domain: spi math

text:
A M Kabaasha

J E Van Zyl

O Piller

MODELLING PRESSURE -LEAKAGE RESPONSE IN WATER DISTRIBUTION SYSTEMS CONSIDERING LEAK AREA VARIATION

Keywords: Pressure, Leakage, N1-Equation, FAVAD-Equation, Pressure Management

Research has shown that area of leak openings varies linearly with pressure under both elastic and viscoelastic conditions. However, hydraulic modelling of water distribution systems currently uses a power function that does not explicitly take into account leak area variation with pressure. This function is commonly known as the N1 power equation.

This paper investigates the possibility of explicitly incorporating leak area variation into hydraulic modelling of water distribution systems. The FAVAD (Fixed And Variable Area Discharges) equation, which takes into account leak area variation with pressure, was incorporated into the widely used Global Gradient Algorithm (GGA) in EPANET.

A stochastic model was developed to generate and distribute leaks in a standard water distribution system. It was based on existing theories about leakage in water distribution systems, taking into account physical properties of the system.

Thereafter, hydraulic simulations were run using both FAVAD and N1 equations.

The results show that the N1 power equation estimates the total system leakage with an error of up to 20%. On a specific junction, the leakage estimation error was found to be as high as 60%.

INTRODUCTION

N1 Power Equation

Hydraulic modelling tools, such as EPANET, model pressure-dependent demands using an emitter function, which is a power function: Pressure management is the application of techniques to reduce excess pressure in an isolated zone of a water distribution system, with the aim of reducing leakage levels. The above function, which is also commonly known as the N1 power equation, is used widely by water utility practitioners when implementing pressure management.

Q = Ch %& (
The N1 power equation has the following disadvantages:

• It is an empirical equation and therefore not founded on fundamental principles of fluid mechanics.

• The values of C and N1 are not constant for a given system, but depend on the pressures at which they are being estimated [START_REF] Van Zyl | Predicting the pressure-leakage relationship: technical paper[END_REF].

• The equation is dimensionally awkward, since the units of C include the variable N1.

Author-produced version of the article published in : 14th CCWI international conference, Computing and Control in Water Industry, 7/11/16, Amsterdam

FAVAD Equation

Hydraulically, leaks are orifices and therefore should adhere to the orifice equation with fixed exponent of 0.5: Using finite element analysis under linear elastic deformation, [START_REF] Cassa | Predicting the pressure-leakage slope of cracks in pipes subject to elastic deformations[END_REF], showed that leak area varies linearly with pressure irrespective of leak type, loading conditions, pipe material and section properties. Some experimental studies like Ferrante et al., (2012 and[START_REF] Cassa | Predicting the pressure-leakage slope of cracks in pipes subject to elastic deformations[END_REF], have also confirmed these results. [START_REF] Ssozi | Numerical Investigation of the Influence of Viscoelastic Deformation on the Pressure-leakage Behaviour of Plastic Pipes[END_REF] also investigated the impact of viscoelasticity on pressure -leakage relationship using finite element analysis and found a linear relationship. Thus, the relationship between leakage area and pressure may be described with the equation;

𝑄 = 𝐶 ) 𝐴 2𝑔ℎ (2
𝐴 = 𝐴 . + 𝑚ℎ (3)
where A = leak area at head h, A 0 = initial leak area (area of leak under zero pressure conditions) and m = head-area slope (potential of pipe to increase leak size under pressure).

Replacing the leak area (A) in equation 2 with equation 3, results in the FAVAD (Fixed And Variable Area Discharges) equation;

𝑄 = 𝐶 ) 2𝑔 𝐴 . ℎ ..2 + 𝑚ℎ &.2 (4)
The FAVAD equation was first introduced by [START_REF] May | Leakage, Pressure and Control[END_REF]. The first term of equation 4 is identical to the orifice equation and describes leakage flow rate through the initial leak area, while the second term describes leakage flow rate through the portion of the leak area that has expanded due to pressure. FAVAD equation provides more realistic description of leakage behaviour for both individual and system leaks than N1 power equation, although it has never been incorporated into any standard hydraulic modelling tools. In this study, the FAVAD equation was incorporated into the widely used implementation of Global Gradient Algorithm (GGA) in EPANET. [START_REF] Schwaller | Modelling the Pressure-Leakage Response of Water Distribution Systems Based on Individual Leak Behaviour[END_REF], developed a stochastic model for generating and distributing leaks in a water distribution system. These leaks adhere to FAVAD behaviour with the two FAVAD parameters, initial leak area (A 0 ) and the head -area slope (m). The same model was used as the basis for this research.

