Optimal power flows J Carpentier Electricit& de France, 3 Rue de Messine, 75008 Paris, France

The state of the art of optimal power flows at the end of 19 78 is reviewed. The different aspects of the problem statement are established in the most complete way possible; then methods for the solution of the static optimization are presented and filed, with a brief description of the most typical ones, trying to show the respective merits of each family, and with a discussion of points of special interest. Finally, dynamic aspects with applications to online dispatch are investigated.

I. In t roduct ion Before tile introduction of optimal load flows, so-called 'economic dispatch' was used to determine the best way to share the real load between several generating thermal units having a total capacity greater than the generation required. Everything was as simple as in paradise : losses were taken into account through B coefficients, control orders were sent directly to the units and nothing happened but savings.

The trouble started around 1961 when the use of networks close to their limits led to a fear of line overloadings; security constraints had to be introduced and optimal power flows were rapidly born. Instead of a 'compact' model consisting of one equation with real powers only, it appeared necessary to consider all the variables defining the state of the system and to solve the economic dispatch and the load flow problems at the same time. An optimal power flow may thus be dethaed as the determination of the complete state of a power system corresponding to the best operation within security constraints. Best operation usually means least fuel cost: security may range from the generation feasibility up to very sophisticated constraints, so that the optimization problem may become huge.

Since 1961, a number of optimal power flow computation methods have been proposed. In this paper, after presenting various possible statements of the problem, our aim is to point out the most typical methods, as they appear today with a thought towards their possible usefulness, and bearing in mind that omissions are always possible. Except for a few papers, the author compelled himself to read (or re-read) each paper mentioned in the references; so, please, forgive him if the list is too limited. Complements to references may be found in the excellent survey on optimal power dispatch carried out by Happ 6s a few years ago.

The author found writing this review very exciting for two reasons: first, looking at 15 years' work corroborated his opinion that sometimes there exist barriers between countries -- researchers from one country working on subjects solved and published elsewhere five or ten years before; secondly, the revolution in computer hardware means that cheap, fast and powerful computers will soon

appear on the market and these will certainly make a contribution in reducing the gap existing between theory and real time applications for optimal power flows. The author hopes that this paper may help in putting researchers in closer contact and in putting theory closer to real time applications.

II. Statement of the problem

II. 1 Preliminary assumptions Before stating the problem, we assume that

thermal generating units were committed (nuclear units being included among thermal units)

hydroelectric plant real generations were fixed

the transmission network was defined, at least pro- visionally.

11.2 Physical control variables In order to find the best operation within the constraints imposed, the following variables are employed:

(1) real generated powers of the thermal generating units

(2) voltages (or reactive generated powers) of the thermal and hydroelectric units and of the synchronous compensators

(3) variable transformer tap ratios for voltage magni- tude and for phase shifts if any

(4) other variable reactive power sources such as capacitors and reactances

(5) in special conditions, hydropower, emergency start-up, load shedding and even network structure changes

As a rule, we shall call these variables the 'control variables' of the problem. In general, only variables (1), (2) and (3) are considered. Let us call:

'complete' or 'real-reactive' optimal power flow a problem with the variables (1), (2) and (3) and possibly (4) and (5)

'real' optimal power flow, the suboptimization problem with the real variables (1) only, the variables (2), (3) and (4) being fixed and (5) possible,

'reactive' or 'reactive only' optimal power flow, the suboptimization problem with the reactive variables (2) and (3) [(4) possible], the variables (1) being settled (except at the slack bus to meet the real power balance).

Vol 1 No 1 April 1979 0142-0615/79/010003-13 $2.00 1979 IPC Business Press 3

11.3 Objective function The 'best operation' consists, in most cases, of minimizing the real production cost F corresponding to the fuel cost directly due to power generation, excluding no load, start-up and shut-down costs. As a rule, F is a polynomial in output power, P, usually of degree less than or equal to 3 and often represented by a piecewise linear function.

In special cases, an environmental cost function replaces the fuel cost function or is mixed with it; load curtailment may also be minimized in some emergency cases.

The considerations above are valuable for the complete and real problems. In the reactive only problem, the losses are usually minimized, but more sophisticated objective func- tions may sometimes be considered.

11.4 Security Security requirements are such that the operation is possible for generating units, transmission lines, transformers and customers - first when the system remains intact, and secondly under contingency conditions, i.e. when one or several elements of the system suddenly trip. According to Dy Liacco 2s,38 the words feasible and secure have been used in the past for the two cases just mentioned. Since the English vocabulary appears to have changed in recent years, here we shall use the usual French designations:

when operation is possible for the intact system, we shall say 'n security' is met

when operation is possible for the system under contin- gency, with k elements tripped (k ~> 1 ), we shall say 'n-k security' is met. Contingencies are usually limited to the loss of 1 element, which corresponds to the case of the 'n - I security'

These labels have the merit of being short and accurate. n-1 security (and sometimes n-k) must be considered because the sudden trip of an element may always occur before any corrective action. Security involves meeting the inequalities linking the control variables of the problems. A list of the most common security inequalities encoun- tere?, in optimal power flows is given below (n or n-k in brackets indicate the corresponding degree of security):

limits on the real power output P of a generator (n)

limits on the reactive power output Q of a generator (n)

limits on a function of these two quantities (e.g. p2 + O2)(n)

limits on transformer tap ratios (n)

limits on bus voltages (n)

limits on currents, apparent power, real power or the phase difference through a line or a transformer (n and n - 1, sometimes n-2)

limits on the generator voltages just after contingency, especially when a generator has tripped (n - l )

The major objective of security requirements is to avoid a general failure in the system, the cost of which is con- siderable: e.g. in France, the general failure of 19 December

1978 lasted only about 3 h but caused a loss of production for the country estimated to be at least the equivalent of 50 years' savings through economic dispatch.

This point leads us to two important questions concerning the problem statement: discussion of the n-1 security and the overconstrained cases. A n-1 (or n-k) security con- straint must be met only if dropping it would cause a general breakdown or at least a sufficient amount of non- delivered power. The 'overconstrained case' happens when, due to constraints, an optimal power flow has no possible solution using conventional variables of types (1) to (4). Then, it is necessary to use a special process, with type (5) variables, such as load shedding or changes in the network structure.

