A Context-free Smart Grid Model using Pretopologic Structure

Guillaume Gu

erard, Souﬁan Ben Amor and Alain Bui

PRiSM CRNS-UMR 8144, University of Versailles SQ, 45 avenue des Etats-Unis, Versailles, France

Keywords:

Smart Grid, Complex System, Pretopology, Game theory, Multi-agent System.

Abstract:

The Power grid evolves, but its structure presents several gaps with the new numerical technologies, renewable

energies and electric vehicles. The literature introduces the concept of Smart Grid, a system which takes into

account the behaviour and the action of its agents. Studying the smart grid through modelling and simulation

provides us with valuable results which cannot be obtained in the real world due to time and cost related

constraints. Nevertheless, due to the complexity of the smart grid, achieving global optimization is not an easy

task. In this paper, we propose a complex system approach to the smart grid modelling, accentuating on the

representation of the structure. Thanks to this combination, the optimization can be achieved on a dynamic

graph taking into account changes and network errors over time, with the ability to detect them.

1 INTRODUCTION

Our society is electrically dependent. The Power Grid

supplies energy to households, businesses, and indus-

tries. Nevertheless, disturbances and blackouts are

becoming common. With the pressure from ever in-

creasing energy demand and climate change, ﬁnd-

ing new energy resources and enhancing energy ef-

ﬁciency have become the priority of many nations in

the 21st century.

The term Smart Grid is coined by Amin in 2005

(Amin and Wollenberg, 2005). Then, the expression

”Smart Grid” has expanded into different dimensions:

some see it as a numerical solution for downstream

counter and mostly residential customers, while oth-

ers have a global vision that transcends the current

structure of the energy market to generate economic,

environmental, and social beneﬁts for everyone. This

article (Gu

erard et al., 2012) includes a survey on

Smart Grid models.

Taking into account all its actors, internal and ex-

ternal features, the Smart Grid is deﬁned as a complex

system of subsystems (Gao et al., 2012). Before at-

tacking optimization problems and network structure

in those systems, we must understand the global and

local complexity in each subsystem.

The articles (Gu

erard et al., 2012), (Gu

erard et al.,

2012) refer to the literature about Smart Grid; the two

articles (Ahat et al., 2013) and (Amor et al., 2014) re-

fer to the complex system approach and the bottom-

up analysis of the Smart Grid; (Gu

erard and Tseveen-

dorj, 2014) exposes the mathematical approach of the

Smart Grid; (Amor et al., 2014) presents the game

theory used for the demand-side management.

In this paper, we focus on the dynamic structure

of the Smart Grid. How to create a graph representing

a dynamic structure? How to solve it? How to predict

consumption and to adjust production in the future?

In the following sections, our model will be pre-

sented, especially the use of pretopology to describe

a complex network. This paper is organized as the

following: in the next section, complex system is in-

troduced, theoretical approach in modelling the Smart

Grid as a complex system is discussed. In Section 3,

we present the details of our Smart Grid model. The

Section 4 is devoted to the pretopologic approach and

the feedback system is exposed in Section 5. Then,

how to plan future consumption and production is de-

scribed in Section 6. The section 7 is devoted to ﬁrst

result and future works.

2 COMPLEX SYSTEM

APPROACH

A system which consists of large populations of con-

nected agents, or collections of interacting elements,

is said to be complex if there exists an emergent

global dynamic resulting from the actions of its parts

rather than being imposed by a central controller.

That is a self-organizing collective behaviour difﬁcult

to anticipate from the knowledge of local behavior

335

Guérard G., Ben Amor S. and Bui A..

A Context-free Smart Grid Model using Pretopologic Structure.

DOI: 10.5220/0005409203350341

In Proceedings of the 4th International Conference on Smart Cities and Green ICT Systems (SMARTGREENS-2015), pages 335-341

ISBN: 978-989-758-105-2

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

(Boccara, 2004).

In (Segel and Cohen, 2001), the authors state that

an autonomous distributed network that process in-

formation adaptively is more effective in describing

the immune system and cellular metabolism. Segel

and Cohens remark can be taken into account by the

majority of complex systems to manage common re-

sources, especially for the Smart Grid.

