Aggregate Feasibility Sets for Large Power Networks

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AGGREGA TE FEASIBILITY SETS FOR LARGE POWER NETWORKS* P. Dersin and A. H. Levis Hd.I.!.,"lrl/l CI/nllllaft' .\1 1/,,111111/11"//1 l 'l,tllllll'

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Abstract. An aggregate feasibility set for large electric power systems is derived. The feasibility set consists of the set of snbstation loads that can be served in steadystate with the available generation withont overloading the transmission lines or transformers. The derivation is based on an aggregate continuum model of large power systems that is obtained from the DC load flow model. Keywords. Power Systems, Large Scale Systems, Load Flow, Feasibility Set AGGREGATE DC MODEL OF THE POWER SYSTEM

INTRODUCTION An integrated methodology that takes into account generation, transmission, and distribution systems simultaneously would be a useful planning and evaluation tool for electric utilities. Such a methodology, based on the evaluation of a hierarchy of attributes that describe the supply of and the demand for electricity independently from each other, has been presented in a series of documents, Dersin (1980},Dersin and Levis (1982a), and Dersin and Levis (1982b). A key issue in this work has been the characteriza tion of electric energy service through the introduction of the feasibility set. This was defined as the set of steady state substation loads that can be served with the available generation resources without overloading the transmission lines or transformers. The feasibility set can also be viewed as a projection of a general security region in the space of loads and generation output levels (Hnyilicza et aI, 1975; Fischl et aI, 1976, Galiana, 1977; Banakar, 1980) on the space of loads.

On the Continuum Viewpoint

The aggregation method involves substituting a geographical description of the power system for the topological (i.e., network-ba.ed) one. Let the geographical domain, D, over which the power system extends, be covered by a square grid. The spacings of the grid are of length .1 , and the elementary meshes of the grid are referred to as .1cells. On each .1-cel1, cell variables are defined in terms of the network variables that refer to branches and nodes contained in the cell. The network description is thereby replaced by an aggregate description in terms of cells variables. The more nodes and branches each .1-ce11 contains, the less detailed the description. Therefore, the aggregation level is defined as the average number nodes in a .1-ce11. The scale, defined as the ratio of h, the greatest distance between any two points in D, to the cell side .1, is an indicator of the number of .1-ce1ls needed to cover the power system and, therefore, of the number of aggregate variables . The proposed approach is applicable to very large systems where, simultaneously, the scale is large and the aggregation is reasonably high; the ratio .1/h will be small and linear expansions will be allowed.

An explicit description of the feasibility set, based on the DC load flow model, has been presented in Dersin and Levis (1982a) in which the requirements of load feasibility reduced to the requirement that a system of linear inequalities have a solution. However, the total number of conditions is 2(n+N}I/(nl}', where Nand n are the number of nodes and branches, respectively. The number of conditions can be reduced substantially, but not enough, by taking into account that at many nodes there is neither generation nor load. Therefore, an aggregate description of the feasibili ty set was sought which would be a good approximation when the systems considered are large. To that effect, a continuum model that corresponds to the DC load flow model is introduced. While the use of continuum models for describing large scale systems is not new --there is a long history, -this particular formulation has been developed for the explicit purpose of deri ving an aggrega te, approx ima te descript ion of the feasibility set.

To simplify the notation, it will be convenient from now on to take the length h as the unit of distance. This is equivalent to representing the power system on a map in such a way that a physical length h corresponds to a unit length on the map. Then for a large system, mode1ed with a large scale, the map representation of the side of a .1cell is much smaller than 1. Consequently, from now on, .1 will be used instead of .1/h. The cell variables, obtained by aggregating the network variables over the respective cells, can be viewed as re.ultinl from the discretization of continuous point functions. Namely, instead of dealing with a tableau of numbers - one number for each .1-ce11 one can deal with a continuous function of the location (x,y) whose average over the kth .1-cell is the kth entry of the tableau. Dealing with continuous functions rather than network data makes it possible to circumvent combinatorial steps.

