A nonserial dynamic programming model with an application to the optimal design of distribution networks

A

_{nonseriai dynamic programming}

model with an application

to the optima! design

of distribution networks

A Nonserial Dynamic Programming (NSDP) forrnulation, aimed at the decomposition and

optimization of interconnected systems, is presented. A new model is Prole for the

system decomposition. kis used to optimize an electrical

_{CrIStribUttOrl}

system and the results

on a 25-node networic are reponed.

El trabajo trata sobre la formulación, aplicación y análisis de un método de descomposición,

basado en los principios de la programación dinámica, a fin de ser usado en la optimes

de sistemas interconectados. Gracias a procedimientos de inmersión y parametrización, la

técnica propuesta logra reorganizar el sistema interconectado en una secuencia y asf aplicar

el

prindpio

de optimaidad

a través de los subsisternas en la misma forma que es apicado a través

de las etapas en programación crinárnica convencional. La aplicación de la técnica produce

mejores resultados en sistemas de estructura esparsa, pero no necesariamente con

interconexiones débiles. El hecho de obtener una solución de lazo cerrado (controles en

función de las interconexiones) propicia su aplicación al análisis de sistemas complejos. La

formulación propuesta se

_{adapta naturalmente a problemas de optimizatión estática. Para el}

caso de sistemas dinámicos de una mayor complejidad algorftmica se propone un método de

descomposición espacio-temporal. La aplicación tratada consiste en la optimización del

diseño de una red de

_{d> bodón}

eléctrica de 25 nodos. Dado un conjunto de nodos con sus

respectivas cargas, las posibles ubicaciones de los transformadores de distribución y las

posibles conexiones entre nodos, el algoritmo desarrollado obtiene el diseño óptimo de la

red a fin de minimizar los costos de instalación de equipos y de pérdidas eléctricas.

José Luis Girrén

Escuela de Ingenierta Eléctrica Universidad Metropolitana

Introduction

Nonserial dynamic prograrnming (NSDP) was first introduced by Bertele and Briaschi _{1 as an} optimi-zation methodology which generalizes dassical Dynamic Programming (DF) formulations. VS ith

respect to standard (serial) DP involving a sequence

Traba4o publicado en fa revista Journal of zhe operada/1d reseordi sodety. Diciembre, 1994.

Jean-Louis Caivet

LAAS CNRS Toulouse, France

of decision processes, NSDP ailows the application

of the Principie of Optimality to a wider class of

problems characterized by a nonserial interaction graph. Rosenthal 2 has proved the optirnality of NSDP arnong nonoverlapping comparison

algo-rithms, defined as procedures which decompose

the problem and appty the Principie of Optimality.

Moreover, many NP-hard probierns including

re-sources allocation, vertex and edge cover, vertex colouring, satis-fiability, set cover, 2-3,4, can be

Roughly, the technique works by arranging

inter-connected subsystems into a sequence and applying

the Principie of Optimality across subsystems in the

same manner as it is applied across stages in ciassical

dynamic programming. At each step, "the

cost-to-go" function is optimized with respect to local

decision variables in terms of parametrized

sub-systems interconnections; this embedding process,

which indeed aliows the sequential resolution of a

family of related problems, requires of the storage

of intermediate results. The efficiency of NSDP

strongly depends on the interaction pattern and on the chosen ordering for the subsystem sequence;

such structural features like sparsity or the

band-width interaction graph play a decisive role in the feasibilIty of the application. A very important (also

NP-hard) problem, defined as the "secondary

opti-mization problem" I, consists in finding an optima! or near-optimal ordering.

where

r

is a scalar objective function ;

= [vi] _{means the system decision vec-}

tor ;

is the feasible control subspace of ;

n _{is the control vector dimension.}

The secondary optimization problem

The overall system will split into K subsystems in

order to be handied by NSDP I. Recall that this

method consists of arranging interconnected

sub-systems into a sequence and applying the Principie

of Optimality across subsystems in the same

man-ner as it is applied across stages in dassical dynamic programming.