The contribution of our approach consists of con-

sidering the Smart Grid as a complex system, solving

the problems at local as well as global level with co-

ordinated methods, presented in (Ahat et al., 2013)

and (Amor et al., 2014). The generality of our ap-

proach allows its applicability in various scenarios

and models, that guarantees the ﬂexibility of the ex-

posed model. The following paragraphs summarize

the proposed approach.

At ﬁrst step, the system should be understood. An

overview brings structural aspects, entities with goals

and behaviours. These one are not randomly dis-

tributed in the system, but according to patterns, and

form distinct groups with their own arrangement.

After analysing the characteristics of the system,

the sub-components are deﬁned. A sub-component

has a structure, objectives and speciﬁc entities, al-

though quantities or position in the system is variable.

As a separate system, it has its own dynamics or an

auto-organization. It is then possible to solve it with

an appropriate optimization method. Through all the

sub-component, a global behaviour emerges.

The sub-components are in interaction, then you

should take into account the I-O data for each method.

The stability of the model depends on local optimiza-

tion, and interactions. It is necessary to optimize each

part of the chain as well as a whole to stabilize the

system.

To prevent system crashes, the model must have

a system of communication and feedback to reach

a global consensus. Moreover, the system is sub-

jected to external pressures. Feedback between sub-

components are essential in maintaining the function-

ality of the complex system.

In the Smart Grid, we aim to optimize the energy

distribution, it also includes the management of pro-

duction, consumption and distribution of the common

resource. Our optimization takes into account the re-

silience and reliability of the network and the research

of minimum cost in a market economy (Ahat et al.,

2013).

3 OVERVIEW AND PROCESS

3.1 A Three Layered Grid

The exposed model has three sub-components: the

T&D, the microgrid and the local level, see ﬁgure 1.

Figure 1: Smart Grid sub-components (from PowerMatrix,

Siemens).

Network transmission and energy distribution net-

work or T&D is a 2-connected structure containing

electricity generators, central-type agents and grid

agents, represented by the two ﬁelds around the center

ring in the ﬁgure 1. Its main role is to deliver energy

to consumption points.

The second level is the link between consump-

tion and energy production, represented by the outer

yellow ring. The microgrid is a broader view of lo-

cal consumers, it is a tree structure representing an

ecodistrict bounded by the upstream substation. Its

role is to distribute energy from substation to con-

sumers. For this, it orders or books an amount of en-

ergy from the T&D network to local consumers.

The outer ring is dotted of local levels. Those con-

sumers are connected to a substation itself connected

to the grid energy. This isolated agent, representing

a residence, factory, etc., supports the consumption

of energy, which is the distribution of energy among

appliances under its responsibility. In other words, a

local level is deﬁned by the area under the control of

a smart meter or other automation/management con-

troller.

3.2 Algorithmics

An iteration occurs every ﬁve minutes. Once data

are updated, the process is decomposed into four se-

quences, see ﬁgure 2. Data are synchronized in order

to let time to compute and to ﬁnd an equilibrium be-

tween each agent of the system (in simulation or in

real network).

Sequence A. to design the intelligent aspect of the

device, a priority is assigned to the devices, and

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Figure 2: Sequential Scheme.

for calculating a consumption value. Indeed, we

use a local knapsack problem, solved by dynamic

programming after data normalization, in order

to ﬁnd a primary ﬁrst optimal resource allocation

(Ahat et al., 2013).

Sequence B. the microgrids book an amount of en-

ergy from producers to consumers using an auc-

tion (Amor et al., 2014). There are two ways

to book energy: a consensus between consumers

and producers, that is a unique game where mi-

crogrids and energy ﬂows are players; and a bid

system with feedback, each microgrid do a local

game with producers. The problem with the ﬁrst

one is the complexity of the problem, impossible

to resolve in a few times for large instance. The

second way has the advantage of time, but do not

guarantee the global optimum. In this model, the

auctions will be adjusted thanks to the feedback

system.