*This work was supported by the Electric EnerlY Systems Division of the U. S. Dept. of EnerlY under Contract No. ACOl-77-ET29033.

Definina algregate load and generation variables is rather straightforward; an aggregate 2 163

2 164

P. De rsin and A. H. Levis

description of the transmission is more subtle. The load aggregate variable k(A) associated with a given A-cell is defined as the ratio of the total load contributed by the nodes in that cell over the cell area: (1)

Likewise, the cell variable I(A), which refera to the generated output, is defined as (2)

The continuum viewpoint implies that k(A) and I(A) should be regarded .. discretiutions of density functions k(x,y) and g(x,y). Trans.ission Modeling froa the Continuum Viewpoint By relying on the analogy between the DC load flow equations and the re la tions between curents and voltages in a purely resistive circuit (Ellerd, 1971) it is possible to define quantities which are continuum equivalents of the flows fm and the voltage angles li i • In each A-cell, a value for the current density i(A) is defined in terms of the branch power flows f. corresponding to that cell, .. follows: If a branch. is totally or partially contained in a Acell (Fig. 1), its contribution to the i(A) vector for that cell is defined by: i

i

y

1

=

1

~ a

,,

R S

•

(y)

f

and oonsidering E to be constant within each Acell. Denoting - by (r 1 ,r,) and (S1'S,) the coordinates of Rand S, respectively, and by (E ,E ) the components of E, it follows: x

lA

The constitutive relation of the DC .odel, which relatea the real power flow fa in a branch. and the voltage anlles li j , li k at its extre.ity nodes j,k, is: (7)

where b. is the susceptance (Sullivan,1977) of branch m (in .elawatts). The above definitions of i and E, and eq. (7) imply a linear relation betwee;;i and E, to within O(A) ter.s_ This relation is formally analogous to Ohm's law at a point (Kraus and Carver, 1975), and involves a .atrix a, defined on each A-cell, as the followinl theorem indicates. Theore. 1: The constitutive relation (7) of the DC flow aodel, which re la tes branch power flows and node voltage angles, can be expressed in a form analolous to Ohm's law at a point, in terms of cell aggregate variables : (8)

(4)

•

where f is the power (in melawatts) flowing through ~ranch ., R, S, R1 , S, are defined in Figure 1, and '.(x) is equal to +1 or -1, dependinl on whether the arbitrary orientation of branch a (used to define f.) coincides with the direction of increasing x or decreasing XI a.(y) has the corresponding .eaning with respect to the y direction. The total density vector i(A) for a A-cell is obtained by adding up the vectora produced by all branches. in the cell.

2

a -

•

. ----.-l----.,--==r.5...--'"

-<;'

_.5-

~-~-I----~--

~6-----l

r:

oos

,

e

b.ll: sin 9 cos 9

••

•

b.ll: sin

,

cos

..]

e

•

where b. is the sUsceptance of the portion of branch. in the celll ll. is the length of branch m relative to A (thus, a dimenaionleu number! ll. - RS/AI and 9. is the angle that defines the orientation of branch ., i.e.,

•

=

-1

tan

if branch. connects nodes Rand S • The proof is given in Dersin, (1980) ch. 5.

II

~

_ ___

a

The DC aodel includes the flow conservation law, a continuum expr.ssion of which can be given by div(!) = g(x,y) - k(x,y)

Figure 1.

e

b.ll: sin

•

(9)

9

s .~.-. ~

-

~

where teras that vanish with A are neglected. The conductance .atrix a takes a value on each A-cell, which la defined - in terms of the dilcrete trans.ission para.eters in that cell by:

RS

as a• (x) f •lA

x

(5)

Aglregation Procedure within a A-cell

The node voltage anglea 6 are now apprOximated by assuming that they vary linearly within the A-cell aa functions of the position coordinates, or, equivalently, by setting the electrical field E

(10)