METHODOLOGY

Incorporating FAVAD concept into standard hydraulic modelling

The GGA in EPANET's hydraulic solver uses an emitter function, equation 5, to model pressure dependent demands like leakage;

𝑄 3 = 𝑘(ℎ) 73 (5)
𝑄 3 = emitter flow rate

E e = emitter exponent k = emitter coefficient h = pressure head.
Author-produced version of the article published in : 14th CCWI international conference, Computing and Control in Water Industry, 7/11/16, Amsterdam This emitter function, was utilised to incorporate the FAVAD concept into standard hydraulic modelling. According to the FAVAD concept, total leakage through a leak is the sum of leakage through both the initial and expanding leak areas.

Therefore, a second emitter function was added to the existing one (see equation 6), in which case the first/original emitter function models leakage through fixed leak area while the second/added one models leakage through expanding leak area.

𝑄 8 = 𝑘 & ℎ ..2 + 𝑘 9 ℎ &.2 (6)
𝑄 8 = total junction leakage flow rate

k 1 = first emitter coefficient (𝐶 ) * 2𝑔 * 𝐴 . ) k 2 = second emitter coefficient (𝐶 ) * 2𝑔 * 𝑚) h = pressure head.

Stochastic model for generating and distributing leaks to a standard water distribution system

A stochastic model was used to generate and distribute FAVAD leaks with random parameters to a standard water distribution system. From this random but realistic distribution of leaks, 200 systems with different leak distributions were generated.

One of International Water Association's water distribution system performance indicators, the ILI (Infrastructure Leakage Index) was used to model leakage level. The ILI is defined as the ratio of current annual real loss to unavoidable annual real loss [START_REF] Lambert | Ten years' experience in using the UARL formula to calculate Infrastructure Leakage Index[END_REF]. In this study, an ILI of 16 was considered realistic for a typical water distribution system.

The parameters that were statistically generated or calculated are; discharge coefficient, initial leak area, and head-area slope.

Because the leak generation process is described in detail in [START_REF] Schwaller | Modelling the Pressure-Leakage Response of Water Distribution Systems Based on Individual Leak Behaviour[END_REF], only the main points are highlighted below:

• discharge coefficient (𝐶 ) ) was modelled using a normal distribution.

• initial leak area (𝐴 . ) was modelled using lognormal distribution bounded by zero for background leaks, and a normal distribution for potentially detectable leaks since they can be easily found and repaired quickly in a system.

• head-area slopes (m) for background leaks with N1 close to 1.5 were calculated using equation 7. These background leaks have initial areas that close completely under zero pressure and their leakage number is 100, [START_REF] Cassa | Predicting the pressure-leakage slope of cracks in pipes subject to elastic deformations[END_REF]. Given the fact that it is not clear what percentage of background leakage could be due to this type of leaks, a fraction of 20% was used.

• head-area slopes of remaining background leaks plus potentially detectable leaks were calculated using a generalized power function of initial leak area, based on finite element study by [START_REF] Cassa | Predicting the pressure-leakage slope of cracks in pipes subject to elastic deformations[END_REF].

𝑚 =

; < * = > ?

Numerous leaks equivalent to an ILI of 16 were generated. Although the generating process was random, it was ensured that an accuracy of plus or minus 0.001% was archived. A leak was then randomly picked and allocated to a pipe. The allocation of leaks to pipes took into account system physical properties. A pipeline with high length was considered to have high probability of having leaks. Cumulative length of all pipes in the system was calculated, and a number between zero and the final cumulative length was randomly generated and used to identify the pipe on which to allocate the leak.