Sequence C. about the routing problem, nodal rule

or Kirchhoff’s circuit speciﬁes that at any node

in a circuit, the sum of currents ﬂowing into that

node is equal to the sum of the currents ﬂowing

out of that node. An electrical circuit is equal to a

graph in which a junction is a node, and physical

connection corresponds to an edge. Routing prob-

lem is equivalent to the known Max ﬂow problem.

Gale’s theorem shows the existence of a solution

in a network of offers and requests (Gale, 1957).

The Max ﬂow at Min cost problem is solved by

Busacker & Gowen (Section 4.2). To recalcu-

late the entire ﬂow is not necessary. The residual

graph removes overﬂows between two updates,

optimizing the computation time of the optimal

ﬂow, see ﬁgure 3.

Sequence D: energy is distributed by knapsack prob-

lem, according to the last auctions. The un-

consummated energy is redistributed among non-

used devices at upper scale. Each device’s priority

is updated according to the result of the ﬁnal dis-

tribution.

Figure 3: Updating of routing.

4 HOW TO UPDATE THE T&D

STRUCTURE THANKS TO

PRETOPOLOGY THEORY

4.1 Notion of Proximity

Complex systems manifested complex characteristics

which are not found in simple networks, and thus

called complex networks. The complex network the-

ory has become a major interest area in complex sys-

tem study and provides mathematical tools to model

the structure of complex systems (Newman, 2003).

The pretopology is a mathematical theory ahead

of the conventional axiomatic topology, which allows

us to express the structural transformations of sets

AContext-freeSmartGridModelusingPretopologicStructure

337

of interacting elements such as building coalitions

among a population phenomenon alliance process of

tolerance, acceptance and the emergence of collective

behaviour. Pseudo-closure and closure are two func-

tions used to model basic operations in complex net-

work theory (Belmandt, 2011). The topology is a par-

ticular case of the pretopology.

The complex networks such as the Smart Grid

change at each time. It is therefore important to model

these functions by taking into account the dynamics.

A complex network is seen as a family of pretopology

on a given set as shown in the ﬁgure 4. The advantage

of this theory is the separation of each criterion in a

pretopologic space to simplify modelling. The over-

all adhesion function is deﬁned as an aggregation of

several spaces. In this manner, a modiﬁcation in a

space is instantaneous. The pretopologic spaces may

include various types of relationships, such as metric

spaces, binary spaces or valued spaces.

Figure 4: A family of pretopology.

The Smart Grid is a complex system, it is gov-

erned by a number of qualitative and quantitative cri-

teria. The voltage of the power lines is an example,

but it is also possible to take into account the electri-

cal leakage based on the length of lines, installations

wear, weather, etc.

Finally, pretopology is a very useful modelling

tool in the context of complex systems to manage ac-

quaintances between agents and be able to follow the

dynamics of relationships between them (Petermann

et al., 2012).

4.2 Routing Problem

In order to deﬁne undercharged, or overcharged lines

during the sequence C, a pretopologic analysis is con-

ducted. For example, let be three pretopologic spaces

, a

and a

. Each edge has three levels of ﬂow cor-

responding to under-load, normal load and overload.

The ﬁgure 5 presents the pretopologic family.

Under-load is possible on an edge if it exists in the fol-

lowing logical space a

∩ a

named [1], same for

overload in the logical space (a

∪ a

)∩ a

named [3].

By default, any edge carries an average load named

[2]. The ﬁgure 6 presents the ﬁnal graph.

Figure 5: Pretopologic spaces.

Figure 6: Network according to pretopologic spaces.

The constructed network is a fully connected

graph which edge’s capacity depends on pretopologic

spaces. Each edge (i, j) is characterized by:

• d

i j

the maximum capacity;

• l

i j

the minimum capacity;

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• c

i j

the unit cost of the ﬂow in the edge. The cost

may vary in function of the total ﬂow. Edge is

duplicate with different costs related to capacity.