Based on the concept of spatial dynamic program-ming 5, a somewhat different model is presented for the solution of the main (primary) optimization

problem. New variables subsets (decisions set and

parameters set) are introduced to characterize the

serially interconnected subsystems resulting from the secondary optimization problem.

The plan of the paper is as follows. The next section

describes the proposed NSDP model. Then, a very simple illustrative example is solved analytically and

numerically. Finaily the proposed model will be

applied to the optimal design of electrical distribution

networks. The algorithm has been conceived to treat either the primary electrical network with its

related distribution substations or the secondary

network with distribution transformers. Several

data sets are used in order to test the model

performance. The application has also been used to

illustrate the influence of the partition and the

subsystem ordering (secondary optimization problem) on the algorithmic complexity of the main problem solution.

NSDP formulation

Problem statement and notations

Let the main optimization problem be formulated

as follows:

Indeed, there existo many possible decompositions

which ensures the yalidity of the dynamic pro-gramming process 4. 6,7. The secondary optimiza-tion problem chooses, between diese different

decompositions, that one which minimizes the

computational effort of the associated primary problem solution. Bertelé and Brioschi proposed

various algorithms which exploft essentially

graph-theoretical properties for determining optimal or "near-optimal" schedules.

VV hen de_11ing with th'.: application, we propase another approach for this secondary optmization

problem in order to reduce two key factors

con-stituted by the number of functional equation

evaluations (time complexity) and the number of

stored resuits (space complexity).

The primary optimization problem

• Subsystem definition

V\ e propase to define the following sets in order to characterize the serially interconnected subsystems

resulting from the "secondary optimization prob-lem" solution.

Subsystem k variables set

Vk = {vi occurring at step k} _{k = I; 2, ... , K}

More precisely, this definition concerns variables

which are involved in at least one of the following decomposftion features:

opt v r (v) ₍₁₎

subject to :

with an objective function sequentially separable into the form:

ri (m i , s i , r2(m 2, 52, rK(mK, sK) ))

Subject to standard monotonicity assumptions on the separating functions r k, the problem is then decomposable by DF.

Further, the functional equations, which involve the

computation of the cost-to-go functions :

I k(sk) = opt _r

r;lc(mk, sK) mK

consist in the following recurrente:

For k = K-1, ,

I,

salve:

1k(sk) = opt mk rk(m k, sk, Ik+ (sk+ i)) ₍₂₎

with

IK(sK) = opt mK rK(mK, SK)

Optima' local variables m;Ak are thus determined as functions of parameters sk Finally, step I gives the nonparametrized solution for ITI;Al (51 = O)). Af-terwards, calculations of the optima) values of parameters siA_{k+i and substitutions of m;"k(si") k+i )} through a forward sequential processing enable the optima] trajectory to be reconstructed.

• Algorithmic complexity

For the general case where the solution of functional equations (2) is perforrned over a grid of discretized

points, the time complexity can be measured by the

number of evaluations of the cost-to-go function, i.e.:

K n s (k) n m (k) NE = 1 [fi «si)

fi

q(mi)

₁

k=t i=1 _i=1

(2a)

with

denoting the number of quantized

values for the element • of the respective vectors s k and mk ; denoting the dimension of the cor-responding vector •.

• the separated constraint set at step k ;

• the broadcasting of information through sub-system k required by the sequential arrangement of

subsystems.

Subsystem

k

decisions set

Mk = { vi E Vk I vi

E

_}

or, equivalently:

Mk = Vk

n

Vc

;k-1 for k = 1, 2, ... , K where i denotes the complement of

"-elements will be arranged in a subvector rn k.

Subsystem k

parameters

set

Sk = { Vi E Vk

_{1 MEMk}}

or, equivalently:

Sk _=Vk

_f}

_{Vk-1 for k= 2, 3, ... , K with S i =}

₀

Sk-elements will be arranged in a subvector s k.

• The composite system "realization"

It results from the previous definitions that the

overall system is embedded in a sequence of

sub-systems parametrized by the vector sk having the interpretation of a (spatial) state.

In this way, the system is governed at "stage" k by the vector mk having the interpretation of a local control.