11.5 Extensions of the statement for static optimization In the static optimization considered so far, extensions to special cases or refinements may sometimes be present in the problem statement, e.g.:

valve point loading

sparse generating capacity

environmental considerations

corona losses

HDVC lines

multiarea operation optimization

11.6 Dynamic despatch and control Optimal power flows are of basic importance for long-range planning and daily scheduling, but their final purpose is real-time secure and optimal operation. In static optimiza- tion, real and reactive loads are the given quantities. In real time, loads change, giving a dynamic aspect to the problem.

Three ways of solving this problem may be given, in order of increasing refinement:

(1) Perform static optimal power flows periodically and use the usual control devices. This process presents an important drawback: without any addition, frequency control (or classical B coefficient dispatch) are inconsistent with the matching of line flow constraints. Similarly existing or planned voltage secondary regulations do not seem to meet the reactive constraints.

(2) Perform static optimal power flow periodically and compute a by-product as 'participation factors' allowing control devices to meet the constraints; this corresponds to following a foreseen trajectory for the loads. This process represents progress but does not allow for unforeseen changes in the loads.

(3) Perform static optimal power flows periodically and compute a by-product allowing control devices to meet the constraints corresponding to the actual loads. The latter process is not classical. We shall see later that a whole family of optimal power flow solutions gives the elements necessary to build the by-product. Processes (2) and (3) correspond to two statements of the 'dynamic optimal power flow'.

4 Electrical Power & Energy Systems

III. General features of the solutions

II I. 1 Some historical breakthroughs The history of optimal power flows may be followed in the references, which are arranged in chronological order. For optimal power flows, fundamental prehistory corresponds to economic dispatch, a basic synthesis of which was given by Kirchmayer 2 in 1958. Security was not taken into account, as also in the work of Squires 4 which certainly was the first attempt to solve load flow and economic dispatch at the same time. In 1962, n security appeared in a fundamental work from the present author, the so- called 'injections method 's was introduced, where the optimal power flow problem with security was stated, optimality relations established and solved by a fairly crude method.

Then, after some years of little activity, 1968 saw the arrival of four new important methods, appearing simul- taneously: the Dommel and Tinney reduced gradient method12; the Sasson Hessian method13; the Wells linear programming method H and the author's differential injec- tions method 14 using the generalized reduced gradient. The basic new concepts contained in these methods are still valuable today and, to the author's knowledge, the decade following was mainly dedicated to improvements both on the theoretical and practical sides, the application of the optimal power flow in real time being undertaken now.

111.2 Method families Among the various solutions reviewed, the main difference appears to be the 'compactness' of the method. We shall say a method is 'compact' if, directly or as a by-nroduct, it provides a model of the system expressed only versus the (physical) control variables defined in the problem state- ment, with security inequalities, if any, all being included in the model. Thus, the classic economic despatch is com- pact. Hessian and Dommel Tinney reduced gradient are noncompact, since their system models use all the variables. Compactness generally involves greater complexity but allows wider applications.

Starting from this viewpoint, we shall review in turn:

classic economic dispatch

general static optimization noncompact methods

general static optimization compact methods

Peculiar aspects of static optimization methods are then investigated:

special purpose extensions

discussion of the n-1 security

overconstrained cases

Finally dynamic optimal load flows and control will be considered.

111.3 Main notations and symbols We shall use the following notations and symbols:

i ~ [0, NI Vi Oi

bus number; 0 refers to the slack bus voltage magnitude at bus i voltage phase angle at bus i

Pi sum of the generated real powers at bus i, if any

Qi sum of the generated reactive powers at bus i, if any

Ci real load at bus i D i reactive load at bus i I i real power injection at bus i = Pi - - C i K i reactive power injection at bus i = Qi - Di Pik real power generated at bus is given by the

generator k (actual or fictitious for piecewise linearization)

C sum of the real loads of the system R i 'reactive generalized injection' in i, used in a

load flow process. R i may be equal to V t or K i = Qi - O i or to another quantity when men- tioned

T subscript for 'transposed'. Usually vectors without T will be column vectors

V column vector with N rows Vi: same definition for all the variables above with index i

E = W = , J = = = Jacobian (2N x 2N) ' ~E J'" J'

p = p(W) = p(E) Total real losses of the system Z, Z m, Z M Variable, minimal and maximal corresponding

values AZ variation of Z ~21 constrained quantity in the line or trans-

former 1 for the intact system; basically the current;when mentioned, apparent or real power flow, or phase angle difference

g2~ value of f21 after the element d of the system tripped

T1 upper limit for ~1 T' 1 upper limit for g2 d (usually f2 d > ~~l) pg variable transformer tap ratio, for voltage

magnitude ok variable transformer tap ratio, for phase shift u vector of the (physical) control variables of

the system (here control ~ independent) x vector of the state variables of the system,

defined as the variables different from the control variables (here state * dependent)

X : [~] vector of the variables of the system (whole set)

F = F(X) = F(x, u) objective function, usually operation total fuel cost

g(JO = 0 t g(x, u) = 0 j (physical) load flow equations

h(X) ~< 0 } h(x, u)

with

~ Pi = C + p(P) i

(2)

Basically, by a separate computation, tile losses p are expressed versus the generated real powers Pi using a quad- ratic formula p = p(P) such as:

~}p - - = 2 E BliP/+ B aPi i

(3)

Lagrange's method gives the optimality conditions:

dL=0 with L=F(P) - X(~/Pi -P C)

which gives:

aF/a& - ?, = constant

l - (~p/~iP i ) (4)

Equations (4) are the so-called 'coordination equations' which are straightforward to solve taking (3) into account. The B coefficients are computed using various approxima- tions and may be employed over a very wide region, a few sets of coefficients being sufficient for all kinds of opera- tions.

The method clearly allows dynamic dispatch, which is a very important quality and is certainly the main reason for its extensive use. Various improvements and extensions have been carried out, such as:

extension to multiarea economic dispatch 3,24,6s

improvements in the computat,ons of tile loss quadratic formula 10

direct computation of the 'first order differential !osses' OP/~Pi as proposed by Happ 48 by the transposed Jacobian:

a# Op _ jT I __ (5) OP O0

The main drawback of this simple method is the absence of security: this concept is now introduced.