Modelling leakage flow rate with both FAVD and N1 equations

Before applying pressure management

Leakage was first modelled before applying pressure management, using both FAVAD and N1 equations. As the distributed leaks adhere to the FAVAD approach, each one of the 200 randomly generated systems, was simulated first using the

Author-produced version of the article published in : 14th CCWI international conference, Computing and Control in Water Industry, 7/11/16, Amsterdam FAVAD approach, and then N1 approach. This was done by converting from the FAVAD to N1 format. Junction leakage and pressure were obtained in both approaches.

The conversion of leaks from FAVAD to N1 approach was done in the following steps:

• Calculate system leakage exponent (N1 value) at MNF (Minimum Night Flow) conditions,

• Calculate junction leakage coefficients at MNF conditions,

• Calculate the emitter coefficient and exponent and model it using the first emitter in the system (the second emitter is set to zero).

In order to calculate system leakage exponent, at MNF conditions, a data set of system leakage and AZNP (Average Zone Night Pressure), were determined. Pressure in the system was then reduced slightly and another set of system leakage and AZNP determined. From the two data sets, system leakage exponent (N1) was calculated using equation below;

𝑁 & = ABC D E D F ABC GH<I E GH<I F (9)
where Q 1 and Q 2 are system leakages and AZNP 1 and AZNP 2 are average zone night pressures before and after pressure reduction respectively.

With system leakage exponent and leakage at each junction during MNF conditions, junction leakage coefficient was calculated using equation below;

𝐶 = J ? < E (10)
where Q is the junction leakage flow rate at MNF conditions, h is the corresponding junction pressure head and N1 is the system leakage exponent.

With leaks converted to N1 approach, an extended period hydraulic simulation was done and leakage flow rates plus pressure heads were determined.

After applying pressure management

Pressure management was then applied by reducing the level of water in the reservoir, from 125 to 75m, for both FAVAD and N1 approaches. An extended period hydraulic simulation was done and leakage plus pressure heads determined at each of the junctions.

Description of standard water distribution network used in the study

The network used in this study was adapted from EPANET's "Net3.net". The original "Net3.net" system has two sources, and it operates under pumping conditions. In this study the pumps were removed and only one source maintained, as shown in Figure 1. The head at the source, was adjusted to ensure that the range of pressures in the system under gravity were close enough to the original system that operated under pumping conditions. Total length of pipes in the system is 60 km, with pipe diameters ranging from 200 to 800mm.

Critical node

Before leaks were distributed into the standard water distribution system, a node was identified whose pressure was minimum during peak demand conditions. This node is referred to as a "critical node", and it is used in this study to give a representation of how leakage varies on an individual junction.

Author-produced version of the article published in : 14th CCWI international conference, Computing and Control in Water Industry, 7/11/16, Amsterdam

RESULTS AND DISCUSSION

Modelling FAVAD and N1 leakage at varying pressure values

A single period simulation was run during peak demand conditions, with both FAVAD and N1 approaches. Leakage was then modelled before and after applying pressure management. Different levels of pressure management were considered.

Calculation of system N1 was done at the highest pressure, before pressure management. It is therefore evident that at the same pressure where the N1 calculation was done, leakage modelled with both FAVAD and N1 are close enough, as shown in Figure 2. However, as pressure is reduced further, significant differences in leakage are observed. This is because the N1 formulation is an empirical formulation that works in a given range of data set. If used outside the calibrated range of data set, the solution will be erroneous.

Volumetric percentage error in system leakage modelling after pressure management

An extended period simulation was done with both FAVAD and N1 approaches after applying pressure management. The error in system leakage was expressed as a percentage. Negative sign indicates that N1 approach underestimates leakage flow rate.

In about 30% of the systems simulated, the N1 approach was found to underestimate system leakage flow rate in the range of 11.2 to 9.6%. Some systems had a percentage error of up to 20%, see Figure 3.

