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Synthetic Dynamics Power Flow

J. Jardim, Senior Member IEEE and B. Stott, Fellow IEEE

Abstract—This paper introduces an interesting and promis-ing methodology for solving difficult power flow problems. The Synthetic Dynamics Power Flow (SDPF) approach is based on the simple principle that the dynamic response of a stable power system always decays to a stable equilibrium point. We there-fore create and solve a stable artificial dynamic model whose quiescent state is the solution of the desired power flow problem. In theory, the method should converge any steady-state-stable power flow model, and it should never produce unstable solu-tions. Experience to date with a prototype is encouraging, and further research and development seem warranted. At present, the method is computationally rather slow. Its potential value is on problems where standard power flow methods fail or might converge to unstable points.

Index Terms—Electric power systems, network analysis, power flow calculations, nonlinear simultaneous equations, dy-namic stability, synthetic dynamic modeling, reliable conver-gence.

I. NOMENCLATURE

x system state variables, principally bus voltages V/θ

y system decision variables: generator voltage magnitudes and angles, and controlled quantities such as transformer taps T, phase shifter angles φ, and shunt admittances B

n integration step number

h integration step length

K gain constant

′ symbol for first derivative w.r.t. time, e.g. y' = dy/dt

sp subscript indicating "specified" target or limit value

II. INTRODUCTION

After half a century of intensive research, development and practical application, the power flow calculation often re-mains difficult. When a power flow program fails to produce a solution, it is rarely clear whether the program/method is deficient or whether the system as modeled has no stable so-lution. Conversely, standard power flow methods sometimes converge to steady-state-unstable solution points, and this dangerous situation is usually difficult to detect without fur-ther analysis.

It would therefore be of considerable practical value to have a strong power flow solution method that reliably solves stable problems and identifies unstable ones. This question is ad-dressed by the present Synthetic Dynamics approach.

J. L.A. Jardim is with ONS, the operator of the national power system, in Rio de Janeiro, Brazil B. Stott is a consultant in Scottsdale, Arizona, USA, working principally with Nexant, Inc

A. The Nature of the Power Flow Problem

The power flow problem is deceptively complicated. Any formal analysis of its solvability and solution uniqueness has thus far been impossible, except with grossly simplified mod-els. Since this paper presents a new solution method, it is appropriate to review some of the associated issues.

The power flow model is characterized by nonlinear equa-tions that are differentiable and smooth in between numerous (in a large power system) points of discontinuity. Such dis-continuities are mostly due to the modeled limits, target ranges, dead bands, priorities, impedance variations, relative response rates and discreteness of the automatic MW and voltage controls.

The idea that a power flow calculation produces a "snapshot" of system operation is largely an illusion. In fact, standard power flow data tells nothing about the characteristics of the system's automatic controls that in real life make them inter-act in harmony with each other. Rather, such data routinely specifies controls that are mathematically incompatible, inde-terminate, or mutually very sensitive. This provokes conver-gence difficulties and solution non-uniqueness.

In short, the power flow problem formulation is inherently "fuzzy" and, more often than not, it is ill posed. As a result, the developer of each specific power flow program faces multiple choices regarding the exact interpretations and im-plementations of the power flow controls and other modeling features. Each program ends up with its own unique combi-nation of modeling subtleties and algorithm heuristics that will include (a) tunable solution-guiding parameters, (b) automated decisions (mostly transparent to the user) on the rejection, prioritization, sharing, etc. of ambiguous and/or conflicting controls, and (c) a variety of other arbitrary mod-eling assumptions, mostly concerning the controls and their limiters.

Adding to the above the fact that most algorithms are strongly sensitive to the solution starting point, it is inevitable that no two power flow programs will routinely produce identical solutions or have equal convergence reliability, except on trivial problems.

B. Synthetic Dynamics Power Flow (SDPF) Method

This paper describes a new power flow approach with a very attractive property—its apparent ability to obtain a solution whenever the power flow problem is steady-state stable, and not otherwise.