V. General static optimization methods I1: non- compact secure optimal power flow methods

v. 1 General characteristics The problem statement includes inequalities, so that Lagrange's method is theoretically no longer applicable; the conditions for optimality must come either from the application of the theorem of Kuhn and Tucker I or from putting the projected gradient of a convenient function equal to zero. The noncompact methods apply these pro- cesses directly to a system model including the whole set of variables, without building an intermediate 'reduced model' limited to the control variables u. They are in general relatively easy to program and may exhibit high performance often due to the sparsity of the physical load flow equations. However, they are generally not so well

suited to accept constraints as tile compact methods and are not really practical for extension to real tinre operation.

V.2 Direct Kuhn and Tucker methods In these methods, the optimality conditions are established through the Kuhn and Tucker theorem and solved directly (whereas in all the other methods the gradient criterion is used). The earliest was the "injections' method.

V.2.1 Injections method. In tile early sixties, the following problem was presented by the author:

Minimize F(P) under the R)llowing constraints:

li(O, V) Pi + Ci : 0 i physical load flow or

Ki(O , V) Qi + Di : 0 i 'injections' relations, V i

Q~" ~ Qi

V.3 Sasson Hessian methods V.3.1 Principle. In these methods, which are able to handle the complete real reactive problem, network equations g(X) = 0 and the inequality constraints h(X)

Concerning tire use of penalty functions to handle con- straints, it is true for this reduced gradient method and for the previous Hessian method that constraints are nret in a 'soft' manner, with a relatively large tolerance zone; this is certainly acceptable in many cases but causes problems for some difficult security problems.

V.5 Peschon GRG method V.5.1 Principle. A number of other noncompact methods were proposed and it is impossible to be exhaustive. A special mention must be made of the application of the generalized reduced gradient (GRG) method 9,16 of Peschon 27 and others. Peschon, whose previous research with Tinney and others (among whom CuOnod had origin- ated the reduced gradient method just studied) presented GRG in 1971 and compared it with the Dommel Tinney method.

The GRG is a general convex programming method. Several versions exist. Peschon used the general program, which may roughly be summed up as follows.

(1) At each step, the constraints C(Z) = b are linearized,

such as:

A dZ = db = 0 (19)

(2) The linearized constraints are handled exactly as in linear programming: the variables Z are partitioned into basic (or dependent) variables Z and nonbasic (or inde- pendent) variables Z: A dZ + Ad2 = 0. The (square) basis A is inverted by recurrent formulae, e.g. the product form of the inverse, and the reduced gradient G(Z) = dF(Z)/d2 is computed:

OF bF C(2) =--z - A - '~ - : - (2o)

az az

(3) G(Z) is projected onto the feasible region to take an account of the bounds on Z, which gives G'(TZ)

[f G'(') = 0, stop. Otherwise go to (4)

(4) Compute

= -c ' (2 ) {21)

and

(22)

which defines the variation

AZ = ph

meeting the linearized constraints.

(23)

(5) p is taken as the minimum of:

P l minimizing F

P2 such that a variable Zj reaches one of its bounds

P3 such that a variable Zk reaches one of its bounds

This gives a value of Z: Zj = ZI, ZI

(6) 'Nonlinearities in the constraints are taken into account and Z" is conrputed again to meet the nonlinear constrainls. which gives a final value:

Z =7~2,2 j (23)

such that

C(Z) = C(22, Zl) = b (24)

This step uses the general Newton method to solve a set of nonlinear equations. It is called 'RDD' ('Rentrde dans le domaine' which means 'reentering tile feasible region')

(7) l fp = pj or P2 go to (2). I fp = ,P3, ,~k has reached one of its bounds. Change the partition Z, 2" and consequently the basis A, exchanging Zk for a convenient nonbasic variable 2 /and go to (2).

Moreover, to get more accuracy and speed in (2) tile reduced gradient G(2') is computed replacing F by F f,S, tt being the dual variable row vector and S the nonlinear part of C in the neighbourhood of Z. This is called Beale's correction.

This is tile genuine GRG algorithm, the principles of which were used for optimal power flows by Peschon and the present author, 14"31,3s,ss,s6,77,78 and should not be con- fused with some language extensions referring lo completely difl'erent algorithms. 37

A series of tests on convex programming carried out by Colville around 1968 showed that GRG is one of the mosl secure, high performance, general methods in convex programming.

Peschon applied tile general GRG program to the complete real-reactive optimal power flow with variable transformer tap ratios. He applied GRG to the whole problem, Z being the vector X including all tile variables. This work was done essentially fl)r demonstration, and he did not write a special GRG code but used the general form.

V.5 .2 Results and discussion. Peschon carr ied out tests on networks of up to 30 buses. For these cases, despite the fact that he used a nonspecialized GRG program, tie found that GRG was well suited to optimal power flows, giving the same performance as tile Domnrel Tinney reduced gradient. He also mentioned that with a specially written GRG code, results should be much better.

This experiment is of high interest for two reasons first its results and second the history of GRG which is rather entertaining. GRG was conceived by the present author in 1964 (the reader may, by the way, recognize here the use of Newton's method), not in order to solve a general mathe- matical program but the optimal power flow problem, the first internal name of GRG being 'differential injections'; (optimal power flow method which will be presented in the following section). In 1965, it was perceived that differ- ential injections could give a good by-product for general mathematical programming which the author called GRG and which operated at the end of 1965. At that time, Abadie joined the author and from 1966 worked alone on GRG, improving the general code, while the author, appointed in Greece, kept working on differential injections only. This

8 Electrical Power & Energy Systems

explains why the GRG general code fitted so well with opti- mal power flows in Peschon's experiment, GRG being a by-product of optimal power flow and not the reverse as may be believed. This also corroborates Peschon's opinion about the efficiency of a specialized program for two reasons: on the one hand, in the GRG-differential injec- tions algorithm, special additions and short-cuts exist which make it much more efficient; on the other hand it is applied only to the physical control variables of the problem. This shows that our field of study may sometimes be useful to more general applied mathematics; the most unfortunate thing being that circumstances prevented Peschon and the author having direct contact.

Peschon did not limit himself to the usual optimal power flows but altered the program to perform optimal load curtailment under emergency conditions, which is an appli- cation which deserves the greatest attention.

