (f) the objective function seeking to maximize the capacity relaxation tightness. This means that the contingency OPF model must tighten the capacity relaxation to the maximum level and yield the optimal solution of the contingency case (reflecting both preventive and corrective controls) if and only if the contingency case is feasible. Otherwise, the model must stop at some tightening level that is not maximum, yielding a feasible solution - requiring the utilization of removed elements. This solution, while not maximum, is helpful in identifying which network parts require usage of the removed elements and to what extent. Thus, the feasibility of the contingency case can be checked by solving the contingency OPF model generated by the five step process and verifying if the optimal value of the objective function is exactly equal to the maximum possible relaxation tightness.

Two families of bus capacity relaxation variables are now introduced, defined as tf and tf , as are two families of the branch capacity relaxation variables, defined as tf and tf , such that:

1≤ tf < bigM, 1≤ tf < bigM where the value of one is the maximum tightness level for all these variables. The objective function is then given as:

V(N) V(N)

v^* = min £ (** - tf + 1 - Z," )+ £ (* - tf + \ - Z; )

i=l i,j=

(i,j) E(N)

The bus capacity relaxation for P_ig (the same relations apply for Q_ig , P_u , Q_u ) is then given by: t^lb■ P mm . ≤ P < t^ub■ P max , " if Z^e = 0

(\ - t^lb p ■ < P < (t^ub - l)- P , if Z^e = 1

This constraint form only applies if the base case capacity bounds are positive. If the capacity bounds are negative, the constraint exchanges the lb and ub capacity relaxation terms. The branch flow capacity relaxation is given by: t^lb - ² mm . ≤— P.² + ²≤ t^ub■ F.² max ,' if Z^e = 0

(l - t^lb ) J^■ F z./²mi .n≤ P.² + Q z ² < (t^ub - 1) J- F² max ,' if Z^e = 1

The SC OPF workflow itself involves two iterative loops: (1) an inner loop, which is a contingency search routine; and (2) an outer loop (also variously referred to as a "global loop"), which is a preventive control modification routine. By utilizing this inner/outer loop workflow, the size of the SC OPF at each iteration remains reasonable, since all of the thousands of possible contingencies are not collected and analyzed at once. Instead, "severe" contingencies are first identified in the inner loop, with their "signatures" added gradually to the SC OPF. The updated SC OPF is then re-solved in the outer (global) loop, with the process continuing in this manner. In each iteration, there will be fewer and fewer "severe" contingencies in the inner loop.

By virtue of having first applied the above-described convexification to parameters such as the voltage law, and have created a set of discrete values, various tools that lend themselves to discrete analysis can now be employed in dealing with the overall SC OPF analysis. Indeed, as described in detail below, the discretization provides the ability to use various decomposition techniques, variously referred to as column generation decomposition. Specific techniques such as employing Benders cuts, primal/dual decomposition, or Dantzig- Wolfe decomposition to the processing in both the inner and outer loops has been found to significantly contribute to improving the SC OPF numerical performance and opens the potential for permitting growth in the size of the SC OPF problem being handled.

There are two contingency sets involved in the SC OPF workflow: (1) the master set of all contingency cases to be analyzed; and (2) the SC OPF subset, a dynamically growing set of contingencies that is updated with selected contingencies in each iteration of the inner loop. The inner loop uses the above-described single contingency OPF model and solves the contingency OPF and determines if it is feasible (i.e., if it converges). If it is not feasible, the case is defined as "severe" (under the current parameters of preventive control) and the process moves to the outer loop. If the contingency OPF is feasible, the inner loop retrieves the next contingency to be analyzed and re-iterates itself.

FIG. 9 is a flowchart of this process, and begins at step 90 with the solution of the "base case" (i.e., with the binary vectors couple yZ^e , Z ^m ) having a zero value. The master set of all possible contingency cases is then developed (step 92), where as mentioned above this master set may easily contain thousands of contingency cases. At this point, the "inner loop" of the process is entered (step 94) to find successive severe contingencies to be added to the solution of the base case problem. The inner loop itself begins by selecting a contingency from the master set (step 96) and performing a solution to determine if it is feasible (step 98). If it is feasible, the process continues by checking to see if there are any other remaining

contingencies (step 100) and, if so, another contingency is selected (step 102) and the process loops back to step 96 to solve the contingency OPF for this selected contingency. Once all contingencies have been evaluated, the query at step 100 will instead "end" the SC OPF process. The looping back to step 96 will continue until an infeasible contingency is found at step 98, defined as a "severe" contingency (step 104). At this point, the process enters the outer, global loop. The 'signature' of this severe contingency is then added to the SC OPF subset (step 106) and the SC OPF problem is re-solved (step 108), including the updating of the preventive control in the master set (step 110). The global loop returns to step 92 and the process continues in the same manner.