Volumetric percentage error at individual junction

In order to understand the volumetric error on individual junctions, leakage on the critical node was considered in both FAVAD and N1 approaches. Figure 4, shows the percentage of networks with corresponding volumetric percentage errors. Also in this graph, the negative sign means that the N1 approach underestimates leakage flow rate. About 56% of the systems simulated, indicated that the percentage error is in the range of 46.25 to 33.75%. Some systems showed an underestimation of 60% while others showed an overestimation of up to 56%.

CONCLUSION

This paper investigated the possibility of incorporating leak area variation with pressure into standard hydraulic modelling. This was achieved through implementation of the FAVAD concept into the widely used global gradient algorithm in EPANET.

Simulation results showed that using N1 equation outside a pressure range used in calculating the N1 value, results in leakage estimation error. FAVAD and N1 equations gave similar leakage flow rates, when pressure was very close to the one used to calculate system N1. With further reduction in pressure, which is always the case in pressure management, the percentage error between the two approaches increased significantly.

The use of N1 equation in estimating leakage reduction by pressure management, should therefore be done with caution. This is because pressure management always involves reducing pressure, while the N1 value is calculated at high pressure. Author-produced version of the article published in : 14th CCWI international conference, Computing and Control in Water Industry, 7/11/16, Amsterdam

  ) Q = leakage flow rate, C d = discharge coefficient, A = leak area, g = acceleration due to gravity and h = pressure head.

Figure 2 :Figure 3 :

 23 Figure 2: Leakage variation at critical node with varying pressure heads


Fabrice Boissier
email: fabrice.boissier@gmail.com

Irina Rychkova
email: irina.rychkova@univ-paris1.fr

Jelena Zdravkovic
email: jelena.zdravkovic@dsv.su.se

Extending Enterprise Modeling for Decentralized Organizations

New business models imply decentralization of organizations and subsequent change of their management and operation styles. Originally designed for classical (bureaucratic) organizations, many Enterprise Modeling (EM) methods do not explicitly address decentralization and thus require "case-based" adaptations when applied to modern organizations. In this work, we analyze various organizational taxonomies presented in the literature focusing on criteria of organization decentralization. Using two case examples, we demonstrate how the application of the For Enterprise Modeling (4EM) method, efficient for a centralized organization, becomes more challenging for a decentralized organization. We discuss how this method can be extended in order to provide an explicit support for organizational decentralization.

. We selected this method because it covers all the essential aspects of an organization (4EM defines six sub-models focused on goals, rules, processes, actors and resources, concepts, requirements and technical components respectively). A predecessor of 4EM, the EKD approach [3], is proven to be efficient by a number of industrial projects. In addition, the documentation for 4EM is publicly available.

Decentralization is widely discussed in the literature [2][13][17][4]: it is considered as one of the design parameters that predefine performance and evolution of a firm. No general agreement, however, can be seen about the criteria that define (and can be used to evaluate) the (de)centralization of a firm. In this work, we identify the criteria of organizational decentralization provided in the literature, propose their synthetic summary and illustrate them on two examples inspired by real companies: a traditional (centralized) and a decentralized organization. We synthesize our experience and the evidences presented in the literature and identify several challenges that modeling experts can face while creating an enterprise model for a decentralized organization. These challenges are typically handled on the "case basis" and resolved thanks to the experience and often informal communication between practitioners. However, we consider that the integration of the explicit knowledge about decentralization and its specifics into enterprise modeling methods can be very useful. We propose to adapt the EM methods by explicitly decoupling the enterprise models that reflect the corporate (centrally managed) and the local (distributed) elements (e.g., goals, rules, processes, requirements etc.), by providing the relations between these decoupled models and by specifying the coordination and communication processes that would be needed to create and validate these models across an organization. Whereas for traditional, centralized organizations, the local elements will have very little or no impact on the overall EM model, for the post-modern organizations with decentralized structure and distributed decision-making, local models will play a leading role. We illustrate our proposal on the example of 4EM method.