VI. General static optimization methods II1: compact secure optimal power flow methods

Vl. 1 General characteristics Compact methods are characterized by the existence of an intermediate 'reduced model' of the system expressed versus the (physical) control variables, all the constraints being expressed versus the latter variables. This reduced model is a compact representation of the whole system, valuable over a wide region, and allows optimization to be performed. The algorithms include two completely different parts: constructing the reduced model through sensitivity and optilnizing the reduced model subproblem thanks to mathematical programming. Calling these two jobs a 'step', we may distinguish three families of methods, in increasing order of sophistication: one step linear, sequential linear and nonlinear, the two latter methods involving several iterative steps. These methods are in general not so easy to program as the noncompact methods and need more storage location, the reduced problem matrix not being sparse. Computation times may be very short depending on the sophistication either of the method or of the desired results within a given method and in any case increase more than linearly with the number of effective constraints. Consequently, performances are unique for real optimization where the number of effective constraints is always small and comparable with noncompact methods for complete or only reactive optimizations, where the number of effective constraints is greater. By nature, they fit well with n-1 or n -k security and meet constraints 'hardly', i.e. exactly, which is favourable for security. Moreover, the use of the reduced model alone may always be fast; as the complete system behaviour is represented, the reduced model may easily be used for real time operation and control, which is a very important property.

VI.2 One step linear methods VI.2.1 Principle. The principle is very simple: first make the objective function linear or piecewise linear if it was not so already. Then, linearize the network equations. Use these linearized equations to eliminate the state variables (corresponding to the intact system or to a contingency case) from the constraint inequalities and from the objective function if necessary. Add the linearized real power balance relationship: the result is a linear program which is solved by usual dual or primal simplex algorithms.

To linearize, DC load flows or more accurate sensitivity computation techniques (through the Jacobian) may be used: DC load flow neglects the losses whereas sensitivity techniques allow them to be taken into account to some extent.

For the real problem the method is particularly fast and has been used for years. A program for the real-reactive problem has recently been written for use online on a mini- computer.

VI.2.2 Practical aspects and developments. The real prob- lem was initiated by Wells ll and improved by Shen and Laughton. 21 Both may handle n-1 and n -2 security con- straints. Wells considers a global system with a spinning generating capacity constraint which is linear in P; kaughton and Shen do the same and, moreover, meet spinning spare capacity constraints for each generator modelled as a set of linear inequalities in Pik. They use the DC load flow approximation to build the reduced model and the revised dual simplex method to solve the linear program. They are able to solve very large problems very quickly.

To solve a similar problem to Laughton's, Sterling 63 noticed that the constraints have a horizontal basis block angular structure, the satellite elements being the generator spinning spare capacity constraints. So, alter building the reduced model through the DC approximation, he used the Dantzig and Wolfe decomposition algorithm to solve the problem in a primal simplex manner.

Very recently, Stott and Hobson 69 published a paper with a number of improvements:

general n-k security constraints

phase shift angles and HDVC line settings

losses are taken into account approximately, including them in loads

they distinguished between preventive and corrective control, noticing that the normal n-1 (or n-k ) insecure operating state may be subdivided into 'insecure correct- able' and 'insecure uncorrectable', the latter being the only one to be considered in the preventive control

for corrective control, they propose different possible objectives and introduce emergency start-up and load shedding

Concerning computational aspects, they use a symmetric Jacobian to model the intact system state and a DC approxi- mation for contingency. Construction of the" reduced model uses sensitivity and matrix sparsity. Optimization is per- formed with a revised dual simplex method. Numerical performances are excellent for large scale systems (462 buses, 70 controlled generator units, 1 s CPU in IBM370/ 158). Accuracy is better for the intact system than for contingency constraints and the need for an extension of linear programming to convex programming is indicated.

Aschmoneit, Ruhose and Wagner 71 recently solved the complete linearized real-reactive problem with a program written for a minicomputer (up to 150 buses, 50 control

Vol 1 No 1 April 1979 9

variables, 50 constraints; an example with 109 buses, 33 control variables, 19 constraints lasted about 1 min). They used sensitivity and sparsity to build the model and the simplex method. The validity of the linearization is cer- tainly good for the real variables and constraints but seems fairly doubtful for the reactive parl of the problem.

VI.3 Sequential linear methods VI.3.1 Principle. The previous methods may be improved by performing several steps, each step consisting of building the linear reduced model then optimizing it. A typical solution of this kind is the 'Maya' model developed by Merlin 34 for real optimization only.

In Maya, model building uses a simplified load flow with approximated losses p. In phase I, a feasible solution is computed by the dual simplex algorithm, introducing the effective inequalities only one after the other and improving the value o f~ used at each iteration.

In phase II, the objective function, which is piecewise linear, is optimized. The power balance equation is written so as to keep the values of the losses limited to a first order Taylor series development. Primal simplex is used to opti- mize. But, due to the limitation of the first order of the Taylor development of the losses, the coefficients of the real power balance change from the beginning to the end of the linear program. A sequence of linear programs, associ- ated with a dichotomic process, is therefore necessary to converge towards the solution of the optimization: using this process, a nonlinear program is solved by a sequence of linear programs.

VI.3.2 Practical aspects. Maya may rapidly solve large scale (500 bus) real problems with n-1 security. It is currently applied for planning studies where it often plays the part of a subroutine. Real time implementation is foreseen for 1981.

Khan and Pal s9 realized a program very similar to Maya, also with n-1 security. Duran 41 proposed a solution of the real-reactive problem with n security, by sequences of equality-constrained problems, which appears intermediate between one step and sequential linear compact methods.

VI. 4 Nonlinear compact methods VIA. 1 General characteristics. In the nonlinear compact methods, the reduced model is nonlinear and is solved using a general nonlinear programming method, sometimes specially adjusted to the particular nature of the problem being solved.

This gives the general following properties:

the possibility of using the reduced model separately. This is valuable over a wide region and 1nay be used in real time with high performance

The earliest and most typical among these methods is certainly the differential injections method, described now in detail, as an example.

VI.4.2 Differential injections method14,31,35,42,ss, s6, 77,78 Principle. 14,42 The differential injections method is a general method for optimal power flows with various options, the basic one being the complete real-reactive problem with variable transformer tap settings for voltage magnitude, n-1 security for currents in lines, and, which is unique, voltage level n -1 security.

Its main characteristics are:

(a) The nature ofahe reduced model where

the real power balance relationship is developed by a Taylor series up to the second degree (in exactly the same way as in the classic economic dispatch)

the security inequality constraints are linearized

/ ~APi. l~-~oI7 ,Art ,zSd/7'[~]' Id2p A/t= 0

Au ~< 3, k for every selected constraint k [ du o ]

(25)

(b) Tile use of the generalized reduced gradient method (GRG) to optimize the reduced problem

(26)

(c) The computation of first and second order sensitivities to build the reduced model

(d) The intensive use of matrix sparsity properties in Newton's physical load flow and sensitivity computations.