As seen from the above, the SC OPF workflow is complex and there are several key places where its execution can be managed in a variety of ways reflecting different trade-offs between inner loop iterations and global loop iterations, as well as between "type I errors" (checking a feasible contingency) and "type II errors" (not checking an infeasible

contingency).

A few key questions that help to intuitively understand these trade-offs include the following:

1. How many infeasible contingencies should be included in the SC OPF subset per inner loop call?

2. Should any infeasible contingencies be included in the SC OPF subset, or should only a select few of the worst ones be included in the SC OPF subset?

3. Should contingencies be dropped from the SC OPF subset?

4. Should each contingency be checked in the master set again and again after each global loop control update?

5. Should some contingency checks in the master set be skipped if there are a significant number of feasible contingencies in the inner loop at late global loop iterations?

The answers to each of these questions reflect various trade-offs that need to be made, where each option has its advantages and drawbacks. For example, if only one contingency is included in the SC OPF subset, then the size of SC OPF grows slowly but the fraction of infeasible contingencies remaining in the master set after the SC OPF global loop control update will be higher. Or, if the selected contingency signature is somewhat less than the full contingency OPF (which implies smaller size increase and looser coupling of the SC OPF), then that contingency must be checked again after the next control modification.

The outer, global loop uses the SC OPF sub-set of contingency OPF models, indexed by C defined in the (7 , Z"¹) space. As mentioned above, the first contingency OPF in the SC OPF model is the base case, represented by (Z^e=0, Z^m=0).

Contingencies are added to the SC OPF sub-set as contingency 'signatures'. The 'signature' can vary from being represented by the full contingency OPF model to being generated as the 'global Benders cut' by applying Hyperplane Separation Theorem to preventive controls in the contingency OPF. This choice is not obvious and is one of the key things regulating the trade-off between the size of the SC OPF and the ratio of outer loop iterations vs. inner loop iterations. In one analysis, it is possible to use the full contingency OPF as the contingency 'signature' as it guarantees that at least this contingency will always be feasible in the next round of inner loop iterations.

The novel idea that is presented here is the development of a more advanced umbrella framework for the SC OPF workflow that will include the whole variety of choices mentioned in a formal way and use advanced decomposition techniques (such as Bender cuts and column- generation/Dantzig- Wolfe decomposition, discussed below) to fully exploit them. This framework can be seen as an extension of the above-described SC OPF scheme, with more detailed sub-workflows for each step.

In accordance with the methodology of the present invention, each contingency OPF in the SC OPF sub-set uses the contingency OPF model definition from the prior art process described above, with the exception of the third ("modifying") step (which was fixing preventive controls to be equal to the base case values). Instead, for each contingency OPF in the SC OPF sub-set, the preventive controls are set to be equal to the corresponding preventive control of the first contingency OPF in the sub-set. The principal difference is that the preventive controls of all contingency OPFs in the sub-set are no longer fixed to be equal to given levels, but rather are coupled to be changed in a synchronous manner. For example, if scheduling the generated active power, P_ig , is used for preventive control, then the SC OPF constraints added at the "modified" step are given as:

Finally, the SC OPF is different from the single contingency OPF model by its modified objective function. First of all, the SC OPF objective function includes the sum of the objective functions for all contingencies:

Zy '^c ) + penalty

The penalty term corresponds to the objective of minimizing the total number of controls modified. This term involves weighed sum (with weights reflecting controls priority) of penalties for modifying each of the control. The individual penalties are step-functions:

0, control did not change

penalty(control) =

1, control changed

In accordance with the present invention, it has been recognized that massive size and complexity of the SC OPF model can be simplified by using column-generation decomposition. Contingency cases can be thought of as columns that are generated by the inner loop infeasibility search that updates the SC OPF sub-set in the manner outlined above. Therefore, this set of 'severe' contingencies can be defined as the master problem basis column set, with the global SC OPF iterations producing preventive control updates as optimal responses to the new columns that are added. As a result, the column-generation methodology provides a mechanism to manage the overall SC OPF workflow execution naturally and transparently.