As is well known, the dynamic response of a stable power system always decays to a stable equilibrium point. The new power flow method therefore solves a synthetic dynamic sta-

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bility model whose quiescent state is the solution of the re-quired power flow problem. The "dynamics" are explicitly designed to be stable, and they are highly damped to encour-age fast asymptotic convergence to the equilibrium point with little or no oscillation. For want of a better name, we call the method Synthetic Dynamics Power Flow (SDPF).

The basic idea originated in the power flow literature many years ago [1]. Although the implementation was claimed to have exceptionally high convergence reliability, it attracted little attention [2] and no apparent follow-up. More recently, the idea was resurrected and re-developed by the first author of this paper, in conjunction with his work on dynamic stabil-ity and power flow [3].

It turns out that the specific method of [1] is quite weak. It uses undamped second-order dynamics to mimic the behavior of synchronous generators. Despite the introduced artificial damping, its dynamic response is inherently oscillatory, does not guarantee stability, and is not conducive to efficient nu-merical integration. In addition, the method's partitioned ap-proach to the numerical solution of the problem's mixed dif-ferential-algebraic equations requires small integration steps to avoid large errors and numerical instability. No automatic controls other than generator VAR limits were modeled.

The present SDPF method uses first-order dynamics and is much more general and powerful. The same class of methods has been used for solving other types of simultaneous nonlin-ear equations. A prime example is the work of Branin [4]. This is by no means identical to the new SDPF method, but it shares similar properties. To quote from the abstract of [4]:

A new method has been developed for solving a system of nonlinear equations g(x) = 0 . This method is based on solving the related system of differential equations …… This procedure endows the method with a much wider re-gion of convergence than other methods (occasionally, even global convergence) …… The principal limitations of the method relate to the extraneous singularities of the differen-tial equations.

The problems of main interest to Branin were quite nonlinear with severe singularities in the equations themselves. It would then seem that such a class of methods should be even more successful on the less nonlinear power flow problem.

III. SDPF METHOD OUTLINE

A. General Expression of the Power Flow Problem

The problem can be expressed as the set of equalities:

g ( x, y ) = 0 (1)

and the inequalities:

ma ( y ) < 0 (2a)

mb ( x, y ) < 0 (2b)

Set (1) contains the system's continuous power flow and con-trol equations in terms of decision variables y and state vari-ables x. Set (2a) comprises simple limits on decision vari-ables such as controlled transformer taps and angles, and

shunt admittances. Set (2b) comprises functional inequalities such as generator VAR limits. The enforcement or release (limit back-off) of any of the inequalities can swap control equations in and out of set (1).

B. Synthetic Dynamic Approach

Let us synthesize a dynamic equation for each decision vari-able:

y′ = z ( x, y ) (3)

such that the quiescent state is the power flow solution.

Let us then perform a time domain solution of the resulting dynamic system using a stable, closed (implicit) step-by-step integration method. Thus, at integration step n, the generic form of the solution of (3) is:

yn = β. h. z ( xn, yn ) + C (4)

where β is a constant coefficient, and C is a term that usually includes previous values of y or y′.

In this time domain solution, we can rewrite (1) and (4) re-spectively as:

f ( xn, yn ) = yn - β. h. z ( xn, yn ) - C = 0 (5)

and

g ( xn, yn ) = 0 (6)

Equations (5) and (6) together comprise a nonlinear simulta-neous set in the variables ( xn, yn ) whose solution is the sys-tem state at the end of integration step n. A predictive for-mula is used to provide good starting values for xn and yn.

In this application, the concept of time t is notional. The so-lution starts from t = 0 and a set of initial variable values. It proceeds in steps of length h in the usual dynamic stability solution manner [5].

C. Solution of Equations

At each time step, the set of nonlinear simultaneous equa-tions (5) with (6) is solved iteratively by Newton's method to provide the current state of the system. The form of the Jacobian equation at each iteration is:

∂f/∂u

∂f/∂u

∂f/∂u

∂f/∂u

∆u

∆y

∂f / ∂y

∂f / ∂x

∂g / ∂x

∂g / ∂y

∆y

∆x

f

g

• = -

Fig. 1: The Jacobian Equation

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The solution of the Jacobian equation by sparse triangular factorization produces the corrections ∆x and ∆y to the re-spective variables.