It can be shown (Derain 1980) that in the H.it of an infinitely large syste. (A~), both formulations, the discrete and the continuous, coincide. A complete continuum for.ulation of the DC load flow problem can now be given: Theore. 2: In the H.i t of an infini te ly large system (A~ 0), the DC load flow problem approach.. the following boundary value problem in the

2 165

Agg r ega t e Feas ibil i t y Se t s f or La r ge Powe r Ne twork s

Co!parisou Between Discrete and Continuous Solution

fuuctiou f>(x.y): (~

div

srad f»

~

A(X.y) - ,(x.y) (11)

(x.y) a

aD

(12)

The followins compatibility condition. which expresses the equality of load and seneration. aust hold : ffD [A(X.y) - S(x.y)]dxdy = 0

(13)

The matrix ~ is the location- dependent conductance . as defined by (9) from the discrete datal the quantities A(X.y) and S(x.y) are the load and seneration densities. respectivelYI the seographical domain which contains the power system is denoted by D (with interior DO and boundary aD), the outward normal to the boundary aD is denoted by n.

Proof :

Dersin (1980) . ch. S.

The anresate variablel have been defined la al to be clole to the network variablel they approximate when the level of aaaresation il at its lowest (one node per cell) and the scale is lar,e (s .. ll A). ,This aeans that ix and iy a;e cloae to fa' A(x.y)A close to L1' and S(x.y)A close to Gi • The analosoul property for the voltase analel f>i and their continuoul approximation f>(x.y) can be proved directly. by cODlparins the partial differential equation (11) with the discrete load flow equationl in the cale of the lowest a,areaation level and for a Ipecial clall of networkl that conlistl of rectanaular sridl. The phYlical network can then be uled to dilcretixe the partial differential equation. Le •• to replace it by a Iystem of finite-difference equations. which is found to be precilely the syltem of the DC load flow equations. The low assresation level is conlidered first with each A-cell containins only one node. (FiS. 2). Uaina the notation N.W. S. E (North. West. South. Ealt) to refer to the nodes neishborinS a siven node O. the result for the homoaeneous ilotropic cale can be stated as follows. Theorem 3 : For the network of Fig. 2. where all branchel have the same lusceptance b. the equations latilfied by the node variables f>i are :

Isotropy and Homoseneity If the a(x.y) matrix correponds to an isotropic medium. Le•• if ~(x.y)

a(x.y) I

~ [A(O) - a(O)]

where I is the identity matrix. then (11) reduces to:

a

+

where [a(O) - A(O)]A' is the injection at node O.

r---"T-N"-T-- -T --i

(a'~

I

ax

=

I

(14)

A(X.y) - S(x.y)

If

furthermore. homogeneity prevails. Le.. in (14). a(x.y) is independent of the location. the partial differential equation becomes Poisson's:

1

T~~ 6

--t-

1

I

L~_

1

--+-

I

1

L_

1

,

+

ay

,

(1ta) [A(X.y) - S(x.y)]

with the boundary condition

o

for (x.y)

_..1._

Fiaure 2. 8

aD

so that the classical Neumann problem il obtained. A straishtforward example of a network that leads to an isotropic medium is provided by a square grid. More seneral networks can also sive rise to isotropic -- and sometimes homoseneoul-- continua. when certain statistical properties occur which refer to the denlity of the transmission (in miles per square mile) and the orientation of transmission lines. These propertiel involve (1) variation of the transmillion denlity (in miles per square mile) around a conltant al the a"re,ation level increases, (2) uniform distribution of branch orientations. The homoseneous-isotropic cale il important because it leads to explic it relul la on the fealibility let.

