Holomorphic Embedding Load-flow method: Difference between revisions

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The [[Power_flow_study|load-flow]] calculation is one of the most fundamental components in the analysis of power systems and is the corner stone for almost all other tools used in power system management and simulation.

The '''Holomorphic Embedding Load Flow Method''' (HELM)

The load-flow equations can be written in the following general form:

Based on a holomorphic embedding technique, HELM provides the operational solution to the (multi-valued) load-flow problem in real time.

{{NumBlk|:|<math>\sum_k Y_{ik} V_k + Y_i^{\text{sh}} V_i = \frac{S_i^*}{V_i ^*}</math>|{{EquationRef|1}}}}

where the given (complex) parameters are the admittance matrix {{math|<var>Y_ik</var>}}, the bus shunt admittances {{math|<var>Y_i</var>^sh}}, and the bus power injections {{math|<var>S_i</var>}}.

TheTo [[Power_flow_study|load-flow]]solve calculationthis is one of the most fundamental components in the analysis of power systems and is the corner stone for almost all other tools used in powernon-linear system management and simulation. Prior to the development of HELM,equations previousnumerical methodsiterative wereschemes basedmay onbe numericalapplied iterativesuch schemesas such as [[Gauss-Seidel]], [[Newton-Raphson]], or variants (such as homotopic continuation methods<ref>E. L. Allgower and K. Georg, "Introduction to Numerical Continuation Methods", SIAM Classics in Applied Mathematics 45, 2003.</ref>). All of these iterative methods suffer from a fundamental set of problems that limit their reliability<ref group="note">This is a general phenomenon affecting the Newton-Raphson method when applied to equations in ''complex'' variables. See for instance [[Newton's_method#Complex_functions]].</ref>. Iterative Load-flow methods become increasingly unreliable as the electrical grid progresses towards voltage collapse as visualized by the position on the PV/QV curves. By contrast, HELM is non-iterative, deterministic, and non-equivocal. This breakthrough was possible through the understanding of the ability to apply techniques associated with [[Algebraic Curves]] to the load-flow problem. The resultant HELM always ensures the computation of the operational solution to the (multi-valued) load-flow problem. HELM’s equation structure allows competitive computational costs. It is scalable for all sizes and complexities of grids. The HELM algorithm provides new capabilities such as model-based scenarios, assessment of data quality and guided restoration. HELM allows the utilization of more accurate state estimation, solving some of the fundamental problems of data reliability in electrical grids.

Also the solutions of the load-flow equations are multivalued<ref group="note" name="multsol">It is a well-known fact that the load flow equations for a power system have multiple solutions. For a network with {{math|<var>N</var>}} non-swing buses, the system may have up to {{math|2^<var>N</var>}} possible solutions, but only one is actually possible in the real electrical system. This fact is used in stability studies, see for instance: Y. Tamura, H. Mori, and S. Iwamoto,"Relationship Between Voltage Instability and Multiple Load FLow Solutions in Electric Power Systems", '' IEEE Transactions on Power Apparatus and Systems'', vol. PAS-102 , no.5, pp.1115-1125, 1983.</ref>, and iterative methods are unable to select among the large set of mathematical solutions, the unique solution that is currently active in a physical power network. This special solution, fulfilling the operational

constraints of the controlling devices of the power network, is the only one providing the description of the current physical state. Based on a holomorphic embedding technique, HELM provides the operational solution to the (multi-valued) load-flow problem in real time.

= Background =

Traditional load-flow algorithms were developed based on three foundational approaches: the Gauss-Seidel method<ref>J. B. Ward and H. W. Hale, "Digital Computer Solution of Power-Flow Problems," ''Power Apparatus and Systems, Part III. Transactions of the American Institute of Electrical Engineers'', vol.75, no.3, pp.398-404, Jan. 1956.

* A. F. Glimn and G. W. Stagg, "Automatic Calculation of Load Flows," ''Power Apparatus and Systems, Part III. Transactions of the American Institute of Electrical Engineers'', vol.76, no.3, pp.817-825, April 1957.

* Hale, H. W.; Goodrich, R. W.; , "Digital Computation or Power Flow - Some New Aspects," ''Power Apparatus and Systems, Part III. Transactions of the American Institute of Electrical Engineers'', vol.78, no.3, pp.919-923, April 1959.</ref>, which has poor convergence properties, but very little memory requirements and it is straightforward to implement; the full Newton-Raphson method<ref>W. F. Tinney and C. E. Hart, "Power Flow Solution by Newton's Method," ''IEEE Transactions on Power Apparatus and Systems'', vol. PAS-86, no.11, pp.1449-1460, Nov. 1967.