The overall system ís then viewed as a "dynarnical"

system whose realization is characterized by the

input-output representatíon of subsystem k given in Figure I.

• The functional equations

According to previous definitions and observing that

opt m r (m, s)

K

K

U

Mk

= U

Vk

k=1 k=1 By considering a stationary situation with respect

to step k, one obtains the foliowing upper-bound

NE =

_{K q, qm}

n s (k)

=

n

q(s) F-I

whatever k

33

one can rewrite the main optimization problem (1)

with aS :

and a similar definition for q,„.

Similar!), the space complexity can be measured by the number of results (optimal decisions) to be stored, :

K ns(k)

NR =

E

[II q(si) niii(k) (21) k=2 i=I

and for the stationary case :

NR =

(K -

I) q, n„,

Then, NSDP makes both time and space complexity

linear with respect to the number of subsystems K.

An illustrative example

For this purpose, let us revisit the simple three

variable optimization problem considered by Larson, McEntire and Steding 8.

Max V 1 V2 V3 Vj2+0

Step 2 : Sotve, for I2(vi) :

_v2 ;1 _v2;2 Max V i V2 <1

1 2(v 1 ) = TI

;2) v1 (1 -v\o(2;1))

with v2 =

1r( 1f(1 ;2) (

I -v10(2;1)))

Step 1 : Salve, for I :

Max \f( I ;2) v 1 ( I -v o(2;1)) vi .+0

-4

= kf(\.(3);9) with vi = \f(\r(3);3)

• Recovery of

the

optimal solution

Step 1 : v i = If(\r(3);3)

subject to V2; i +V2;2+V2;3 S 1

Step 2 :

v2 = tr( V(1:2) ( 1 -v o(2; 1))) = \f(\r(3);3) Thus, choosing the decomposition depicted by

Figure 1 induces the foilowing subsets definition :

Vi v1}, _{V2={V ,V2}, V3 -7-{V ,V2,V3}} Mi={vi}, M2=-{v2}, M3=-{v3}

S1= 0, S2={vi}, S3={vi,v2}

The simplicity of the problem makes the functional

equations easily solvable analytically.

An analytical saludan

• Dynamic programming cakulation

Step 3 :Sub/e, for 13(v1,v2) :

Max V 1 V2 V3

v30

Step 3 :

v3 =

tr(1-v\o(2; 1 )-v\o(2;2)) =

\f(r(3);3)

To complete this illustration, let us consider now

the basíc DP computational procedure corre-sponding to the solution of functional equations

over a grid of discretized points.

A numerical solution

We assume that the variables are quantized in

uniform increments of 12 Thus = {O, 1/2, I} is

the set of feasible decisions . The basic computa-tional procedure runs as follows.

• Dynamic programming calculcrtion

subject to V2;1 4-V2;2-1-V2;3 S 1 Step 3 : Salve, for 13(v1,v2) : 13(V1,V2) = VI V2 <1 -v2;1-v;2

with v3 = <1 -2;1 -V0;2

MaXV1 V2 V3 V3_1«.)

V

1

1

2

(

y

1

1

O

with v

₂

= O

O

whatever v

2

= 0, 1/2, 1

1

2

1

3

v

O

whatever v3 = O, 1/2, 1

1/2

O

with

3

= O or 1/2

1/2

_O

_{with v = O or 1/2}

3

1

_with

3

= O

1

with v = O

3

1

1/2

unfeasible

1/2

1

unfeasible

1

1

unfeasible

Step 2 : Solve 1 2(v j ) =

max 13(v1,v2)

V2

Step I :

Solve I I

= max 12 (v 1

)

vl

= \f( I ;8) with

= \f( I ;2) .

• Recovery of the optimo! sohrtion

See the shaded arrays within the previously built

tables

17"

Step I :

v i = \f(1;2) .

Step 2 :

v2 =

\f(

I ;2) .