This paper is organized as follows: In Section 2, we discuss the literature on the organizational structures and enterprise modeling methods. In Section 3, we identify and synthesize the criteria of centralization/decentralization and present two examples that illustrate the typical characteristics.We discuss how the enterprise modeling in these organizations can be conducted and identify some challenges related to the decentralization in the latter case. In Section 4 we propose some extensions to EM methods in order to address these challenges. We illustrate our proposal on the example of 4EM. Section 5 presents our conclusions and defines various objectives for future work.

Introduction

Cloud computing, mobile computing, social networks, peer production, open innovation -are examples of paradigms and technologies that enable new business models today. Adhering to these models, modern organization tend to abandon bureaucratic organizational structures based on formal planning and centralized decision-making and to adopt less formal, decentralized structures and management styles.

Since organizations rely increasingly on their information and communication technologies (ICT), any organizational transformation implies (and depends on) a transformation of its ICT. This is often addressed as Business-IT alignment.

While many examples of progressive decentralization in the organizations can be seen (e.g., diversification, adopting flat management style, coopetition, virtual and networked organizations), very little is known about how decentralization impacts the organizational IT.

In order to continuously support Business-IT alignment in organizations seeking decentralization, the latter needs to be explicitly addressed by enterprise models, enterprise architecture, IT governance and other related disciplines. In this work, we are focusing on Enterprise Modeling (EM).

Our motivating question is: How the existing methods of Enterprise Modeling (EM) support the decentralization?

2 Related Work

Organizational Structure and Decentralization

The terms centralization and decentralization often refer to the power over the decisions made in the organization. According to Henry Mintzberg [START_REF] Mintzberg | The structuring of organizations: A synthesis of the research[END_REF], when all the power for decision making rests at a single point in the organization -the structure should be called centralized ; to the extent that the power is dispersed among many entities, the structure should be called decentralized. Another type of organization may be defined between these two [START_REF] Zdravkovic | Requirements for it governance in organizations experiencing decentralization[END_REF]: the federation, where some decisions are top-down while some are lateral.

Military organizations are typically examples of centralized organizations. They have an explicit hierarchy, with responsibilities and decision making power clearly defined and fixed for the positions within this hierarchy. Many public organizations have a hierarchical structure with federated decision making, where decisions are made by a group of individuals (a board or committee) appointed by the authority or government. This also applies to their IT. Centralized and Federated organizations are very stable and robust but they cannot respond easily to change and are slow to act.

A number of organizational structures supporting decentralized decision making have recently emerged and became popular. These structures are often addressed as "post-modern" organizations. These organizations are grounded on the principles of social P2P [START_REF]Valve handbook for new employees[END_REF], implementing peer-production, peer-trust, peerreview, and peer-vote mechanisms for decentralized communication and decision making. The examples of post-modern organizations include Collaborative Network (CN), Virtual Organization, Coopetitions, and Sociocratic organizations. They distinguish from both centrally and federally governed organizations and from anarchies.

New branches that incorporate some of Endenburg's principles of sociocracy include holacracy. The examples of holacratic organizations include Sun Hydraulics, Valve, W. L. Gore, GitHub, Zappos. Among the core principles behind the post-modern organizations are self-organization and peer-to-peer (P2P) that were extensively studied in the literature.

While decentralization is widely addressed in the organizational science literature, very few works discuss the impact of organizational structure and decentralization on the organizational IT. In [START_REF] Ross | Enterprise architecture as strategy: Creating a foundation for business execution[END_REF], the authors specify four operational models (Diversification, Coordination, Replication and Unification) that refer to different organizational structures of a company. Authors show that an operational model defines the requirements of process standardization and integration and thus has a direct impact on the enterprise architecture of the company.

In [START_REF] Weill | IT governance on one page[END_REF] the same authors propose a framework to help firms design and communicate IT governance. This framework considers five major areas in the IT where the decisions have to be made and six decision-making archetypes that reflect the organizational structure of a company.