* S. T. Despotovic, B. S. Babic, and V. P. Mastilovic, "A Rapid and Reliable Method for Solving Load Flow Problems," ''IEEE Transactions on Power Apparatus and Systems'', vol. PAS-90, no.1, pp.123-130, Jan. 1971.</ref>, which has fast (quadratic) iterative convergence properties, but it is computationally costly; and the Fast Decoupled Load-Flow (FDLF) method<ref name="FDLF">B. Stott and O. Alsac, "Fast Decoupled Load Flow," ''IEEE Transactions on Power Apparatus and Systems'', vol. PAS-93, no.3, pp.859-869, May 1974.</ref>, which is based on Newton-Raphson, but greatly reduces its computational cost by means of a decoupling approximation that is valid in most transmission networks. Many other incremental improvements have been published. The underlying technique in all of the existing methods remains an iterative solver, either of Gauss-Seidel or of Newton type. There are two fundamental problems with all iterative schemes of this type. On the one hand, there is no guarantee that the iteration will always converge to a solution; on the other, since the system has multiple solutions<ref group="note">It is a well-known fact that the load flow equations for a power system have multiple solutions. For a network with {{math|<var>N</var>}} non-swing buses, the system may have up to {{math|2^<var>N</var>}} possible solutions, but only one is actually possible in the real electrical system. This fact is used in stability studies, see for instance: Y. Tamura, H. Mori, and S. Iwamoto,name="Relationship Between Voltage Instability and Multiple Load FLow Solutions in Electric Power Systemsmultsol", '' IEEE Transactions on Power Apparatus and Systems'', vol. PAS-102 , no.5, pp.1115-1125, 1983.</ref>, it is not possible to control which solution will be selected. As the power system approaches the point of voltage collapse, spurious solutions get closer to the correct one, and the iterative scheme may be easily attracted to one of them because of the phenomenon of Newton fractals: when the Newton method is applied to complex functions, the basins of attraction for the various solutions show fractal behavior<ref>[[Newton's_method#Complex_functions]]</ref>. As a result no matter how close the chosen initial point of the iterations (seed) is to the correct solution, there is always some non-zero chance of straying off to a different solution. These fundamental problems have been illustrated for the two-bus model<ref>[http://www.elequant.com/products/agora/demo/iterativeloadflow/ Problems with Iterative Load Flow], Elequant, 2010.</ref>. Although there exist homotopic continuation techniques that alleviate the problem to some degree<ref>V. Ajjarapu and C. Christy, "The continuation power flow: A tool for steady state voltage stability analysis", ''IEEE Trans. on Power Systems'', vol.7, no.1, pp. 416-423, Feb 1992.</ref>, the fractal nature of the basins of attraction precludes a 100% reliable method.

The key differential advantage of the HELM is that it is fully deterministic and unambiguous: it guarantees that the solution always corresponds to the correct operative solution, when it exists; and it signals the non-existence of the solution when the conditions are such that there is not any solution (voltage collapse). Additionally, the method is competitive with the FDNR method in terms of computational cost. It brings a solid mathematical treatment of the load-flow problem that provides new insights not previously available with the iterative numerical methods.

The patented HELM load- flow algorithm was invented by Antonio Trias<ref>

{{cite patent

| country = US

Line 26 ⟶ 41:

}}</ref>. It is implemented as industrial-strength real time and off line packaged [[Energy_management_system|EMS]] applications for management and analysis.

= Background =

The load-flow problem, also known as [[Power_flow_study|power flow]], computes the steady state of three-phase balanced AC power networks with complex injections expressed totally or partially in terms of power.

Traditional load-flow algorithms were developed based on three foundational approaches: the Gauss-Seidel method<ref>J. B. Ward and H. W. Hale, "Digital Computer Solution of Power-Flow Problems," ''Power Apparatus and Systems, Part III. Transactions of the American Institute of Electrical Engineers'', vol.75, no.3, pp.398-404, Jan. 1956.

* A. F. Glimn and G. W. Stagg, "Automatic Calculation of Load Flows," ''Power Apparatus and Systems, Part III. Transactions of the American Institute of Electrical Engineers'', vol.76, no.3, pp.817-825, April 1957.