Step 3 :

Applicttion tod eoptkral design of an elestricd

distribution network

Problem statement

During the last two decades many efforts have been devoted to _{the development of numerical algorithms}

for the optima! design of electric distribution

net-works. As mentioned- by VN, illis and Northcote-Green 9, "many of these procedures use optimiza-tion or other high speed search techniques. There

have been a number of different approaches, each

aimed at different levels of the overall distribution problem".

1-he practica' problem of energy flow considered in this paper could be addressed to either primary or

secondary networks. However, restrictions on

substation locations (area costs, site availabiirty,. )

limit the interest of the former case and places the

challenge on the distribution transformers location

within the secondary network. Then, the problem

under interest can be described as follows :

Oven a rneshed load nodes set, the posible transforrner

(respectively, substation) locations and the transformer

(substationj action radius, which will be defined further, find the radial cable layout, the section of ad cable

segments and the number, location and _{capacity of} distribution transformers (substations) in arder ta

mini-mize costs related to active power _{transpon and}

transformer (substation) and cable supplying.

Our investigation makes use of a rectangular-shaped

grid. A different load value, which may represent

the aggregation of severa! actual load points, can be

assigned to each node. Feasible transformer

loca-tions, not fixed a priori, that broads the scope of

optima' design, can be attributed to -for

example-hatf of nodes (referred as biack nades) distributed in an alternate pattern; the rest (referred as white

nades) will be merely load locations.

Thus, the problem can be formally stated in the following way :

min

_{E {c}

₁

_-(t(i))

₊

_{Ida) Ecs(5(j))}

₊

_I

9(I)›

0c04.1),t(i).5(i)) ICE _ic110) IELQ,D

subject to

qrs„(t(i)) S X

E

x(i,j,l) q(l) S q,,,x(t(i))

VE/

;

lEf IEL(PA)

cL(50))]}; (3)

(4)

q ;,„‘„(50)) 5 E x(i,j,1) q(1) S

qo ;.(s0)) d iE/, VjEJ(i) ; ₍₅₎ IE1-(i,1)

{Kip (s(j)) d(j)

E

x(i,j,l) q(1)} S % Vmax ViEl,V ; iEr(1.19 IcE1-0,9

(6)

viere:

1 set of feasible transformer locations (biack nodes) ;

f(i) set of cable segments constituting the differentfeasible paths starting from transformer i ; L(i,j) set of load nodes reachable from transformer i through segment j ;

j(i) set of cable segments direcriy connected to transforrner i ; L'(i) set of terminal load nodes reachable from transforrner 1 ;

J"(i.,1') set of segments _{joining transformer i and terminal node ;}

t(i) selected nominal capacity of transformer i ;

c-r(-) transformer cost function ;

d(j) length of cable segment j ;

s(j) selected section of cable segment j ;

cs(•) cable segment cost function ;

x(i,j,1) binary decision variable valued as follows:l, if transformer i feeds node load I through segrnent j and O, otherwise ;

q(1) load value at node 1 ;

cL(•) cable losses cost function ;

q,„,(•) minimal transformer capacity (as a function of nominal transformer capacity) ;

q,x_{(•) maximal transformer capacity (as a function of nominal transforrner capacity) ;}

minimal cable segment capacity (as a function of cable segment section) ;

q :max_{(•) maximal cable segment capacity (as a function of cable segment section) ;}

K0(•) _{cable dirwibution coefficient function ;}

% Vmax maximal voltage drop.

The overall objective function (3) contains three parís

(a) the transformers cost evaluated by a

dis-crete function (decision Cable), from their

respective operating range determined by

the feeding constraints (4) ;

(b) the cables costs similarly evaluated from their section range determined by the

operating constraints (5) ;

(c) the cable losses cost evaluated by a qua-.

dratic function of power flow.

37

Maximum voltage drop is checked through

con-straints (6). Additional concon-straints deal with

net-work radiality and topology. indeed, as it ís usual in

distribution planning, a radial solution in which each

load is fed from a single transformer throughout a

unique path, will be found. Moreover, all load nodes belonging to a given path are assumed to be fed by

Let us note that the latter constraints, which are

given by a logical formulation, can be easily coded in

a Dynamic Programming approach.