Until the quiescent condition is reached, there is no need for high accuracy convergence. We are not interested in the tra-jectory of the dynamic response, but only the eventual equi-librium state. The use of a very high stability integration method with step control (see section V.D) limits the error propagation.

IV. SYNTHESIS OF A STABLE DYNAMIC MODEL

Ideally, the dynamic model in (3) must be designed to guar-antee asymptotic convergence to the system equilibrium point, if the latter exists. Because the system is nonlinear we can base the design on the Multi-Input-Multi-Output Circle Criterion Theorem [6]. Then, assuming that the system nonlinearities satisfy the sector condition, a simple integral control for every controlled variable is a conservative ap-proach to guarantee global convergence. In the case of a sin-gle-input-single-output nonlinear control system, satisfying the sector condition means basically that the nonlinearity (e.g., power versus angle characteristic) belongs to the [0,K] sector.

It can easily be seen that the above is true for a simple two-generator model with no MVAR limits. Fig. 2 depicts the MW flow versus angle between the generators. Considering the origin at the stable equilibrium point (Ps, δs), the system nonlinearity (Pe) clearly belongs to the [0,K] sector in the range [δu+2π, δs]. But, any δ> δu or δ< δu+2π is also in this range because δ belongs to a cylindrical subspace. Addition-ally, the nonlinearity around (Ps, δu) does not belong to the [0,K] sector, characterizing it as unstable for the proposed control system. Thus the simulated trajectories will move apart from it.

Fig. 2: Two-Generator P x δ Characteristic

For a multi-machine system with hard nonlinearities it is dif-ficult, if not impossible, to prove that the same property holds globally. On the other hand, the existence of a stable equilib-rium point indicates that the nonlinearity satisfies the sector condition in a neighborhood of the equilibrium point. Then the integral control model guarantees asymptotic stability for the region of attraction of the equilibrium point.

Because of the model's very high stability properties, the val-ues of the gains in the integrator equations are not critical unless there are control conflicts. The method works with wide variations in their values. The objective is to super-damp the system's dynamic behavior, since it runs no risk of "synchronous instability" (using the transient stability anal-ogy). In fact, even if a "machine" pole slips, it will eventu-ally resynchronize if there is an equilibrium point. At the same time, there is considerable scope for investigating the values of gains that produce the smoothest, fastest converg-ing, overall dynamic response.

V. SDPF METHOD DETAILS

A. Dynamic Modeling

This section describes the synthetic dynamic modeling ap-proach for the most common power flow components. As explained above, each controlled power system device is rep-resented by a simple integrator model, involving a feedback controller acting to maintain a quantity such as MWs or volt-age at its specified value.

1. Generators

Here, there are two main choices for the synthetic dynamics:

Model 1: The terminal voltage components θ and V become the decision variables.

Model 2: The generator is represented by a voltage behind an arbitrary reactance (which can be very small). The inter-nal voltage components δ and E become the decision vari-ables. Because the present implementation is an adaptation of an existing dynamic stability program, it was more convenient to use this model.

The active power dynamics for a generator with MW output P(x,y) are then represented by the differential equation:

θ′ = Kp ( Psp - P(x,y) ) (7a) or

δ′ = Kp ( Psp - P(x,y) ) (7b)

This ensures that, when the dynamic system has reached equilibrium, the generator has the specified MW output.

Similarly, the reactive power dynamics are represented by:

V′ = Kv ( Vsp – Vc ) (8a) or E′ = Kv ( Vsp – Vc ) (8b)

where Vc is the controlled voltage, whether at the generator terminal or at a remote bus.