The quadratic character of equation (25) is fundalnental for it gives validity to lhe reduced model over a large region and practically guarantees convergence. Philo- sophically, this reduced model, as well as the whole method, appears as a direct extension, in the presence of constraints, of the classical economic dispatch, to which it is reduced (with exact B coefficients) in the real power only option if no constraints are effective.

The basic algorithm is as follows:

(1) settle the initial values of the control variables u

(2) by a Newton's load flow compute the state variables x a sophisticated program

(3) often a lower speed and greater storage location than for

linear programming techniques

an outstanding accuracy and an exact solution concerning the optimum as well as constraints meeting (4)

a secure convergence, due to the use of a general strong nonlinear programming technique

analyse the constraints and select those which are outside, on or near their limits, which gives the set of the 'selected constraints' to be included in the reduced problem. This includes n- 1 security analysis

build the reduced problem by sensitivity computations, including first dp/duo and second d2p/du2o 'differential losses', the latter being computed only at steps (1) and (2) [only at step (1) for the real only option].

10 Electrical Power & Energy Systems

(5) from the second step, if the new reduced problem is identical with the previous one, stop; otherwise go to (6)

(6) optimize the reduced problem by GRG and go to (2)

In (3) the degree of accuracy of the n-1 security analysis for line currents may be chosen by the user, ss from a linearized option up to an exact computation. The latter is not so time consuming as might be thought, because effective constraint 'angling' is first performed with the linearized option and accurate computations performed only for selected constraints.

The n-1 voltage level security ss guarantees after a generator trip changing the reactive power balance, that voltage at each generator bus will remain within acceptable limits.

In any case, the after-trip variables are expressed versus the control variables before tripping, such coefficients as 'distribution factors' for line currents and similar coefficients for voltages being computed and stored to be used in the constraints, which appeared sufficient for accuracy since at the last step Au ~ 0

In (4), sensitivities are computed using the transposed Jacobian, as shown for the first differential losses as early as 1962 (reference 5)

In (6), GRG is specialized to the reduced model the constraints of which are linear except for one which is quadratic. Compared with the general version, presented in paragraph 5.5.1, changes are as follows:

Point (5): Pl minimizing G is computed directly, which avoids a long search. Point (6): RDD is carried out at each GRG iteration only by solving a second degree equation. One, two or three times per GRG, a Newton load flow allows computation of the actual value of the constraint slack variables and an RRD process is performed to meet the actual nonlinear constraints, i.e. to compensate all the approximations due to the truncated Taylor developments of the reduced model: tile latter process is equivalent to having an exact convex reduced model, which makes the method exact. Moreover, as in the general GRG code, the optimization itself is preceded by a phase 1 finding a feasible point in a dual simplex manner.

Practical aspects and extensions. The method has been improved for 10 years, and four main options now exist : real-reactive (ID), real only (IDA), reactive only (IDR) and decomposed real-reactive (IDD).

The existence of ID, IDA and IDR allowed a study of real reactive decoupling possibilities s6 to be carried out. It was shown that only the reduced model was worth decoupling. This gave the IDD option, in which one builds two reduced models, one for the real and one for the reactive variables and constraints. This option needs less storage location and tends to replace ID.

The average number of steps is 3 to 4, except for the real only option for which it is 2 to 3.

The convergence is very secure; there is no empirical coefficient to adjust and the method is straightforward-

portable. The necessary storage locations are relatively large for reactive and complete options.

Computation durations, although not so short as for linear methods, remain acceptable: in 370/168, from 2 s for the real option of a 100 bus and 30 control variable system to a few minutes for a 480 bus system with 220 control variables in a real-reactive option in a nearly overcon- strained case. As yet, the programs have been used mainly for studies concerning five different countries.

A reactive only option was run on a minicomputer pre- pared for the Algerian system; it has been running with simulated data for one year and is just being installed now in Alger for real-time use: for a system with about 100 buses and 35 control variables, the response time is 1 min when alone and may reach 3 min due to other simultaneous tasks.

Mainly for data and computer reasons, the programs have not yet been used for scheduling or in real time in France, the present-day objective being real time implementation for 1981.

Promising extensions are possible towards combination with secondary voltage regulation, 78 discussion of the n-1 security,77 overconstrained cases and the use of the reduced models for real time 35,78 are in view.

VI.4.3 Other compact nonlinear methods. As it seems to be the most advanced and is well known by the author, the differential injections method has been presented in detail, but many other searchers found excellent algorithms, especially Sjvelgren and Bubenko, s3,62 lnnorta, Marannino and Macenigo, s8 all of whom as in IDD solve the complete real-reactive problem with n-1 (line) security using separate real-reactive reduced models.

Sjvelgren uses a linear real reduced model solved by dual Simplex; the reactive reduced model is nonlinear; Sjvelgren tried two methods to solve it and Zoutendijk successive linear programs and quadratic programming: he found quadratic programming the better method.

In Innorta's real-reactive decoupling, the reactive model has the interesting special property of handling the current limits. Sophisticated mathematical programming methods are used: Griffith and Stuart algorithms, Dantzig and Wolfe decomposition and Rosen projected gradient methods. Both Sjvelgren and Innorta methods handle large scale systems.

Wollenberg and Stadlin, s2 in an excellent paper, solve the real problem with n security by Dantzig and Wolfe decom- position for nonlinear programming. They also compute constrained participation factors for online operation and give very interesting solutions for the overconstrained case.

Nabona and Freris 61 solve the real-reactive problem with n security and may also perform optimal allocation of spinning reserve by 'quadratic approximation program- ming', the idea of which is the same as Beale's correction in GRG, and Podmore 47 uses the Rosen projected gradient method to solve the real problem with n security.

Vol 1 No 1 April 1979 11

Makala and Laiho v3 and Kopflnan 7s proposed a 'direct search' algorithm. The author is not sure that they always reached tire optimum: Nielsen and Poulsen s7 compared this algorithm with the Dommel and Tinney method and found the latter definitely better.

Finally, a very interesting method but difficult to classify was proposed by Glavitsch 4s for real-reactive online applications: it solves the reduced problem directly from Kuhn and Tucker theorem.