Organizational Structures and Enterprise Modeling

Our current research is focused on Enterprise Modeling (EM), "a technique that helps to capture the different elements and structures of an enterprise as well as to visualize the inter-dependencies between the elements" [START_REF] Sandkuhl | Enterprise modeling. Tackling Business Challenges with the 4EM Method[END_REF].

The type of organizational structure can affect the process of enterprise modeling: the answers to the questions "which types of enterprise models do we need?", "who will be involved in modeling/validation?", "how/in which order the model(s) need to be created?" can vary depending on the organizational structure. For example, a problem owner and a project leader can be formally identified within a centralized organization whereas in a decentralized organization this is not always the case. Moreover, the information required for modeling can be formalized and controlled by one designated team/expert/role in the organization or can be non-formalized, distributed within a community. The decision making becomes a collaborative process that can involve conflicts of interests (individual vs. team vs. corporation). Without a proper technical support and facilitation, such a process can take more time and efforts in a decentralized environment.

To reduce the risks of project failure due to lack of expertise, the EM methods need to be enhanced providing better guidance in case of decentralized organizations.

4EM

4EM (For Enterprise Modeling) is an EM method and framework developed in the academia [START_REF] Sandkuhl | Enterprise modeling. Tackling Business Challenges with the 4EM Method[END_REF] [START_REF] Bērziša | Capability as a service in digital enterprises[END_REF]. The 4EM method is comprised of three core elements:

-A defined procedure for modeling using a defined notation for representing the modeling product (defined modeling process and product) -Performance of EM in the form of a project with predetermined roles (project organization and roles) -A participatory process to involve stakeholders and domain experts (stakeholder participation)

"The 4EM method uses six interrelated sub-models, which complement each other and capture different views of the enterprise which can also be considered as perspectives" [START_REF] Sandkuhl | Enterprise modeling. Tackling Business Challenges with the 4EM Method[END_REF][Ch. 7.2 p. 77]. Each sub-model contains its own notations and rules to generate a diagram.

-Goal Model shows the goals the organization wants to achieve, or to avoid.

-Business Rules Model is used to define and maintain explicitly formulated business rules, consistent with the Goals Model. Business Rules may be seen as operationalization or limits of goals. -Business Process Model is used to define enterprise processes, the way they interact and the way they handle information as well as material. -Concepts Model is used to strictly define the "things" and "phenomena" one is talking about in the other models.

-Actors & Resources Model is used to describe how different actors and resources are related to each other and how they are related to components of the Goals Model and Business Processes Model. -Technical Components & Requirements Model is focused on the technical system that is needed to support the goals, processes, and actors of the enterprise.

Enterprise modeling (EM) projects are conducted in response to some problems identified in a company.

The EM process defined by 4EM includes 10 steps [START_REF] Bērziša | Capability as a service in digital enterprises[END_REF], and involves various actors (usually Problem owner, Domain expert, EM project leader, EM facilitator, and Tool expert). In each step, one of the actors is typically assigned as a Responsible, and one or several other actors as Participants. This is illustrated in Table 1 taken from [START_REF] Persson | Towards defining a competence profile for the enterprise modeling practitioner[END_REF]: For example, in (1), the Problem Owner -an organizational actor that will benefit from solving the problem -is assigned as a responsible for the step. The EM project leader ensures the communication between the domain experts and the rest of the project team. The problem owner and the EM project leader should discuss the problem boundaries, prospective solutions and outcomes.

4EM does not prescribe a particular list of sub-models to be focused on or the order in which they should be created -this is also a part of the EM project. Steps (3)-( 9) can be replicated for all the sub-models defined by 4EM.

The original version of 4EM provides a set of generic concepts not particularly tailored for decentralization. In the next section we discuss how 4EM can be applied in the case of a decentralized organization.

Applying 4EM on Centralized and Decentralized Organizations

In this section, we first explain criteria used to explain centralization and decentralization, then we illustrate application of 4EM for two fictional companies inspired by real companies and studios3 : one with centralized and one with decentralized organization.