0

1

--+I

S 1 _ .1._ - -+I

(15)

1 _ ..l._ 1

I

1

ax

(16)

I I

_..l._

1

-+E 1 1 -"1"-

1 1 -r

I

_..J 1 I

-..J 1

I --I I

I

_ .1._

_...l

Lowelt Aagregation Level

Equation (16) is the finite-difference approximation. with Itep A. of the Pais Ion equation (15). Accordinaly. the followins bound holds: (17) where C is a constant independent of A. Proof : See Derain (1980). The bound (17) is specific of the Poislon equation (Wedland. 1979). Conlider now a hiaher :ssresation level. when each A-cell would contain k nodel instead of one. The relult of theorem 3 il then weakened. as indicated in the followins corrollary. Corrollary 3: For a rectanaular arid network where the assresation level is cholen luch that each A-

2 166

P. Der sin a nd A. H. Lev i s

cell contains k' nodes. the followina bonnd holds for the error between the value of the continuous point fnnction I) at any point (x.y) of the A-cell and the arithmetic averaae of the node variables in the cell.

(x .y) and (xk'Yk) are respectively the coirdinates of j and k. the extremitiel of branch m. Thus. the P fnnction satisfiel the homoleneous oquation (~Irad

div (18)

The constant C is independent of A and O(A) denotes a quantity which vanishes with A. The proof relies on the ore. 3 and the t r ian, le inequality. These approxi.ation results can be extended to the ,eneral (inhoaoaeneons. anisotropic) case (Dersin. 1980). THE AGGREGATE FEASIBILITY SET The Aaaresate Coefficients The feasibility set was introduced in Dersin and Levis (1982a) to denote the set of substation loads than can be served in steady state with the available aeneration resouroes without overloadinl the transmission lines or transformers. When the distribution of power flow thronah the network is represented by the DC load flow (Sullivan. 1977) then the feasibility set can be desoribed by a system of inequalities which are linear in the real bus loads. Therefore. the set is a convex polyhedron in the space of substation load vectors. The defininl inequalities are:

(21)

P) - 0

An explicit assumption is now made about the scalo

and the aalrelation level : the scale is hrle. 10 that A is 1 . . 11 .. compared to the extent of the domain D and the allreaation level il hilh enoulh 10 that the lenlth ~ of a branch is Imaller than A. To tranllate thOle allDlDptionl into simplificationl in the analytical expre .. ion for the lolntion P(x.y). the two branch extremitiel (x ·. y j ) and (xk'Yk) are modeled as thouSh they were in f inltely clole to eaoh other. Then the diltance ~ il .. de to 10 to zero while the intenlity F of the lource goel to infinity in luch a way that the product F~ remainl conltant. lay M. Then. the function P(x.y) reachel a lim i t. which il the potential created by a dipole (Webster. 1957) of moment M located ar the lilllit location (~.11) and directed alonl the Itrailht line frolll lource to sink. AI the proco.. jUlt described is carried out. r 1 and r. tend to a cOlDIDon value r (Fig . 3). and P reachel a limit. For an arbitrary number" of lin,ularities. P(x.y)

1

2na

2

(22)

(Mmcolem/rm) + C.

m

The parameters M1 • M•• • ••• M". and C. in the dipole approximation are deterlllined to within an arbitrary lIIultiplier by requirina P(x.y) to vanilh on the 11 nodel of the snbset A.

msB

(I,, )

for t

> 0 and t < 0

(19)

where (1) Li is the real bus load at bus i (in meaawatts). Si is the maximDlD aeneratins capacity at bns i (in meaawatts). and 'm is the .axiaDlD phase an,le bound across branch ml (2) A is a subset of the set of node indices and B is a subset of the set of branch indicesl (3) For eaci compatible choice of A and B. the coefficients Pi and a (zr) are determined as the solntion of I linear alaebraic systeml furthermore. the Pi coefficients constitute the solution of a DC load flow in the network with the branches of B removedl (4) The factor t is arbitrary and can be positive or neaativel (!S) the notation ( )+ refers to the positive part of that quantity. The aglrelate description of larle power systems will be used to derive an allreaate feasibility set. First. one obtains cell variablel Pk which are approximations of the coefficients Pi in (19). A continuous function P(x.y). the solntion of the continuum DC load flow cosrelpondinl to the discrete DC load flow of which Pi il the solntion. satisfies the followinl eqnation: 11

div

(~

grad P) -

2

I

f

Fisure 3. Dipole lIIodel of a branch In the hOllloleneoul-isotropic case. P(x.y) il a harmonic fnnction. Therefore. there exists an analytic function of the complex variable z=x+iy of which the imaginary part is aP(x.y) and u(x.y) its real part . Let the branch let B conlilt of one lingle branch • •odeled al a dipole and let the boundary aD of the domain be far enoulh from the dipole location \. If the dipole has lIIolllent M and orientation y. the correlponding functions P and a are obtained al followl:

a(x.y) + ia P(x.y)