VII. Particular aspects of static optimization methods

VII. 1 Special purpose extensions Global spinning reserve and sparse generating capa- city 11,21,61,63 may naturally be introduced into the prob- lem statement, adding linear or quadratic constraints. HDVC lines may be introduced with the same formulae as phase shifters. 69 Coronna losses 73 correspond to ground inductances varying linearly with voltages in the normal voltage range, which may give optimal voltage levels not as high as usual. Valve point loading was performed by Fink 2 with classic economic dispatch. It is combinatorial and does not seem to have been generalized with security constraints. Environmental considerations 23,46, so arise from the necessity to limit some thermal plant effluents such as nox and to limit thermal pollution. Both are increasing functions of the generation power produced and the problem may be restated so as either to minimize pollution or limit it or better, combine it with fuel cost in the objective function, e.g. through corresponding tax values. Finally, multiarea operation has been solved in an excellent way for classic economic dispatch 3,24,6s but remains a challenge when security is consideredfl 8

VII.2 Discussion of the concept of n - 1 security

If n -1 security is met, then the sudden trip of one element will not provoke a general failure of the system, n 1 security constraints prevent the trip of one element from causing the trip of a second element. Sometimes, it may happen that the trip of this second element has no other effect on the network, and thus tire trip cascade stops. In the latter case, the n -1 security constraint was not useful and moreover prejudicial because it increased the generation cost. This was recently perceived 69,77 and a theoretical solution proposed by the author in reference 77. When a n -1 constraint is effective, on one hand the corresponding operation overcost is estimated, e.g. through the correspond- ing dual variable, or on the other hand the trip cascade is simulated, e.g. by n- k contingency and the resulting non- delivered power computed. Multiplying the cost of this non- delivered power by the probability that the first considered element trips gives the mathematical expectation of the possible damage if the n-1 constraint is not met (enormous for a general breakdown, comparable to the overcost for a small amount of nondelivered power, zero if the cascade effect stops). The comparison of the operation overcost and of the damage-expected-cost then allows a choice of keeping or dropping the considered n -1 constraint. This solution is still at a theoretical stage but may be implemented easily with a fast n-k contingency analysis method: corresponding recurrent fornmlae, allowing computation of n k contin-

gency quickly once n 1 has been carried out are given in reference 77, and should be compared to those given in reference 69.

Finally, this analysis allows the abandonment of nonusefid n 1 constraints, giving rise to savings and also avoiding overconstrained cases.

V l 1.30verconstroined cases Overconstrained cases are of prime importance. They happen .when the constraints are such that the feasible region is empty and no solution exists. Then the problem is what to do.

As in the Wollenberg and Stadlin s2 analysis, one may try to reduce the 'overloads el', the amount by which the con- straints are surpassed. For this purpose, they minmrized functkms of e:

~aie i, Maxei, ~,aie 2 i

and found the latter objective led to the most reasonable solution. Peschon 2v searched for the optimal load curtail- ment meeting the constraints. Stott and l lobson 69 searched for enrergency start-up and load sheddings meeting the constraints. Front the author's point of view, the solution should be as follows: drop the n - I non-useful constraints as in section VII.2, then meet all other n-- 1 (and of course n) constraints by load shedding, emergency start-up, and even structure changes whenever possible.

This problem is certainly the most important in optimal power flows today;often, in such emergency situations, dispatchers have not the same scientific attitude as model builders: they usually gamble to avoid little load sheddings and usually win (but the corresponding probability is not unity), Fast and secure automatic load shedding (and inci- dentally start-upt computation programs and control devices would certainly be the solution to this problem.

VIII. Dynamic optimal power flows and control As stated in the problem statement in 11.6, the final objective is real time secure and optimal operation of the system.

Corresponding to point (1) of this statement, if the optimal power flows program is fast enough, this may be carried out periodically and the usual control devices employed. This gives an approximate solution and is better than nothing. For this case as well as for cases 2 and 3, real - reactive decoupling is valuable, ()wing to the system feed- back. Real power implementation is straightforward. An example of reactive power implementation of this kind is given by the Alger dispatch centre, already mentioned.

An important improvement concerning real power consists, as a by-product of an optimal power flow, in computing 'participation factors' sharing forecasted load variations between the various plants of the system. 32,44, .':;2,64

In this direction, Patton 44 in particular takes an account of 'change-related costs' due to power changes in the plants. Wollenberg and Stadlin s2 compute 'constrained participation factors' taking account of the line current security constraints, which is a very important improve-

12 Electrical Power & Energy Systems

ment. Adler and Fisch164 base the computation of these constrained participation factors on the worst case bus load variation forecasts.

To guarantee the matching of constraints, the latter solution is good as far as the actual load variations remains the same as forecasted. To take account of the actual load variations, the classic economic dispatch might be extended to use the reduced models of compact optimal power flows on the one hand, and data coming from state estimation on the other, to perform a secure control meeting the security constraints. An example of such a possible application was proposed by the author for real powers as using the real power reduced model of the differential injections method. It was even possible to introduce dynamics into this process, which thus became a secure generalized frequency control process.

The use of the reactive reduced model of compact optimal power flows makes it possible to build a voltage control regulation. The author is studying such an application 78 leading to a tertiary voltage control: primary control is given by the usual local generator voltage control; secondary voltage control will, as in France, incorporate five to ten generation buses, the reactive powers of which are linearly dependent (which cuts down computation durations for the differential injections method, changing only the variable R i in the physical load flows; for secondary control see also reference 74); the reduced model would give a tertiary secure and optimal control.

IX. Conclusion Since their introduction in the early 60s, optimal power flows have developed and even seem to provide savings 31,6s,66 all the more interesting as fuel costs increase.

But their existence was due to a need for security, and in this field, even if considerable work has been performed, progress is still necessary,-especially for real time applica- tions: developments based upon applications of the reduced models of the compact methods will certainly be very helpful, as well as a deeper analysis of the security con- straints discussion and of overconstrained cases.

Finally, better contacts between researchers and a better understanding of others' methods would be very profitable ou the world scale: the author cannot help claiming his strong faith in the usefulness of this Journal to meet this objective.

Acknowledgments The author thanks his wife Rosine whose kindness and understanding allowed him to write this paper, especially for the most studious Christmas Day of her life. Thanks are also due to his friends from Electricit6 de France who helped him in a period of 'overconstrained' health, especi- ally Mrs Ferrier, Mrs Bruneau, Mr Girard and Mr Kalil for their Saturday working, and all his foreign friends whose friendship considerably helped him. He also wants to thank in advance all those who will write him to mention an error, an omission or a new development, or to ask for any information related to this review.

References

Kuhn, H W and Tucker, A W 'Non linear program- ming' Proc. 2nd Berkeley Symposium on Mathematics, Statistics and Probability University of California Press, Berkeley, California (1951)

2 Kirchmayer, L K Economic operation of power systems Wiley, New York (1958)

3 Kirchmayer, L K Economic control of interconnected systems Wiley, New York (1959)

Squires, R B 'Economic dispatch of generation directly from power system;voltages and admittances' AIEE Trans. PAS Vol 52,.Part III (1961), pp 1235-1244

Carpentier, J 'Contribution ~ I'~tude du dispatching ~conomique' Bulletin de la Soci~t~ Frangaise des Electriciens Ser 8, Vol 3 (August 1962)

Carpentier, J and Siroux, G 'L'optimisation de la pro- duction ~ I'EIectricit~ de France' Bulletin de la Socidt~ Francaise des Electriciens, Ser. 8, Tome IV, No 39 (March 1963)

7 Carpentier, J and Canal, M 'Ordered eliminations' Proc PSCC 1, London (1963)

8 Carpentier, J 'Application of Newton's method to load flow computations' Proc. PSCC 1 London (1963)

Carpentier, J and Abadie, J 'Generalisation de la m~thode du gradient r~duit de Wolfe au cas de con- traintes non lin&aires' Proc. 4th IFORS Meeting Boston (1966)

10 Dopazo, J F, Klitin, O A, Stagg, G W and Watson, M 'An optimization technique for real and reactive power allocation' Proc. IEEE Vol 65 (1967), pp 1877-1885

11 Wells, D W 'Method for economic secure loading of a power system' Proc. lEE Vol 115, No 8 (August 1968} pp 1190-1194

12 Dommel, H W and Tinney, W F 'Optimal power flow solutions' IEEE Trans. PAS Vol 87 (1968) pp 1866- 1876

13 Sasson, A M 'Nonlinear programming applications to power systems' Proc. Symposium Helors-IFORS Athens (1968)

14 Carpentier, J, Cassapoglou, C and Hensgen, C 'Injections diff&entielles, une m~thode de r~solution g~n&ale des probl~mes de dispatching economique sans variables enti~res utilisant le proc~d~ du gradient r~duit g~n~ralise' Proc. Symposium Helors-IFORS Athens (1968)

15 Carpentier, J, Cassapoglou, C, Bellon, J L, Bonneau, P and Xirokostas, D 'Injections totales, une m~thode g~n~rale de r~solution des probl~mes de dispatching economique avec variables enti~res, tenant compte exactement des co~ts de marche ~ vide des groupes' Proc. Symposium Helors-IFORS Athens (1968)

Voi 1 No 1 April 1979 13

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

Abadie, J and Carpentier, J 'Generalization of the Wolfe Reduced Gradient method to the case of non- linear constraints' in Optimization Academic Press (1969), pp 37-47

Shen, C M and Laughton, M A 'Determination of optimum power system operating conditions under constraints' Proc. lEE Vol 116, No 2 (February 1969), pp 225-239

Sasson, A M 'Nonlinear programming solution for the load flow, minimum loss and economic dispatch problems' IEEE Trans. Vol PAS-88 (April 1969), pp 399-406

Sasson, A M 'Combined use of the Powell and Fletcher Powell nonlinear programming methods for optimal load flow' IEEE Trans. Vol PAS-88 (October 1969), pp 1530-1537

Fink, L H 'Economic dispatch of generation via valve- point loading' IEEE Trans. Vol PAS-88, No 6 (June 1969), pp 805-811

Shen, C M and Laughton, M A 'Power system load scheduling with security constraints using dual linear programming' Proc. lEE Vol 117, No 11 (November 1970)

Despotoric, S T, Babic, B S and Mastilovic, V P 'A rapid and reliable method for solving load flow problems' IEEE Trans. PAS-90 (1971), pp 120-130

Gent, M R and Lamont, J W 'Minimum emission dispatch' Proc. PICA (1971), pp 27-38

Aldrich, J F, Leuer, J F and Happ, H H 'Multi-area dispatch' Proc. PICA (1971 ), pp 39-47

Dy Liacco, T E, Wirtz, B F and Wheeler, D A 'Auto- mation of the CEI system for security' Proc. PICA (1971), pp 93-102

Sasson, A M, Viloria, F and Aboytes, F 'Optimal load flow solution using the Hessian matrix' Proc. PICA (1971), pp 203-209

Peschon, J, Bree, D W and Hajdu, L P 'Optimal solu- tions involving system security' Proc. PICA (1971), pp 210-218

Jaimes, F J and El Abiad, A H 'Optimization by a sequence of equality constrained problems' Proc. PICA (1971), pp 219-227

Henser, P B and Cory, B J 'Solution of the minimum loss and economic dispatch problems including real and imaginary transformer tap ratio' Proc. PICA (1971), pp 228-236

Sasson, A M, Aboytes, F, Cardenas, R, Gomez, F and Viloria, F 'A comparison of power system static optimization techniques' Proc. PICA (1971), pp 328- 340

31 Carpentier, J, Cassapoglou, C, Brellas, I~1 and Margari- tidis, P 'Elaboration des incidences de nouvelles methodes de calcul elaborees en Grace sur la securite et I'~conomie de I'exploitation ainsi que sur I'~quipe- ment d'un dispatching automatique' Proc. Symposium on the automation of dispatch centers, CEE paper GE 71-16364 (October 1971)

32 Bechert, T E and Kwatny, H G 'On the optimal dynamic dispatch of real power IEEE Trans. PAS Vol PAS-91 (May-June 1972), pp 889-898

33 Cory, B and Henser, P 'Economic dispatch with security using nonlinear programming' Proc. PSCC 4 (1972)

34 Merlin, A 'On the optimal generation planning in a large transmission system (The Maya model)' Proc. PSCC 4 (1972)

35 Carpentier, J 'Results and extensions of the methods of differential and total injections' Proc. PSCC 4 (1972)

36 Adielson, T 'Determination of an optimal power flow by iterative suboptimizations' Proc. PSCC 4 (1972)

37 Velghe, J 'Optimal control of real and reactive power flow under constraints' Proc. PSCC 4 (1972)

38 Dy Liacco, T 'Design considerations for an advanced system operation center' Proc. PSCC 4 (1972)

39 Stott, B and Alsac, O 'Fast decoupled load flow', presented at the IEEE PES summer meeting Vancouver, Canada (July 1973)