Criteria of Decentralization

Criteria according to Mintzberg: In [START_REF] Mintzberg | The structuring of organizations: A synthesis of the research[END_REF], the five organizational structures are defined: simple structure, machine bureaucracy, professional bureaucracy, divisionalized form, and adhocracy. These organizational structures are defined based on their coordinating mechanism, the key part of the organization and type/degree of decentralization: MI1 The key part of the organization. One (or combination) of the following six components can be a key part: the strategic apex or top management, the operative core or the workers performing actual tasks and providing direct services, the middle line or middle management, the technostructure or analysts who design, plan, change, train the operating core, the support staff or the workers providing indirect services. MI2 The prime coordinating mechanism.

The major method the organization uses to coordinate its activities: Direct supervision where one individual is responsible of the work of others, standardization of work process, standardization of skills, standardization of output, mutual adjustment when work is coordinated through informal communication. MI3 The type of decentralization used.

Mintzberg specifies the extent to which the organization involves subordinates in the decision-making process. Three types of decentralization are identified: Vertical decentralization is the distribution of power down the chain of command, or shared authority between superordinates and subordinates in any organization. Horizontal decentralization is the extent to which non administrators (including operators sometimes) make decisions, or share authority between line and staff. Horizontal decentralization is related to the coordinating mechanisms, with direct supervision corresponding to more horizontally centralized, and mutual adjustment to more horizontally decentralized organizations accordingly. Selective decentralization is the extent to which decision-making power is distributed between different units within the organization.

Criteria according to Morgan: In [START_REF] Morgan | The future of work attract new talent, build better leaders, and create a competitive organization[END_REF], five types of organizational structure are identified: (1) Hierarchical organizations, based on multiple management layers, top-down communication and rigid rules for decision making; (2) Flatter organizations, seeking to reduce the number of hierarchical levels and to support both top-down and bottom-up communication flow; (3) Flat organizations, where all entities are equal and no hierarchy is defined (4) Flatarchies that can be found in between hierarchies and flat organizations, with temporarily formed hierarchies out of flat teams; (5) Holacratic organizations focusing on distributed decision making while giving everyone the opportunity to work on what they do best.

Morgan distinguishes the organizational structures based on those criteria [START_REF] Morgan | The future of work attract new talent, build better leaders, and create a competitive organization[END_REF]:

M1 A number of hierarchical levels that the organization defines.

Centralized organizations tend to have more levels and as a consequence more elaborated chain of commands and coordination protocols; M2 Flexibility of hierarchical levels.

Centralized organizations tend to have fixed, not evolving hierarchies; M3 Flexibility of decision making rules.

Centralized organizations tend to have decision making power concentrated in the top hierarchical levels, coordination and decision making rules are formalized and fixed. For the decentralized organizations, according to Morgan, decision making rules are not fixed and the power is distributed between entities.

Criteria according to Luthans: Luthans described 3 criteria [START_REF] Luthans | Organizational behavior : an evidence-based approach[END_REF].

L1 Geographical/territorial concentration or dispersion of operations. The extreme centralization case would be a single huge factory creating the final product from raw materials. In contrary, extreme decentralization case implies one factory per transformation or service. L2 Geographical/territorial concentration or dispersion of functions.

The most centralized case would describe a facility containing the whole functions like production, sales, finance, and direction (typically one facility with factory and headquarter inside, like in the beginning of the XX th century, or more recently small industries). L3 Extent of concentration or delegation of decision making powers.

Decisions may be taken by a single person for everything (CEO), partially delegated (middle management), or even shared by the whole employees (workers' cooperative).

Criteria according to CIGREF : The CIGREF (Club Informatique des Grandes Entreprises Françaises -large french corporations IT workgroup) defined 4 levels of centralization Decision making is kept within one person at the top hierarchy or the whole workforce, the two dimensions "quantity" and "hierarchical level" are taken into account. PS2 Structure of communication lines (by hierarchy, or direct communication).

Communication between teams are either made by a strict process involving hierarchy, or without any formal method and people exchange directly information. PS3 Forms of coordination (vertical coordination, or lateral coordination).