=

a-1 (20) with boundary conditionl .. Itated in Theorem 2. In (20). I) refers to the Dirac impulse fnnctionl

iy

M 211

z=r

+ C

1 211

z=r

m

+ C

e

where C il an arbitrary cOlllplex conltant. derivation. lee Dersin (1980).ch 6.

(23) For the

Once the P(x.y) function is known. the a(x.y) function can be obtained frolll it by intesration. usina the Cauchy-Rielllann equations. Frolll thil

2 167

Agg r egate Feasi bil i t y Se t s f o r Large Powe r Networks

prop.rty. it can b • • hOWD that the a(x.y) fnnction approximate. the a(zk) co.ffici.nt. in (19) in th: .... way a. P(x.y) approximat.. the Pi co.ffic i.nta. Wh.n the a •• uaption. of homoa.neity and isotropy are remov.d . a fnnction a(x.y) can .till b. d.fin.d. which i. r.lat.d to P(x.y) by a .y.t.m of two lin.ar fir.t-order partial differ.ntial .qnation •• In the case of several dipole. (,>1). the .nalytic fnnc tion is obta iDed a. • .uaa. tion ov.r .11 the dipol •••• n.loaon.ly to (22). The Fe •• ibility Condition.

condition., th... can b. writt.n a. follows : I:

2([t pk]

A (.1) - [t pk]+ .k(.1») .1 k

k-1

n ~

2f(r t

j

)

IM j I It I

> 0 and t < 0

wh.re pk is the av.ras. of the siven by eq. (25) over .11 k.

Theor.m 4: In accordance with the continnon.-.p.c. model ins of load. seneration .nd the nod. and r.sion variable •• the feasibility condition. ar. mod.l.d by the followins intesr.l in.qualiti •••

ff {A(X.y)[t P(x.y)]-.(x.y)[tP(x.y)]+)dxdy > O.

t

<0

(24)

One .nch pair of ineqnaliti.. ari.e. for each choice of an intes.r /I where 0 ~ , ~ q, • node .ubset AC (1 ••••• q) with, nod •• , • dhtribution of , dipole. loc.ted at r ...... r/l. with siv.n ori.ntation. Y." ... Y, .nd moment. M..... M, to be determined. Given A .nd the dipole di.tribution. function P(x.y) is expre •• ed by :

the

,

P(x.y)

~2!a

2

Mk

k-1

Im [ei'Ykr] z - k

+ C

•

(25)

The , dipole moment. Mk and the con.tant C. are d.termiD.d to within an arbitrary mul tipler t by !equirins th.t P vani.h on the /I point. z - ai of A. Accordinsly. P(x.y) is d.termin.d to within .n arbitrary mnltipli.r t. Equation (25) i. v.lid only if the dipol •• are far enonsh from the boundary aD .0 that bound.ry condition (ap/an ~ 0) is .utomatic.lly satisfi.d, if this i. not the c •••• a harmonic function u(x.y) mn.t be add.d to fnlfill the boundary condition . The variable f i. a bonnd on the masnitud. of a differ.nc. of vo'tas. phase anslo.. If branch m connects nod •• j and k. the constraint i.: (26) The bonnd fm is mod.1od by a sc.lar po.itive function f(x.y). which acts .s a bound on the masnitud. of the electrical field:

function P(x.y)

