Holomorphic Embedding Load-flow method: Difference between revisions

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The '''Holomorphic Embedding Load-flow Method (HELM)''' is a solution method

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

for the [[Power_flow_study|power flow]] equations of electrical power

systems. Its main features are that it is

[[Direct_method_(computational_mathematics)|direct]] (that is,

non-iterative) and that it mathematically guarantees a consistent

selection of the correct operative branch of the multivalued problem,

also signalling the condition of voltage collapse when there is no

solution. These properties are relevant not only for the reliability

of existing off-line and real-time applications, but also because they

enable new types of analytical tools that would be impossible to build

with existing iterative load flows (due to their convergence

problems). An example of this would be decision-support tools

providing validated action plans in real time.

The HELM load flow algorithm was invented by Antonio Trias and has

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

been granted two US Patents<ref>

{{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>}}.

To solve this non-linear system of equations numerical iterative schemes may be applied such as 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>.

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.

= 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" name="multsol" />, 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 35 ⟶ 29:

| gdate = 2009-04-14

| invent1 = Antonio Trias

}};

* {{cite patent

| country = US

| number = 7979239

Line 43 ⟶ 37:

| gdate = 2011-07-12

| invent1 = Antonio Trias

}}</ref>. A detailed description was presented at the 2012 IEEE PES General

| assign1 = Aplicaciones en Informatica Avanzada, S.A.

Meeting, and published in <ref name="helmpaper">A. Trias, "The Holomorphic Embedding

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

Load Flow Method", ''IEEE Power and Energy Society General Meeting 2011'', 22-26 July 2012.</ref>.

The method is founded on advanced

concepts and results from Complex Analysis, such as holomorphicity, the

theory of Algebraic Curves, and Analytical Continuation. However, the

numerical implementation is rather straightforward as it uses standard

linear algebra and Padé approximation. Additionally, since the

limiting part of the computation is the factorization of the

admittance matrix and this is done only once, its performance is

competitive with established fast-decoupled loadflows. The method is

currently implemented into industrial-strength real-time and off-line

packaged EMS applications.

= Background =

The [[Power_flow_study|load-flow]] calculation is one of the most

fundamental components in the analysis of power systems and is the

cornerstone for almost all other tools used in

[[Power_system_simulation|power system simulation]] and

[[Energy_management_system|management]]. The load-flow equations can

be written in the following general form:

{{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>}} representing

constant-power loads and generators.

To solve this non-linear system of algebraic equations, traditional

load-flow algorithms were developed based on three iterative

techniques: 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 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 exist; however, the

underlying technique in all of them is still 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" 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>, 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 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>. 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 of iterative

loadflows have been extensively documented

<ref>R. Klump and T. Overbye, “A new method for finding low-voltage power flow solutions", ''in IEEE 2000 Power Engineering Society Summer Meeting,'', Vol. 1, pp. 593-–597, 2000.

* J. S. Thorp and S. A. Naqavi, "Load flow fractals", ''in Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 2, pp. 1822--1827, 1989.

* J. S. Thorp, S. A. Naqavi, and H. D. Chiang, "More load flow fractals", ''in Proceedings of the 29th IEEE Conference on Decision and Control, Vol. 6, pp. 3028--3030, 1990.

* S. A. Naqavi, ''Fractals in power system load flows'', Cornell University, August 1994.

* J. S. Thorp, and S. A. Naqavi, S.A., "Load-flow fractals draw clues to erratic behaviour", IEEE Computer Applications in Power, Vol. 10, No. 1, pp. 59--62, 1997.

* H. Mori, "Chaotic behavior of the Newton-Raphson method with the optimal multiplier for ill-conditioned power systems", in ''The 2000 IEEE International Symposium on Circuits and Systems (ISCAS 2000 Geneva), Vol. 4, pp. 237--240, 2000.

</ref>. A simple

illustration for the two-bus model is provided in

<ref>[http://www.elequant.com/products/agora/demo/iterativeloadflow/ Problems with Iterative Load Flow], Elequant, 2010.</ref>. Although

there exist [[Homotopy|homotopic]]

[[Numerical_continuation|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 for all electrical

scenarios.

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 no 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 a 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.

Line 89 ⟶ 186:

so that the right-hand side in ({{EquationNote|2}}) can always be calculated from the solution of the system at the previous order. Note also how the procedure works by solving just [[System_of_linear_equations|linear systems]], in which the matrix remains constant.

A more detailed discussion about this procedure is offered in Ref. <ref>A. Trias, name="The Holomorphic Embedding Load Flow Method as an enabler of reliable real-time applicationshelmpaper", ''IEEE Peprint'', 2011.</ref>.