If however the generator is operating at a fixed VAR output Qsp (e.g. it is on a VAR limit), eq.(8) is replaced by:

V′ = Kq ( Qsp – Q(x,y) ) (9a) or E′ = Kq ( Qsp – Q(x,y) ) (9b)

sδ uδ

Pe Ps

δ

P

πδ 2+u

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2. LTC Transformers

An LTC transformer with tap T controlling a terminal or re-mote bus voltage V has the differential equation:

T′ = Kt ( Vsp – V ) (10)

3. Controlled Phase Shifters

A phase shifter whose angle φ regulates the MW flow P through itself becomes:

φ′ = Kφ ( Psp - P(x,y) ) (11)

4. Controlled Shunt Reactances

A variable shunt admittance B controlling a bus voltage V becomes:

B′ = Kb ( Vsp – V ) (12)

5. Other Controls

The above approach can be applied to any other controlled devices, such as SVCs, controlled series reactors, FACTS, and HVDC links, as well as to system-wide controls such as area interchange. Special modeling features can readily be incorporated. For example, the generator VAR limits as above can be continuous or piecewise, and functions of volt-age. Control targets, dead bands, etc. can be interpreted and incorporated as desired.

6. Modeling Options

The approach offers considerable flexibility for variants, since in principle a given control can be modeled either dy-namically or it can be included in set (6) in the usual power flow manner (i.e. incorporated into the Jacobian equations, or adjusted in between integration steps). Generally, dynamic modeling seems to be the most effective.

The same applies even to non-controls. Consider for example the modeling of loads. We could represent their voltage components as decision variables in the manner of (7) and (9). At present, however, we have chosen to incorporate loads in the normal Newton manner in the power flow equations (6). Another example of a non-control that could be modeled dynamically is distributed slack. The relative merits of the modeling approaches need to be studied.

B. Nonlinear Equation Solutions

At each integration step, equations (5) and (6) are solved si-multaneously, using the Jacobian matrix equation of Fig. 1. Vectors x and g in Fig 1 represent the voltages and mis-matches respectively of the network buses. Therefore ∂g/∂x is the familiar Newton power flow Jacobian matrix.

The following refers to our implementation using generator model 2. The situation with model 1 is somewhat different, since the generator mismatch equations in (6) will be re-placed by their counterparts from (5).

Vector y contains all the decision variables δ and E, and T, φ, B, etc. Each element of vector f corresponds to (5) with the right-hand side of an equation such as (7) - (12) substituted for function z.

The matrix rows and columns of Fig. 1 that pre-border the Newton power flow Jacobian matrix ∂g/∂x are extremely sparse and their diagonal elements are never zero. Therefore the factorization of each such row and column is trivial, and except for certain remote controls no matrix fill-in occurs. An option would be to construct only the standard power flow Jacobian equation, and embed into it the factor terms from the pre-bordered rows and columns. As desired, inci-dence symmetry can be preserved by adding dummy zeros.

Since the step-by-step time domain "simulation" proceeds in relatively small steps, each solution based on Fig. 1 con-verges very quickly. Frequently only one iteration is needed. In fact, the Jacobian matrix factors are often reused (dishon-est Newton method). However, if convergence is slow or not obtained, the matrix is reformed and refactorized, and/or the step length is reduced.

C. Inequality Constraints

The inequalities (2) are tested at the end of each integration step. When a differential variable subject to (2a), such as in (10) - (12), violates its limit it is simply pegged at the limit. When a functional inequality (2b) is violated, an equation swap takes place. For example, when a PV generator VAR limit is violated, eq.(9) is substituted for eq.(8). The normal power flow rules for limit back off apply.

D. The Integration Method

The present implementation uses predictor-corrector Back-ward Difference Formulas [7] with automatic step-length control. As indicated in Section III.B, the correctors are im-plicit. They are stiffly stable, basically meaning that they can handle high-frequency oscillations without resorting to rela-tively very small time steps. The step-length control is par-ticularly important because (a) it carefully avoids numerical instability, and (b) it can expand the step length h when the dynamic response smoothes out, to reduce the overall compu-tation.