Coordination is achieved by an exchange of information about the state of the work, or by standards accepted by all teams. Within vertical coordination, management transfers information and plans objectives with specific rules and standards, besides horizontal coordination, people align themselves by exchanging information directly and create work team if it is necessary. Strategic apex and Operating core (Decision making power and Production) MI1

Case 1 : Application of 4EM on Centralized Corporation

Centralized Corporation is an American developer, publisher and distributor of video games (inspired by Electronic Arts, Take 2, and many others). Centralized Corporation is producing and distributing video games through the dematerialized retail store; it also publishes games in a physical form (gameboxes, DVD) for distribution in regular stores, using subcontractors for packaging production. The competitive advantage comes from the studios and subsidiaries specialized in different themes and franchises (dedicated studio sports, adventure, etc). Strategic decisions (financial goal, supporting or discontinuing games, retail strategies, and so on) are taken by a CEO and a board of studio directors, technical decisions (design, packaging, and so on) are taken within each studio (see MI1-L3-PS1-PS2). Hierarchical levels are limited (Corporation level, design studio level, project level) and fixed (M1-M2), allowing each level to report formally to its direct supervisor (L3-PS1-PS2). Standardization of skills and outputs are used for coordination, and direct communication are used within each studio, forming a limited vertical decentralization (MI3-L3-PS1-PS2). The board of directors and the operating core (designers) are the key parts of the company (MI1).

Enterprise Modeling for Centralized Corporation: Consider a project, integration of a new IS supporting design and development of games in Virtual Reality (VR) format. This project requires creation or modification of (some of) the 4EM submodels (see [START_REF] Sandkuhl | Enterprise modeling. Tackling Business Challenges with the 4EM Method[END_REF]). It will be conducted following the process described in section 2.3. In the Centralized Corporation, a studio director (if the system is requested by one studio only) or a board of directors including CEO (if the system is demanded by a group of studios) can be considered as a Problem Owner. A middle manager will be appointed as a EM project leader. The project leader will create a transverse team of domain experts (from different studios) if necessary and will supervise the project. An internal or external EM experts will join the team to ensure that the methodology (4EM in our case) is respected and that the models are understood by all the stakeholders. The process defined by 4EM will be implemented with respect to the guidelines: for example, at the step 3)Plan for modeling session the project leader will set up the meeting with the studio director and with the 4EM expert (EM facilitator) in order to plan a specific questions to be addressed during a modeling session and to identify the domain experts (e.g., managers, developers, VR experts, designers) to participate in the modeling sessions. At the step 7)Conduct modeling session, the 4EM experts will produce the models together with the appointed domain experts. Creating different models involves different domain experts: for example, elaboration of a Goal model will require a participation of studio directors and CEO; business managers in the studios will be involved in definition of Business Rules and Processes; technical experts will be participating in Process and Concept modeling; specification of technical Requirements and preparation of deployment plan for Technical Components will involve technical experts, developers and testers. In the Centralized Corporation, the domain experts will be defined and appointed by a project leader.

Case 2 : Application of 4EM on Decentralized Corporation

Decentralized Corporation is an American digital distributor of video game, and game developer (inspired by Valve Corporation [START_REF]Valve handbook for new employees[END_REF]).

Decentralized Corporation is distributing video games through its own dematerialized worldwide retail store, it also creates games. It is specialized in the development of an international platform providing games, updates, multiplayer support based on friend lists, worldwide ranking, through cloud technologies. Independent developers are invited to propose their projects on the platform, and even to put their games on it for publication.

Concentration of operations and functions: The company is located in one building (Headquarters) where all the teams are placed 4 . The teams are working together on different projects; the employees are focusing on game design, developing, testing, marketing and sales. The non-core functions such as infrastructure management (data center, development and testing environment), network and security management are outsourced to the external service providers or used "in the cloud" (see L1-L2).

An employee of the Decentralized Corporation can choose a project or a team to work with; the roles and the competences required for these roles are defined, but the team members can switch the roles freely. If an employee wants to launch a project, he can create a team. No micromanagement exists: for example, the founder of the company works within his own team. