Classifications

• • H—ELECTRICITY
  • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
  • H02J—ELECTRIC POWER NETWORKS; CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
  • H02J3/00—Circuit arrangements for AC mains or AC distribution networks
  • H02J3/001—Arrangements for handling faults or abnormalities, e.g. emergencies or contingencies
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06F—ELECTRIC DIGITAL DATA PROCESSING
  • G06F2111/00—Details relating to CAD techniques
  • G06F2111/04—Constraint-based CAD
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06F—ELECTRIC DIGITAL DATA PROCESSING
  • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
  • G06F2119/06—Power analysis or power optimisation
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06F—ELECTRIC DIGITAL DATA PROCESSING
  • G06F30/00—Computer-aided design [CAD]
  • G06F30/20—Design optimisation, verification or simulation
• • H—ELECTRICITY
  • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
  • H02J—ELECTRIC POWER NETWORKS; CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
  • H02J2103/00—Details of circuit arrangements for mains or AC distribution networks
  • H02J2103/30—Simulating, planning, modelling, reliability check or computer assisted design [CAD] of electric power networks
• • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
  • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
  • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
  • Y02E40/00—Technologies for an efficient electrical power generation, transmission or distribution
  • Y02E40/70—Smart grids as climate change mitigation technology in the energy generation sector
• • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
  • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
  • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
  • Y02E60/00—Enabling technologies; Technologies with a potential or indirect contribution to GHG emissions mitigation
• • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
  • Y04—INFORMATION OR COMMUNICATION TECHNOLOGIES HAVING AN IMPACT ON OTHER TECHNOLOGY AREAS
  • Y04S—SYSTEMS INTEGRATING TECHNOLOGIES RELATED TO POWER NETWORK OPERATION, COMMUNICATION OR INFORMATION TECHNOLOGIES FOR IMPROVING THE ELECTRICAL POWER GENERATION, TRANSMISSION, DISTRIBUTION, MANAGEMENT OR USAGE, i.e. SMART GRIDS
  • Y04S10/00—Systems supporting electrical power generation, transmission or distribution
  • Y04S10/50—Systems or methods supporting the power network operation or management, involving a certain degree of interaction with the load-side end user applications
• • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
  • Y04—INFORMATION OR COMMUNICATION TECHNOLOGIES HAVING AN IMPACT ON OTHER TECHNOLOGY AREAS
  • Y04S—SYSTEMS INTEGRATING TECHNOLOGIES RELATED TO POWER NETWORK OPERATION, COMMUNICATION OR INFORMATION TECHNOLOGIES FOR IMPROVING THE ELECTRICAL POWER GENERATION, TRANSMISSION, DISTRIBUTION, MANAGEMENT OR USAGE, i.e. SMART GRIDS
  • Y04S40/00—Systems for electrical power generation, transmission, distribution or end-user application management characterised by the use of communication or information technologies, or communication or information technology specific aspects supporting them
  • Y04S40/20—Information technology specific aspects, e.g. CAD, simulation, modelling, system security

Definitions

• 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.
• SC-OPF Security-Constrained Optimal Power Flow
• 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.
• 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.
• N-1 security 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.
• 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.
• corrective rescheduling three different modes of control adjustments that affect the SC OPF are defined: (1) preventive mode; (2) corrective mode; and (3) preventive/corrective mode.
• 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.
• 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.
• SC-OPF Security-Constrained Optimal Power Flow
• 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.
• 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.
• 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.
• 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.
• 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.
• FIG. 10 is a block diagram illustrating the applicability of the convexification process to the SC OP problem.
• 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.
• 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
• 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
• AC branch 20 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
• 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:
• 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).
• AC to DC converter AC to DC converter
• DC to AC converter DC to AC converter
• 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.
• 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.
• V(N) is the set of buses
• E(N) is the set of branches.
• V(N) the following subsets are distinguished: AC bus set V(AC) and DC bus set V(DC).
• 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.
• box constraint i.e., lower and upper bound constraint values
• 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 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:
• 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 set of AC branch constraints includes: (1) the "voltage law” (VL), defined as follows:
• the branch flow cone (BFC) constraint consists of the following second-order (Lorentz) cone constraint:
• 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.
• 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:
• tap choice variables w tk are restricted to be in [0; 1] box.
• tap choice variables can also be restricted to be binary by either of the following two methods:
• the first of two methods implies using a "mixed integer programming" (MIP) solver and is more expensive computationally while providing more responsive transformer control.
• MIP mixed integer programming
• 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:
• the set of "area” constraints involve aggregation of branch power flows into area interface flows:
• 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: ' ⁇ ' ⁇ ⁇
• V DC - (cos(a) + cos( ))
• V DC V AC ⁇ cos(a + ⁇ ) - ⁇ ⁇ ⁇ R com
• 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 ( ⁇ , ) 2 + 23 ⁇ 4 + B a 'P a + B i r g Q 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.
• 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
• 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.
• 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.
• 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.
• 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, R 2 and R 3 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, M 2 and M 3 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.
• 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.
• 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, M 2 and M 3 in FIG. 5), an accurate coverage of the ring sector is provided.
• 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).
• 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:
• FIG. 8 contains a flowchart outlining these steps as they are used in the convexification process.
• 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.
• the process of the SC OPF solution should be as fast as solving the DC approximation.
• BFB branch flow bilinear
• 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.
• 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.
• 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.
• any contingency C can be parametrically represented by a set of four indicator vectors:
• Bus Elimination indicators as defined by the set: ⁇ z e i e ⁇ 0,1 ⁇ , Vz e V(N) ⁇
• Bus Modification indicators as defined by the set: ⁇ z" ⁇ - e ⁇ 0,1 ⁇ , Vz e V(N) ⁇
• all of these indicator vectors have a zero 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.
• 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.
• N system graph
• the parameters associated with the "modified” element change their parameter values.
• 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:
• 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 Z 2 ; (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".
• 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).
• 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.
• 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.
• 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.
• the contingency OPF model contains the base case situation and, therefore, is guaranteed to have one feasible solution (that is, it is always converging).
• 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.
• 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.
• bus capacity relaxation variables 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:
• 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.
• an inner loop which is a contingency search routine
• an outer loop also variously referred to as a "global loop”
• 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.
• 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.
• 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.
• 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
• step 100 the query at step 100 will instead "end” the SC OPF process.
• step 98 an infeasible contingency is found at step 98, defined as a "severe" contingency (step 104).
• step 104 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).
• step 108 the SC OPF problem is re-solved
• the outer, global loop uses the SC OPF sub-set of contingency OPF models, indexed by C defined in the (7 , Z" 1 ) space.
• 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.
• 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.
• the SC OPF constraints added at the "modified" step are given as:
• the SC OPF is different from the single contingency OPF model by its modified objective function.
• the SC OPF objective function includes the sum of the objective functions for all contingencies:
• 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:
• the column generation technique will seek the several most important columns to update in the master set.
• 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.
• Benders cuts (or another column decomposition technique) on the outer loop to accelerate the outer loop iterations.
• 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.
• 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.
• the preventive control levels are much more well shaped and decomposition converges faster.
• 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.
• 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.
• 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.
• IPM interior point method

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