VI. SDPF METHOD PERFORMANCE

The method has been prototyped by adapting the relevant parts of an existing industrial dynamic stability program [3]. The program reads several standard ASCII power flow data file formats, and is self-dimensioning to handle any size of power system. The SDPF program has been run with a lot of power flow data and has been subjected to some very severe tests. It has converged in almost all cases where a solution is known to exist (there were some problems with extreme starts such as all voltages zero, but even this may be an implementation issue). In this section, we provide a few illustrations.

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A. Test Systems

Table 1 summarizes the test systems, which are all real cases. "Brazil" signifies the high voltage network of Brazil. "N.A. West" and "N.A. East" respectively signify the Western and Eastern Interconnections of North America. "PARs" indi-cates controlled phase angle regulators (phase shifters).

TABLE 1 TEST SYSTEMS

Brazil N.A.

West N.A. East

Buses 3,410 9,146 40,020

Generators 289 1,336 5,006

LTC transformers 481 901 4,488

Controlled PARs 0 13 41

Controlled shunts 11 54 226

B. Examples

On most large problems, a conventional power flow program will only converge with a good starting point. Most pro-grams use a previous solution or a non-iterative starting proc-ess (e.g. one fast decoupled iteration). But in many difficult cases, even this does not work.

To test the reliability of the SDPF program, we have used three other power flow programs as benchmarks. Two of these are very widely used industrial power flow programs (which have to be nameless), and one is the Newton power flow of [3], which is usually more reliable than the other two.

Table 2 demonstrates the remarkable convergence powers of the prototype SDPF method and program. In the table, "Benchmark Programs" refers to the three power flow pro-grams mentioned above. In the "starting point" column:

• "Near" means starting from an existing solution or a non-iterative starting process (this varies).

• "Flat" means starting with each bus voltage and LTC transformer tap at 1 per-unit and each bus and PAR angle at zero

• "Extreme" means that successive buses are given starting voltage angles of 0, 180, 0, 180 and so on, with all volt-ages and LTC taps at 1 per-unit.

A solution is signified by � and solution failure is signified

by �. On each test system, none of the benchmark programs solve except from a starting point that is near to the solution. The SDPF method converges every case. C. Trajectories

Fig. 3 is a plot of the bus voltages against integration "time" following an Extreme start on the Brazilian system. The starting point subjects the system to an enormous shock, and

some voltages initially sink near to zero while others rise to near 2 per-unit.

TABLE 2

TESTS FROM DIFFERENT STARTING POINTS

starting

point Brazil N.A. West

N.A. East

Near � � �

Flat � � �

Benchmark Programs

Extreme � � �

Near � � �

Flat � � �

SDPF Program

Extreme � � �

Most of the voltages are super-damped, and they decay rap-idly and monotonically to their solution values. Several volt-ages stabilize slowly—the reason and a cure for this are under investigation. The bus angles behave similarly.

Since the SDPF method does not have quadratic conver-gence, it is at a disadvantage compared with a Newton power flow when starting near to the solution. That is, SDPF still takes more than a few integration steps to reach an accurate steady state.

Fig. 3: An Extreme Voltage-Versus-Time Plot

D. Computing Efficiency

The SDPF method as prototyped is not competitively fast, because of the number of integration steps it requires. For a very rough comparison, let us assume that, when it solves, a Newton power flow will take something like 15-20 iterations on a large power flow problem (because of the control limit-ing activity). Each such iteration requires a Jacobian matrix factorization. On the same problem, the SDPF method will typically take between 50 and 150 integration steps. About half these steps require Jacobian matrix factorizations, and the others only need repeat solutions. So then the SDPF solu-tion effort is equivalent to between two and five Newton solutions. For difficult problems that Newton cannot solve, SDPF can sometimes take double the above number of steps.

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E. Comments

We have consistently observed over a period of time that the convergence power of the SDPF method is exceptional. A simple explanation draws on the analogy with physical power system dynamics. Thus, if our artificial dynamic model is mathematically stable, and our solution method is numeri-cally stable, then we expect a quiescent state to be reached if the power flow model is stable.