40 Takahashi, K, Fagan, J and Chen, M 'Formation of a sparse bus impedance matrix and its application to short circuit study' Proc. PICA (1973), pp 63-69

41 Duran, H 'A simplex like method for solving the optimum power flow problem' Proc. PICA (1973), pp 162-167

42 Carpentier, J 'Differential injections method, a general method for secure and optimal load flows' Proc. PICA (1973), pp 255-262

43 Carpentier, J 'Total injections method, a method for the solution of the unit commitment problem including secure and optimal load flow' Proc. PICA (1973), pp 263-268

44 Patton, A 'Dynamic optimal dispatch of real power for thermal generating units' Proc. PICA (1973) pp 403-411

45 Glavitsch, H 'Economic load dispatching and corrective rescheduling using on-line information on the system state' Proc. PICA (1973), pp 412-420

46 Lamont, J W and Gent, M R 'Environmentally-oriented dispatching techniques' Proc. PICA (1973), pp 421- 427

14 Electrical Power & Energy Systems

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

Podmore, R 'Economic power dispatch with line security limits' IEEE Trans. PAS (January-February 1974), pp 289-295

Happ, H H 'Optimal power dispatch' iEEE Trans. PAS (May-June 1974), pp 820-830

Alsac, O and Stott, B 'Optimal flow with steady state security' IEEE Trans. PAS (May-June 1974), pp 745-751

Delson, J K 'Controlled emission dispatch' IEEE Trans. PAS (September-October 1974), pp 1359-1366

Rashed, A M and Kelly, D H 'Optimal load flow solution using Lagrangian multipliers and the Hessian matrix' IEEE Trans. PAS (September-October 1974), pp 1292-1297

Wollenberg, B F and Stadlin, W O 'A real time opti- mizer for security dispatch' IEEE Trans. PAS (September -October 1974), pp 1640-1649

Sjelvgren, D V and Bubenko, J A 'Decomposition techniclue in a security related optimal power flow' Proc. PSCC 5 (1975)

Despotovic, S T 'Optimal power dispatch with line security limits' Proc. PSCC 5 (1975)

Carpentier, J 'System security in the differential injections method for optimal load flows' Proc. PSCC 5 (1975)

Carpentier, J, Saminaden, V, Boull6, D, Girard, R and Nguyen, V T 'Real and reactive decoupling possibilities in optimal load flows: a compact version of the differential injections method' Proc. PSCC 5 (1975)

Nielson, J and Poulsen, W 'Comparison of different numerical methods for economic load dispatch under security related constraints' Proc. PSCC 5 (1975)

Innorta, M, Marannino, P and Mocenigo, IV] 'Active and reactive power scheduling with security and voltage constraints' Proc. PSCC 5 (1975)

Khan, M A and Pal, M A 'Security constrained opti- mization of power systems' Proc. PICA (1975), pp 61-67

Vuong, G T and Robichaud, Y 'Emergency control of a power system using on-line security constrained optimization' Proc. PICA (1975), pp 68-70

Nabona, N and Freris, L L 'Optimum allocation of spinning reserve by quadratic programming' Proc. lEE Vol 122, No 11 (November 1975)

Sjelvgren, D 'Application of mathematical program- ming to electric power system expansion planning' PhD Thesis Dept of Electric Power System Engineering, the Royal Institute of Technology, Stockholm (1976)

Sterling, M J H and Irving, M R 'Constrained dispatch of active power by linear decomposition' Proc. lEE Vol 124, No 3 (March 1977)

64

65

66

Adler, R B and Fischl R, 'Security constrained econ- omic dispatch with participation factors based on worst case bus load variations' IEEE Trans. PAS-96 (March-April 1977), pp 347-356

Happ, H H 'Optimal power dispatch. A comprehensive survey' IEEE Trans. PAS-96 (May-June 1977), pp 841-853

Barcelo, W R, Lemmon, W W and Koen, H R 'Opti- mization of the real-time dispatch with constraints for secure operation of bulk power systems' IEEE Trans. PAS-96 (May-June 1977), pp 741-757

67 Bala, J L and Thanikachalam, A 'An improved second order method for optimal load flow' IEEE Trans. PAS-97 (July-August 1978), pp 1239-1244

68 Mamandur, K R C and Berg, G J 'Economic shift in electric power generation with line flow constraints' IEEE Trans. PAS-97 (September-October 1978), pp 1618-1626

69

70

71

72

73

74

75

76

77

78

Stott, B and Hobson, E 'Power system security control calculations using linear programming' IEEE Trans. PAS-97 (September-October 1978), pp 1713-1731

Fernandez, R A, Happ, H H and Wirgau, K A 'The application of optimal power flow for improved systems operations' Proc. PSCC 6 (1978), pp 465-472

Aschmoneit, F C, Ruhose, K H and Wagner, G G 'Steady state sensitivity analysis for security enhance- ment' Proc. PSCC 6 (1978) pp 481-484

Wallach, Y 'Remarks on parallelisation of economic dispatch calculations' Proc PSCC 6 (1978) pp 485-488

Makela, L and Laiho, Y 'Optimal voltage level and reactive power control using an approximated sensi- tivity matrix' Proc. PSCC 6 (1978), pp 489-492

Van Meteeren, H P and Maas, J 'Secondary voltage- reactive power control by constrained least square minimization' Proc. PSCC 6 (1978), pp 493-496

Kopfman, G 'Management of electrical power systems at minimum cost using the sensitivity matrix' Proc. PSCC 6 (1978), pp 497-501

Fletcher, R and Ramsay, B 'Optimal electric power system scheduling by use of a multiplier penalty function method' Proc. PSCC 6 (1978), pp 502-508

Carpentier, J 'Discussion of the security concept in optimal power flows taking into account economical and stochastic aspects' Proc. PSCC 6 (1978), pp 509- 516

Carpentier, J 'Optimal voltage scheduling and control in large scale power systems', proposed paper to the IFA C Symposium on computer applications in large scale power systems, New Delhi ( 16-18 August 1979)

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