For example, the column generation technique will seek the several most important columns to update in the master set. Thus, the inner loop process, using column generation, will not stop with the identification of the first infeasible contingency, but rather continue to find several 'severe' contingencies. This is dictated by adding a properly-specified objective function penalty for the inner loop's single contingency OPF model.

The aforementioned techniques may be applied to improve performance of SC OPF workflow in accordance with the present invention in one or more of the following ways. First, it has been found that "inner" Benders cuts may be used to accelerate the performance of the inner loop at late global loop iterations. That is, if it apparent that there are a large number of feasible contingencies, then Benders cuts (or another one of the column generation

decomposition techniques) can be used to 'skip" the feasibility check of some of the neighboring constraints. Type I errors (i.e., needless checking of feasible contingencies) carry a lot of dead-weight in this scenario that adversely impacts performance to no avail. It is to be noted that the contingency OPF problem carries an explicit dependence on the parameter vector pair (z^e,z^m ). Thus, this vector pair (z ^e , Z ^m ) can be considered as fixed controls and generate a Benders cut using Farkas lemma for the constraint that is violated by a feasible contingency:

Recall that the purpose of the inner loop is to find infeasible constraints and those are given by the constraint above. Thus, if a Benders cut is implemented that cuts off contingencies that violate that constraint, then the search is being narrowed without losing any of the infeasible contingencies.

However, there are three subtle issues to take care/be aware of: (1) vectors (z^e,z^m) are binary and thus continuous Benders cut should be included into a branching procedure for probing next contingencies with respect to cutting plane constraints. Thus, a new dedicated MIP cut-and-branch procedure should be developed for probing contingencies in the inner loop; (2) the contingency OPF model includes implicit dependency on (z^e,z^m) as well (since the voltage law constraint is dropped for the branches eliminated by the contingency). Thus, naive Bender cuts are not applicable and proper Benders cuts implies combinatorial dualization of the contingency OPF; and (3) Benders cuts are generated for the current preventive controls only. That is, after the next outer loop iteration, the new MIP infeasible contingency search procedure must be re-initialized.

Also, it is possible to use Benders cuts (or another column decomposition technique) on the outer loop to accelerate the outer loop iterations. A contingency that is found to be infeasible under preventive control in the inner loop implies violation of the following constraint: v^* (preventive control) = 0 Thus, Benders cuts can be generated for the preventive control using Farkas lemma based on the violation of the aforementioned condition. This will effectively cut-off the current preventive control for the current contingency, which is sufficient to consider it infeasible. Note that this Benders cut can be reinforced if the OPF case is run in the corrective mode and it is defined as feasible. This in rum prohibits using naive Bender cuts, but significantly narrows the scope of preventive control search such that not only infeasible controls are cut-off but feasible ones are constrained. There is one issue to take care/be aware of and that is that naive Benders cuts use only active constraints (through non-zero dual variables). This creates a 'signature' of the contingency OPF problem that is not a complete representation. Actually, the dual variables are functions of the preventive control and are only constant in some local neighborhood of the current preventive control values. Thus, naive global Bender cut's contingency OPF signature will cut off the current infeasible control but does not guarantee that the next iteration of the preventive control (produced by SC OPF taking into account the new outer loop Benders cut) will be feasible under this contingency. Thus, this particular contingency must be checked again and again in every of the next inner loop calls.

However, there is a relatively easy alternative to fix this: include the full contingency OPF problem as the signature into SC OPF, not just the active constraint's Bender cut. This eliminates the need for checking that contingency again and again, but carries a much heavier toll in for the SC OPF problem size.

Additionally, it is possible to utilize Dantzig- Wolfe (dual) decomposition methods to accelerate the SC OPF model. SC OPF as it was formulated has an inherent block structure: each contingency case problem appears by its signature that does not interact with others. The coupling block is fixing the preventive controls at the same level for all cases. Each block can be solved quickly (given the preventive controls levels), since it is equivalent to the single contingency OPF mode run and the result would be feasibility/infeasibility of the problem as well as primal/dual variables that can be used to modify the preventive control levels, if needed. Moreover, with "outer loop" reinforced Benders cuts added, the preventive control levels are much more well shaped and decomposition converges faster.