When there is no stable solution point, the method has the highly desirable characteristic that the dynamic response fol-lows a trajectory that transits the "nearest" stable point and then veers away. This is based on manual observation and comparison with solutions using continuation methods. This nearest point can be identified and captured via an index such as the norm of the decision variables derivatives or of the mismatches. This feature remains to be implemented.

The strength of the approach manifests itself in its ability to continually recover from the "shocks" of large and small dis-continuities such as generator VAR limiting1. Thus, the solu-tion process simply "rides out" step function changes that can upset a power flow method like Newton.

VII. CONCLUSIONS

The SDPF approach is novel and in principle it is very straightforward. All existing tests indicate that it has un-precedented convergence powers, and it shares the property of a power system dynamic stability simulation that it will always gravitate to a stable system operating point, if this exists. Otherwise, the algorithm naturally homes in on (and then departs from) the nearest point to stability, which needs to be identified and captured.

One of the main prerequisites for a usable power flow method is that it should easily handle a wide variety of control mod-els and discontinuities. The flexibility of the SDPF approach seems to excel in this respect.

The SDPF implementation presented here is an advanced prototype that demonstrates the properties and potential of the approach. More development and testing are clearly needed, and seem to be merited. This work is ongoing, with the pri-mary aim of reducing the number of integration steps and incorporating more models. The theoretical basis for the ap-proach is intriguing, and offers scope for further work

On present evidence, an SDPF-based method is not a general alternative to conventional power flow methods, if only be-cause it is not as fast. On the other hand, it should be attrac-tive for routinely difficult problems or as a backup when the primary power flow method fails. In particular, it may find a valuable role in static voltage stability monitoring, where concern about the reliability of the solution is at its greatest.

.

1 After all, dynamic simulation in a real power system readily handles huge discontinuities such as system short circuits

VIII. REFERENCES [1] R.H. Galloway, J. Taylor, W.D. Hogg, M. Scott “New Approach to

Power-System Load-Flow Analysis in a Digital Computer”, Proc. IEE, Vol 117, No 1, January 1970, pp. 165-169.

[2] B. Stott, “Review of Load-Flow Calculation Methods”, Proc. IEEE, Vol. 62, No. 7, July 1974, pp. 916-929.

[3] J.L.A. Jardim, C.S. Neto, W.T. Kwasnicki, "Design Features of a Dynamic Security Assessment System", IEEE PES PSCE Conf., New York, Oct. 2005.

[4] F.H. Branin, “Widely Convergent Method for Finding Multiple Solu-tions of Simultaneous Nonlinear Equations”, IBM J. Res. Develop., Sep. 1972, pp. 504-522.

[5] B. Stott, “Power System Dynamic Response Calculations", Proc. IEEE, Vol. 67, No. 2, Feb 1979, pp. 219-241.

[6] Hassan K. Khalil, “Nonlinear Systems”, Macmillan Publishing Com-pany, New York, 1992.

[7] J. D. Lambert, "Numerical Methods for Ordinary Differential Sys-tems: The Initial Value Problem", Wiley, 1991.

IX. BIOGRAPHIES

Jorge Luiz de Araujo Jardim received his PhD from Imperial College, London, UK. He has worked in Rio de Janeiro in the FURNAS utility com-pany, and at the national electric power research center CEPEL. He is cur-rently at ONS, the operator of the Brazilian national power system. From 1999-2002 he was with BC Hydro in Vancouver, Canada. His main field of interest is power system analysis methods and software, with emphasis on voltage and dynamic stability.

Brian Stott received his PhD from the University of Manchester (UMIST), UK. He taught in universities in England, Turkey, Canada and USA. From 1976-83 he worked at CEPEL, the Brazilian power systems research center in Rio de Janeiro. From 1984-2000 he headed PCA Corp, an Arizona company that has supplied power network analysis software all over the world. Since 2000 he has been a consultant, primarily to the Energy Solutions Division of Nexant Inc. He is a recipient of the IEEE Millennium Medal.