For example, the initial cost is 1 for a ﬂow among

[0, 3], 3 among [3, 6], and 5 among [6, 8]. Three

edges rely i to j. Because the cost function is

stricter increasing, this method does not perturb

the algorithm of maximum ﬂow at minimum cost

only if. A path with an available capacity is called

an augmenting path. At each iteration, the edge

(i, j) is valued at c

i j

if the edge (i, j) is not satu-

rated. The edge ( j, i) is valued at −c

i j

if the edge

(i, j) is not empty;

• x

i j

the ﬂow passing through the edge.

In order to resolve this routing problem, the Bu-

sacker & Gowen algorithm is used, an example is

shown in the ﬁgure 7. The idea behind the algorithm

is: as long as there is a path from the source (the start

node) to the sink (the end node), with an available ca-

pacity on all edges in the path, we send ﬂow along

one of these paths, ﬁlling in priority the path with the

minimal cost. Then we ﬁnd another path, and so on.

Figure 7: Busacker & Gowen algorithm.

5 FEEDBACKS

Various criteria and technical constraints limit the

amount of energy that circulate in each edge. Produc-

tion and consumption must match as best as possible.

The energy produced or the energy consumed (se-

quence B and C) may be different. This is a mis-

management of resources for the consumers, or mis-

management of the energy produced. The difference

between the total energy and the total consumed en-

ergy (equal to Max ﬂow) can be signiﬁcant. A recog-

nition algorithm must locate bottlenecks to perform

feedback between T&D network and each microgrid.

For these, two tests are used:

• The Max ﬂow problem on the graph without ca-

pacity constraints on the edges containing the

sink. The result will provide the maximum un-

constrained consumption.

• The Max ﬂow problem on the graph without ca-

pacity constraints on the edges containing the

source. The result will reﬂect consumer demand

in order to predict future production.

Management on the consumer side, called De-

mand Side Load Management (DSM), aims to in-

crease the efﬁciency of generating by shifting con-

sumption in low consumption period (Saad et al.,

2012). Many devices can temporarily go into a

standby mode (heating) or consumption can be post-

poned. Some devices are also able to stop using en-

ergy during operation (preemption). Approximately

50% of the consumption of residential areas can be

controlled without reducing the comfort (Block et al.,

2008).

To increase the effectiveness of the DSM, it is

assumed that the front part of the infrastructure is

a home automation and every device can be con-

trolled separately by the user and regulation algo-

rithms. Feedback of the sequence C on sequence B,

with rewards or punishments, determines how smart

meter will control automation in order to ﬁnd a better

auction.

The gap between the constrained solution and the

two tests determined how to perform the feedback.

If a microgrid can consume more than its bid, it in-

creases the bid during the next auction. The values

obtained by the graphs are used in the feedback to

punish or reward and adjust the new bids. The ﬁgure

8 shows the bid and the received energy in a microgrid

at each feedback (in this case, the curves are voluntar-

ily exaggerated to see the gap between each one).

Feedback reorganizes the distribution of resources

among the different microgrids. After a limited num-

ber of feedbacks, supply and demand ﬁnd a consensus

throughout the graph as shown in ﬁgure 8.

Figure 8: Equilibrium thanks to feedback in a microgrid.

The feedback seems trivial, but if we consider

thousands of production and consumption places,

then the problem of maximal ﬂow has multiple valid

patterns. Added to a global policy, these attempt to

understand production and consumption behaviours

in order to build valid patterns for future iterations.

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6 GLOBAL DIRECTION

6.1 For the Following Iterations

When searching for the shortfall in production or con-

sumption during the sequence C, the ideal distribution

of production without consumption constraint and the

ideal distribution of consumption without production

are calculated (see previous section). These data al-

low us to understand the evolution of consumption

over time. Prognostics are calculated at the end of

the sequence D as follows:

1. For consumers: consumption of future consump-

tion is the weighted average of the auction con-

ducted. Let z

be the bid made at the feedback

i − 1, the consumption used by prognostic is cal-

culated as follows: Z = 2

∑

i=1

i∗z

n(n−1)

. The latest bids

have a greater impact on the prognostic.