* Hale, H. W.; Goodrich, R. W.; , "Digital Computation or Power Flow - Some New Aspects," ''Power Apparatus and Systems, Part III. Transactions of the American Institute of Electrical Engineers'', vol.78, no.3, pp.919-923, April 1959.</ref>, which has poor convergence properties, but very little memory requirements and it is straightforward to implement; the full Newton-Raphson method<ref>W. F. Tinney and C. E. Hart, "Power Flow Solution by Newton's Method," ''IEEE Transactions on Power Apparatus and Systems'', vol. PAS-86, no.11, pp.1449-1460, Nov. 1967.

* S. T. Despotovic, B. S. Babic, and V. P. Mastilovic, "A Rapid and Reliable Method for Solving Load Flow Problems," ''IEEE Transactions on Power Apparatus and Systems'', vol. PAS-90, no.1, pp.123-130, Jan. 1971.</ref>, which has fast (quadratic) iterative convergence properties, but it is computationally costly; and the Fast Decoupled Load-Flow (FDLF) method<ref name="FDLF">B. Stott and O. Alsac, "Fast Decoupled Load Flow," ''IEEE Transactions on Power Apparatus and Systems'', vol. PAS-93, no.3, pp.859-869, May 1974.</ref>, which is based on Newton-Raphson, but greatly reduces its computational cost by means of a decoupling approximation that is valid in most transmission networks. Many other incremental improvements have been published. The underlying technique in all of the existing methods remains an iterative solver, either of Gauss-Seidel or of Newton type. There are two fundamental problems with all iterative schemes of this type. On the one hand, there is no guarantee that the iteration will always converge to a solution; on the other, since the system has multiple solutions<ref group="note">It is a well-known fact that the load flow equations for a power system have multiple solutions. For a network with {{math|<var>N</var>}} non-swing buses, the system may have up to {{math|2^<var>N</var>}} possible solutions, but only one is actually possible in the real electrical system. This fact is used in stability studies, see for instance: Y. Tamura, H. Mori, and S. Iwamoto,"Relationship Between Voltage Instability and Multiple Load FLow Solutions in Electric Power Systems", '' IEEE Transactions on Power Apparatus and Systems'', vol. PAS-102 , no.5, pp.1115-1125, 1983.</ref>, it is not possible to control which solution will be selected. As the power system approaches the point of voltage collapse, spurious solutions get closer to the correct one, and the iterative scheme may be easily attracted to one of them because of the phenomenon of Newton fractals: when the Newton method is applied to complex functions, the basins of attraction for the various solutions show fractal behavior<ref>[[Newton's_method#Complex_functions]]</ref>. As a result no matter how close the chosen initial point of the iterations (seed) is to the correct solution, there is always some non-zero chance of straying off to a different solution. These fundamental problems have been illustrated for the two-bus model<ref>[http://www.elequant.com/products/agora/demo/iterativeloadflow/ Problems with Iterative Load Flow], Elequant, 2010.</ref>. Although there exist homotopic continuation techniques that alleviate the problem to some degree<ref>V. Ajjarapu and C. Christy, "The continuation power flow: A tool for steady state voltage stability analysis", ''IEEE Trans. on Power Systems'', vol.7, no.1, pp. 416-423, Feb 1992.</ref>, the fractal nature of the basins of attraction precludes a 100% reliable method.

The key differential advantage of the HELM is that it is fully deterministic and unambiguous: it guarantees that the solution always corresponds to the correct operative solution, when it exists; and it signals the non-existence of the solution when the conditions are such that there is not any solution (voltage collapse). Additionally, the method is competitive with the FDNR method in terms of computational cost. It brings a solid mathematical treatment of the load-flow problem that provides new insights not previously available with the iterative numerical methods.

= Methodology and Applications =

HELM is grounded on a rigorous mathematical theory, and in practical terms it could be summarized as follows:

# Define ana specific (holomorphic) embedding for the equations in terms of a complex parameter {{math|<var>s</var>}}, such that for {{math|<var>s</var>{{=}}0}} the system has an obvious correct solution, and for {{math|<var>s</var>{{=}}1}} one recovers the original problem.

# ItGiven this holomorphic embedding, it is now possible to compute univocally power series for voltages as analytic functions of {{math|<var>s</var>}}. The correct load-flow solution at {{math|<var>s</var>{{=}}1}} will be obtained by analytic continuation of the known correct solution at {{math|<var>s</var>{{=}}0}}.

# Perform the analytic continuation using algebraic approximants, which in this case are guaranteed to either converge to the solution if it exists, or not converge if the solution does not exist (voltage collapse).

Line 50 ⟶ 55:

= Holomorphic Embedding =

Consider the following general form for the load-flow equations:

{{NumBlk|:|<math>\sum_k Y_{ik} V_k + Y_i^{\text{sh}} V_i = \frac{S_i^*}{V_i ^*}</math>|{{EquationRef|1}}}}

where the given (complex) parameters are the admittance matrix {{math|<var>Y_ik</var>}}, the bus shunt admittances {{math|<var>Y_i</var>^sh}}, and the bus power injections {{math|<var>S_i</var>}}. For the purposes of the discussion, we will omit the treatment of controls, but the method can accommodate all types of controls. TheFor methodthe usesconstraint an embeddingequations techniqueimposed by meansthese ofcontrols, an appropriate holomorphic embedding amust complexbe parameteralso {{math|<var>s</var>}}defined.

The method uses an embedding technique by means of a complex parameter {{math|<var>s</var>}}.

The first key ingredient in the method lies in requiring the embedding to be holomorphic, that is, that the system of equations for voltages {{math|<var>V</var>}} is turned into a system of equations for functions {{math|<var>V(s)</var>}} in such a way that the new system defines {{math|<var>V(s)</var>}} as holomorphic functions (i.e. complex analytic) of the new complex variable {{math|<var>s</var>}}. The aim is to be able to use the process of analytical continuation which will allow the calculation of {{math|<var>V(s)</var>}} at {{math|<var>s</var>{{=}}1}}. Looking at equations ({{EquationNote|1}}), a necessary condition for the embedding to be holomorphic is that {{math|<var>V^*</var>}} is replaced under the embedding with {{math|<var>V^*(s^*)</var>}}, not {{math|<var>V^*(s)</var>}}. This is because complex conjugation itself is not a holomorphic function. On the other hand, it is easy to see that the replacement {{math|<var>V^*(s^*)</var>}} does allow the equations to define a holomorphic function {{math|<var>V(s)</var>}}. However, for a given arbitrary embedding, it remains to be proven that {{math|<var>V(s)</var>}} is indeed holomorphic. Taking into account all these considerations, an embedding of this type is proposed:

{{NumBlk|:|<math>\sum_k Y_{ik} V_k(s) + Y_i^{\text{sh}} V_i(s) = s\frac{S_i^*}{V_i ^*(s^*)}</math>|{{EquationRef|2}}}}

With this choice, at {{math|<var>s</var>{{=}}0}} the systemright becomeshand theside load-flowterms equationsbecome forzero, the(provided casethat wherethe alldenominator intensitiesis arenot zero.), Thethis operationalcorresponds solutionto for thisthe case being, as is well known, thatwhere all the voltages are equal.

the injections are zero and this case has a well known and simple operational solution: all voltages are equal and all flow intensiti es are zero. Therefore this choice for the embedding provides at s=0 a well known operational solution.

Now using classical techniques for variable elimination in polynomial systems<ref>B. Sturmfels, "Solving Systems of Polynomial Equations”, CBMS Regional Conference Series in Mathematics 97, AMS, 2002.</ref> (results from the theory of [[Resultants|Resultants]] and [[Groebner_basis#Elimination_property|Gröbner basis]] it can be proven that equations ({{EquationNote|2}}) do in fact define {{math|<var>V(s)</var>}} as holomorphic functions. More significantly, they define {{math|<var>V(s)</var>}} as [[Algebraic_curves|algebraic curves]]. It is this specific fact, which becomes true because the embedding is holomorphic that guarantees the uniqueness of the result. The solution at {{math|<var>s</var>{{=}}0}} determines uniquely the solution everywhere (except on a finite number of branch cuts), thus getting rid of the multi-valuedness of the load-flow problem.

The technique to obtain the coefficients for the power series expansion (on {{math|<var>s</var>{{=}}0}}) of voltages {{math|<var>V</var>}} is quite straightforward once one realizes that equations ({{EquationNote|2}}) can be used to obtain them, order by order, by using the power series expansion for {{math|<var>1/V</var>}}, since their coefficients are related by the convolution formulas derived from the following identity:

Line 68 ⟶ 74:

1 & = V(s)V^{- 1} (s) \\

& = \left(\sum_{n=0}^\infty a_n s^n\right) \left(\sum_{n = 0}^\infty b_n s^n\right) \\

& = a_0 b_0 + \left(\sum_{k=0}^1 a_{1 - k} b_k\right) s + \left(\sum_{k=0}^2 a_{1 2- k} b_k\right) s^2 + \cdots + \left(\sum_{k = 0}^n a_{n - k} b_k\right) s^n + \ldots

\end{align}

</math>