1.- HELM MATHEMATICAL FOUNDATIONS

A. Holomorphic embedding of the load flow equations Embed the non linear, constant power contribution, ( s=0 is the no load case, s=1 is the load flow problem to be solved) Every holomorphic embedded power flow problem can be cast into an algebraic curve (a polynomial with polynomial coefficients) through ELIMINATION Algebraic curves can be represented by their power series in a disk with a non null convergence radius.

The closest singularity to the origin determines the convergence radius The formal power series and its reciprocal are used in order to create a recursive deterministic way of computing the coefficients through solving a linear equations system, requiring a nonsingular admittance matrix [Y] B. Analytic Continuation The solution at s=0 is taken as the solutions with all Flows equal to zero and all voltages equal to the Swing voltage. Analytic Continuation beyond the convergence radius is performed using an algebraic approximant to the algebraic curve . B. Stahl’s Theorem (1997) For obvious physical reasons, univalence is required. A mechanism to compute a unique solution of the algebraic curve is provided by the Stahl’s theorem (1997) that proves algebraic curves have a maximal (and therefore unique) analytical continuation, provided by the continued fraction of their power series. These can be computed with Padé Approximants. The univalence is achieved applying the cuts to the s plane. • If s=1 can be reached from s=0 then the problem • has a solution. • If s=1 can not be reached there is no solution.

3. WHAT DOES HELM ALLOW US TO DO A.Real time simulation HELM gives the solution with no equivocation and provides reliable information about the consequences of actions on real time. HELM based tools are currently used by real-time operators for a last minute check of programmed outages or maintenance actions. B. Real Time PV-QV curves computation Computes PV-QV curves in real-time providing the real-time operator with accurate information about The Current State (Base Case) as well as comparisons with contingency cases. The operational limits are displayed to asses the actual situation. C. Solving limit violation problems HELM being non equivocal allows for a reliable and Intelligent search of the Electric State space in order to find the required actions to solve limit violation problems. The current system uses a A* algorithm relying upon a well proven heuristic function that is able to take into account the utility operational criteria that can be user defined with a set of parameters that are taken into account by the heuristic function. Example of a limit violation situation The system provides a choice of plans The plans display their cost (evaluated in terms of the devices used) and the improvement they bring to the problem. The sole manoeuver displayed solves all the overloads . D. GENERATION OF RESTORATION PLANS Actions at breaker level of a restoration plan In case of a Blackout an A* algorithm based on HELM and on a well proven heuristics provides plans to solve blackout problems on a power network. The system manages black start and multiple islands problems. It has successfully passed an operator certification examination at CFE (Mexico) scoring 83 (the level required for operator’s certification being 75)

E. ENHANCED WORK FLOW AND CASE VISUALIZATION HELM-Flow provides bus bar level Planning tools to delayer, analyze, and provide decision support even in non resolved cases.

4. WHERE MIGHT HELM TAKE US FROM A RESEARCH PERSPECTIVE

A. Parallelization HELM computation of the Padé approximant for a node can be performed independently from the computation of the Padé approximant for other nodes. The algorithm is parallelizable to a high degree. This will enable a very fast response even on very large networks. B. Reliable computation of contingencies HELM provides unequivocal answers to power flow problems. HELM is, therefore, the methodology of choice to get reliable answers when analyzing contingencies. For transmission systems, where there is a limited time to recover the N-1 condition after a contingency, massive parallel computing is required to provide the right answer within the allowed time. HELM is well suited for this purpose because of its accuracy/reliability and of its parallelization capability. C. Application to Smart Grids HELM is currently used within the AGORA product in real-time advanced EMS applications for transmission power systems- like limit violation solving and restoration. The adaption of these advanced functions to the context of Smart Grids is the natural step to introduce these proven technologies to the distribution power grids. D. Application to DC Micro-grids The power flow equations for DC grids, with constant power injections are also non linear. They have the same algebraic structure than the AC power flow equations and HELM can be adapted to solve this problem. This opens a new world of possibilities in analyzing and solving the DC micro-grid problems. E. Automatic management of Power Systems The continuously increasing complexity of power systems requires the system operators to have reliable tools available that provide both information about the current state and assessment about the power system management. In normal operational conditions, the system should assess optimized operation and in alert state it should provide the operator with recommendations about the best suited actions to restore normal operation conditions. The recommendations should be based on the current state of the power system and this can only be achieved using a non equivocal power flow methodology coupled with an efficient and intelligent search of the available alternatives. HELM based advanced EMS applications are already doing this on demand. The next step should be to provide the system with the ability to identify the power system state: Normal Operation, Alert/Emergency, Restorative, and providing advice to the operator according to the detected state. Given enough information the system could even activate auto-healing procedures to solve specific sets of problems opening the path to automatic management of power systems.

2.- SIGNIFICANT DIFFERENCES WITH ITERATIVE METHODS

A.Non equivocal If the problem has a solution, i.e. if the point s=1 is within the maximal analytical continuation provided by the algebraic approximant, then HELM computes the solution. If the problem does not have a solution, that is s=1 can not be reached by a continuous path from s=0 without finding a branch cut, HELM indicates that the problem does not have a solution This is a fundamental difference with iterative methods. Converge of iterative methods depends on the choice of the seed point. Non convergence is not a clean cut indication that there is no solution to the problem. B. Provides the Operational solution. The equations of the load-flow problem for a grid with N nodes have as much as 2N solutions, but only one of these solutions can exist on the network at a given time. How, when solving the load flow equations can we pick the right one? Consider the solutions of the two buses case with Z=R+jX and S=P+jQ) where and where |W| is the swing voltage.

If P=Q=0, then σ=σR+jσI=0, and the two solutions are U+ = 1; U-=0. Clearly the preferred solution is U+, if there are no injections, the flows are zero, and all the voltage are equal to the swing voltage. HELM choses (at s=0) the solution where all flows are zero and where all voltages are equal to the swing voltage. The analytic continuation of this solution at s=1 gives the solution of the initial problem, as at s=1 the injections have the values of the injections of the original problem. The solution provided by HELM is therefore continuously linked with the solution at s=0 which is the operational solution. That is the type of solution that all the voltage control devices seek to stabilize. Iterative methods can not control the kind of solution they will provide. C. HELM is reliable even near the voltage collapse Padé Approximants provide reliable values for the algebraic approximant even in the vicinity of the voltage collapse, where iterative methods become highly unreliable. D. HELM provides new analytical tools to explore Load flow problems The use of algebraic approximant to build the solution has enabled to define a new kind of approximant called the sigma approximant that has proved to be extremely useful to point at the nodes that are at the root of voltage collapses cases. E. HELM is computationally efficient. Computation time is the same than Fast Decoupled Newton-Raphson.

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Current State

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