In one study associated with 300-bus SC OPF iterations, it was found that the SC OPF size quickly grows large and numerical performance slows down more than linearly if decomposition is not used, which means that the SLP solver does not exploit the potential decomposition power properly. Thus, the use of column generation decomposition in accordance with the present invention offers a large contribution in improving the SC OPF numerical performance and open more potential for SC OPF size growth. FIG. 10 is a system- level block diagram showing the specific architecture that is employed as a result of utilizing convexification to create a set of discrete variables to solve in the SC OPF analysis. The inputs to the analysis are shown as interacting via OPF API layer 130, defined as OPF solution manager 140, OPF model generator 150 and OPF case processor 160. OPF solution manager 140 and OPC case processor 160 provide input to OPF model generator 150.

In accordance with the present invention, convexifications iterations manager 200 and solver model generator 300 are particularly formed to utilize discrete variables and perform the inner/outer loop, bi-level processing of the OPF problem to arrive at a solution. As shown, the data from OPF model generator is supplied as an input to convexification manager 200, which then works with solver model generator 300 to arrive at the solution. Model generator uses as inputs the various discrete variable tools mentioned above, shown in FIG. 10 as a cutting planes generator 400 (which also interacts with the convexification iterations manager 200), a linear algebra sub-routine module 410 and an interior point method (IPM) solver module 420), which then presents the results through the solver API layer 500.

It is to be understood that this invention is not to be limited by the embodiment shown in the drawings and described in the description, which is given by way of example and not of limitation, but only in accordance with the scope of the claims appended hereto.

Claims

What is claimed is:

1. A security-constrained optimal power flow (SC OPF) method of operating a power system, comprising the steps of:

determining non-convex relations defining attributes of the power system;

performing a convexification on at least one non-convex relation to create a discrete solution;

solving a base case version of the SC OPF; and

iteratively solving contingency cases of the SC OPF using the discrete solution for the non-convex relations, the iteration taking the form of utilizing an inner loop to evaluate contingencies and find infeasible contingencies, and an outer loop to re-solve the SC OPF problem with the infeasible contingency, instituting column-generation decomposition techniques to minimize the iterations performed by both the inner loop and the outer loop.

2. The method of claim 1 wherein the voltage law associated with the AC branch constraints is a non-convex relation that is subject to convexification to create a discrete solution.

3. The method of claim 1 wherein the step of performing the convexification includes the steps of

reformulating the relation using a complex plane representation;

defining anchor points within the complex plane representation; and

creating convex boundary solutions for each anchor point to define as the discrete solution.

4. The method of claim 3 wherein the complex plane representation comprises a ring sector, with a rectangle boundary defined to surround each anchor point.

5. The method of claim 1 wherein the inner loop iterations utilize column-generation decomposition to eliminate the processing of a select number of feasible contingencies.

6. The method of claim 5 where Benders cuts are used to eliminate the evaluation of a set feasible contingencies.

7. The method of claim 5 wherein Dantzig- Wolfe decomposition is used to eliminate the evaluation of a set of feasible contingencies.

8. The method of claim 5 where primal/dual decomposition is used to eliminate the evaluation of a set of feasible contingencies.

9. the method of claim 1 wherein the outer loop iterations utilize column-generation decomposition to cut off the current preventive control for the current infeasible contingency.

10. A system for performing security-constrained optimal power flow (SC OPF) analysis of a power system, the system including

a convexification module for receiving as an input a non-convex relation utilized to define the base case SC OPF and creating as an output a discrete version of the non-convex relation, utilizing bounded variables in [0,1] for analysis purposes; and

a column-generation module for interacting with the convexification module and evaluating an on-going SC OPF flow to find and eliminate certain contingencies from the analysis.

11. The system of claim 10 wherein the convexification module receives as an input the voltage law relation of the AC branch flows and generates a discrete variable output.

12. the system of claim 10 wherein the column-generation module utilizes at least one of the following: Benders cuts, primal/dual decomposition and Dantzig- Wolfe decomposition.