On the other hand, while our problem statement

aims to a more flexible policy than other ones 10,1 ,1 2 by incorporating optimization of trans-former locations, it results in a strongty connected

grid which is known to be nonsuitable for NSDP

solution 3.

To face the resulting computational complexity, we further introduce the concept of "transformer

action radius" under the form of a decentralization

constraint. In this way, the transformer action

radius will be chosen in order to limit the number

of nodes reachable from the transformer node in

any radial direction. Thus, for a rectangular grid, a

given transformer with an action radius taken equal

to r wiil feed at most [1+2r(r-1)] nodes. So for r=1,

only one node (the proper transformer node) is

reachable from the transformer, for r=2, five nodes

are reachable, and so on. Of course, this number

may be limited by such physical features as network

boundaries, maximum voltage drop, maximum

transformer capacity,

An heuristic approach of the secondary problem V'S e have mentioned the importante of choosing a

convenient decomposition in order to reduce the

computational effort associated with the primary

problem solution. This can be summarized through the question

What is the optima! partition of the

problem and what subsystem order-ing must be considered to minimize the algoritiunic complexity of the pri-mary problem?

The answer to this question is not easy; indeed, it

is an NP-problem. To overcome this difficulty, we propose the following heuristics.

Pcrrtitioning

nentially and n, increase linearly. This behaviour

shows that a fine decomposition will generalty impty

fewer evaluations of the cost-to-go functions (NE)

than a coarse one. On the other hand, this choice

must be sufficiently balanced to limit the storage

requirements (NR).

Ordering

As mentioned before, subsystem ordering plays an

importaht role in the complexity of the primary

problem. Once a partition has been chosen, we propose to find the ordering which reduces the

computational effort represented by (2.a) and (2.b).

However, the ordering which minimizes (2.a) is not

necessarily that which minimizes (2.b). So taking

into account computer core-memory limitations,

we stressed to state the secondary problem as follows :

min NE tFE(w)

subject to

NR 5 Smax

with the following notation

w a given ordering for K subsystems ;

E(w) set of K! possible orderings ;

S„,„ maximal number of resufts which can be stored.

Some resufts

Figure 2 shows, for an action radius equal to 3 and

a 25-node network, three different

decomposi-tions.

A branch-and-bound algorithm has been applied to

each of these decompositions and the optima!

orderings which minimize NE (Table I) have been found. These results agree with comments made aboye.

Subsystem size cleariy has a great effect on the

measures NE and NR defined aboye. Thus, a fine

decomposition (few nodes per subsystem) will

generate large values for K and relatively small

values for q n, and in a lesser levet for ch and n m. A

coarse decomposition (many nodes per subsystem)

The primary problem solution

NSDP has been applied to the network shown in

Figure 3. The decomposition is that of Figure 2.b.

Numbers beside nades represent the node load (in kVg; numbers associated to links give inter-node

distantes (in m). Recail that black nodes are

cable types (I/O AWG, 4/0 AWG and 400 MCM) are also considered. All other values for parameters appearing in the problem statement are given in Tabie II.

Figure 4 shows the optirnal design found by the algorithm. The corresponding cost is 1057 k units.

In order to test the optima! policy, two important

disturbances on input parameters have been

stud-ied. The first one is characterized by a high cost on

transformer furniture (see values in table II).

The solution is shown in Figure 5. We check that only four u-ansformers (the minimal number taking into account the action radius) have been retained.

The cost of this solution is 2053 k units.

The second disturbance considera a high cost in power transpon (see c's and c";t. values in table 11). Figure 6 shows the corresponding solution, where

II transformers (over a maximal number of 13) are

needed. Small load values at nodes numbered 5 and

11 (the not retained transformer locations) made

more interesting feeding these loads from

trans-formers numbered 9 and I, respectively. This so-lution costs 1930 k units.

Condusions

A ctynarnic prograrnming mode( has been presented for the saludan of spatially interconnected sub-systems. Based on the separation between decision

variables and communication parameters, it gives a simple way to apply the Principie of Optimality over

the subsystems, once an artificial pattern of serial

connection has been chosen. The algorithm

com-plexity was also analyzed for the general case where

the solution is performed over a finite grid of quantized variables.