2. For producers: prognostics are calculated like the

previous one. The plants do not have the same

ability to adjust production, and need a consensus

among themselves. Currently the model is only

supporting a single producer (with many plants).

Thus, the calculation of upcoming productions

does not consider the competition. We actually

work to how to plan the production with many

producers.

6.2 For Further Iterations

At long term, we must equalize the production curve

of the Smart Grid. Local agents do not have an

overview of the consumption (or production) curve.

In order to smooth the curve, an algorithm, based on

a mathematical function, guarantees the quality of the

solution, or, give advice for feedback.

Bounded function cannot cross some threshold. In

the case of consumption, that means the curve cannot

pass under or over a ﬁxed value. In a perfect Smart

Grid, a bounded function guarantees a continuous

consumption. But, in a real network, the consump-

tion during night or day, or the sector (primary, sec-

ondary, tertiary), or the external factors, or the scale

of the grid, may radically vary. To avoid incoherent

behaviours due to inappropriate management policy,

a function bounded on its slope must be used.

Lipschitz continuity, named after Rudolf Lips-

chitz, is a strong form of uniform continuity for func-

tions. Intuitively, a Lipschitz continuous function is

limited in how fast it can change: there exists a deﬁ-

nite real number such that, for every pair of points on

the graph of this function, the absolute value of the

slope of the line connecting them is not greater than

this real number (see ﬁgure 9). If the consumption

curve is k-Lipschitz, we guarantee that the curve can-

not have a peak demand (depends on the value of k).

Before each feedback between sequence B and C,

we check if the result is Lipschitz continuous. If it is

not, the auctions are adjusted. If it is continuous, and

the solution is eligible, then the sequence D starts.

Figure 9: A Lipschitz function.

7 FUTURE WORKS

To validate the model, instances at local and global

scale have been made. Agents data, like engine

consumption or energy plants production, are imple-

mented thanks to the french national production com-

panies public data and energy distribution data (EDF

and RTE).

In the ﬁrst tests, consumption and production tend

towards equilibrium. Local and renewable energies

are privileged to maximize their proﬁtability. The

model limits the losses of the distance of consump-

tion, and uses the least amount of fossil energy. It

works at any scale and any agents under the condition

that there exists a feasible solution.

The proposed model works for randomized or

parametric Smart Grids. We actually work in the Posi-

tive Energy 2.0 project led by ALSTOM Energy Man-

agement and various companies such as Bouygues or

Renault to validate the model on real projects.

The model will be compared to, electronic, auto-

matic models and simulations. Positive Energy 2.0

project provides two microgrids (one using electronic

networks and one using automatic devices), Embix

provides data from several buildings and plants, and

some microgrid simulations.

The analysis of the ﬁrst results with public data,

the project’s results with private data and the eco-

nomic study will form a forthcoming publication.

The model can be improved. The learning pro-

cess must be developed in order to optimize the auc-

tion and the choice for utility functions. Moreover,

artiﬁcial intelligence will be implemented in order to

adjust each parameter in real time. The Smart Grid

change over time, so the AI must ﬁnd the best con-

ﬁguration in real time in order to keep the Smart Grid

optimized and to avoid local dysfunctions.

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8 CONCLUSION

As Smart Grid can be qualiﬁed as a complex system,

classical optimization methods cannot be applied di-

rectly, due to the computational complexity for time

and memory.

Integrating pretopology offers a better notion of

proximity between the agents. This allows handling

multiple criteria simultaneously using an aggregation.

In addition, the modiﬁcation of a single criterion en-

tails that the update of the pretopology assigned and

not a total restructuring of the network.

More generally, we also demonstrated how to

solve optimization problems in complex systems.

While applying for optimization algorithms directly

in complex systems is nearly impossible, we should

analyse the system and divide them into sub-systems

with deﬁned characteristics, then we should apply for

speciﬁc algorithms and coordinate them using multi-

agent simulation in order to achieve global optimiza-

tion on a deﬁned and dynamic network.

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