Since .ach dipole di.tribution produces a pair of conditions (28) th.re are an infinity of linear inequalities snch a. (28) in the lumped loads A(.1 k ). This iaplies that the feasibility set is now a smooth convex set instead of a conv.x polyh.dron. Bowever. not all constraint. involve all loads nece.sarily: it may occur that pk is zero for some cells k . Ther.fore. there may be points of the boundary where the tans.nt hyperplan. is not defined, the set boundary is then only piecewhe continuou.ly differentiable. CONCLUSIONS Th. formulation of the DC load flow problem as a boundary problem for a partial differential .quation. the coefficient. of which are d.rived frOll the discrete tran.mission data. is nsed to d.rive dir.ctly an assresat. feasibility .et in the sp.ce of lumped lo.d vectors. This fe •• ibility .et i. an aaaroaato in that no di.tinction can be m.de between load p.ttern. which differ only by the internal distribution of the lo.d .. ons bu.es of the .... .1-cell. The method i. diroot in that .11 the n.ce.s.ry calculations are performed .t the .Iareaate l.v.l from aaareaate data: the method does not involve derivina the exact fe.sibility conditions and the .omehow .asres.tina them. The method .pplies to 1arao power networks that .xtend ov.r l.rs. enouah a.oSr.phical dom. i n so th.t a sqnare ,rid can be snperimposed on that domain with most .1-cells not empty and with .1 small with r.sp.ct to the dimensions of the domain. These conditions permit the application of the continuum viewpoint. namely. the snbstitntion of analytical formnlations for combinatorial (i.e.. araphtheoretic) ones. REFERENCES Banakar. M. M. (1980). Analy.is and characterization of aener.l s.curity rea ions in power networks. Ph.D. Thesis. McGill Univer.ity. Montr.al. Canada. D.rsin. P. (1980). On .t.ady-state load feasibility in an electric pow.r network. Ph.D. Thesis. R.port LIDS-TB-991. Lab. for Information and Decision Systems. M.I.T. Cambridse. MA. D.rain. p .. and A. B. Lovh (1982). Feasibility sets for steady-state load. in electric power networks. IEEE Trans . on Power Apparatn. and Syst.ms. PAS-101. pp. 60-73.

m.x on r ~ s. t. II~ 11 ~ 1 max ov.r n

(28)

(hoaoaeneon.-i.otropic

~)

t

•

(27)

As pOinted ont •• rli. v the aSlrelat. feasibility set is d.fined in ~. wh.re I: is the numb.r of cell. that contain lo.d.. The .xpression (24) for the f ••• ibility conditions i. a continuous-spac. approximation of the alar.sat.

Derain. p.. and A. B. L.vis (1982) . Charaoterhation of electricity d.mand in a power systea for service snfficiency evalnation. European 1. of Op.rational R••• arch. Vol. 11. pp. 255-268. Elaerd. O. 1. (1971). Electric Eneray Systems Th.Ory: An Introdnction. McGraw-Bill. New York.

2 168

P . Ders in and A. H. Lev i s

Fiachl. R•• G. C. Ejebe. and 1. A. DeXaio (1976). Identification of power ayatea ateady-atate aecurity reaiona nnder load uucertainty. Proc. 1976 IEEE-PES Su.aer aeetinl. Portland. OR. Galiana. F. (1977). Analytic propertiea of the load flow problea. Proc. IEEE Couf. on Circuita and Syateaa. Phoenix. AZ. Hnyilicu. E.. S. T. Lee. and F. C. Schweppe (1975). Steady-atate socurity r8liona : sot-· theoretic approach. Proc. 1975 PICA Conference.

Irraua. 1. R.. and Ir. R. Carver Electroaalnetica. XcGraw-Hill. New York. Sullivan. R. L. (1977). XcGraw-Hill. New York. Webater. A. Equations of York.

G.

Power Syate.a Planninl.

(1957).

Xa the.a tical

Wendland. W. L. (1979). Plane. Pitaan. London.

(1975).

Partial Physica .

Differential Hafner. New

Elliptic Syate.a

in

the