To circumvent to some extent the curse of

dimen-sionality, an action radius concept has been intro-duced as a decentralization constraint. Thus, the

method has been succesfully applied to the optima!

design of a 25-node network with a 3-node action

radius, which constitutes a valid case from a

prac-tical point of view. CPU time of the arder of

hundred of seconds on a SUN 3/110 can be

con-sidered satisfactorily as well as the reasonable

storage effectively required. This action radius cleariy

appears as a key factor in finding a compromise

between NSDP algorithmic complexity and the physical significante of the distribution network_

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10 35

A

120 KVA

75 KVA

A

45 KVA

400

410

110

20 10 35

A

120 KVA

• 75 KVA

A

45 KVA

Figure 5 - The oprima' design with perturbation

011 CT

10

A

120 KVA

• 75 KVA

• 45 KVA

Figure 6 - The optima' design with perturbation

Of1 Cs

and

CL

41

400

t(i) , V i ' I

0

45

75

120

e

r

0

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KVA

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Action radius = 3

s(j) , V j ' J

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4/0

400

c

5

0

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145

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0.07

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0

0.18

0.17

0.15

units/m

units/KVA

2

m

KVA

KVA

10-5 (KVA m) -1

units/m

units/KVA2 m

Decomposition

_{min NE}

_{Optimal ordering}

5 subsystems

(figure 2.a)

6.51.10 12

1 2 3 4 5

7 subsystems

(figure 2.b)

2.52 . 10

10

1 2 4 5 6 7 3

13 subsystems

(figure 2.c)

2.37.109

5 3 2 1 4 6 7 8 10 13 12 11 9

Table I - Secondnry problem solutions

Table II - Numerical data

Referentes

1. U. BERTELE and F. BRIOSCHI (1972) Nonserial

dynomic programming. Academic Press, New York 2. A. ROSENTHAL (1982) Dynamic programming is

optima] for nonserial optimization problems. SIAM J. Comp. 11, 47-59.

3. G. NEMHAUSER (1966) Introduction to dynomic

programming. iley, New York

4. D. FERNÁNDEZ-BACA (1988) Nonserial dynamic programming formulations of satisfiability. Infor-mation Processing Letters 27, 323-326.

5. P. McENTIRE, C. CHONG and R.E. LARSON (1978) A global optimality theorem for spatial dynamic programming. Presented at theSecond

Lawrence Symposium on Systems and Decision Sci-ences, Berkeley, California, 1978.

6. C. CHONG, P. McÉNTIRE and R.E. LARSON (1978) Decomposition of mathematical program-ming by dynamic programprogram-ming. Presented at the

Second Lawrence Symposium on Systems and Decision Sciences, Berkeley, California, 1978.

7. S. VERDÚ and H.V. POOR (1984) Backward, forward and backward-forward dynamic program-ming modeis under commutativity conditions. Pre-sented at the23rd Conference on Decision and Con-trol, Las Vegas, December 1984.

8. R.E. LARSON, P. McENTIRE and T. STEDING (1979) Foundations of spatial dynamic program-ming. Proceedings of I 8th IFFF Con ference on Deci-sion and Control., San Diego, California, 1979. 9. H.L V11LUS and J.ED. NORTHCOTE-GREEN

(1985) Comparison of severa! computerized dis-tribution planning methods. IEEE Transactions on PAS 104, 233-240.

10. R.N. ADAMS and MA LAUGHTON (1974) Op-tima' planning of power networks using mixed-integer programming. Proceedings of lEE 121,

139-148.

11. D.L. WALL, G.L. THOMPSON and J.E.D. NORTHCOTE-GREEN (1979) An optimization model for planning radial distribution networks.

IFFF Transactions on Power Apparrrtus and Systems

98, 1061-1068.

J.T. BOARDMAN and C.C. MECKIFF (1985) A branch-and-bound formulation to an electricity distribution planning problem. IFFF Tronsacdc.is on Power Apparatus and Systems 104, 2112-2118.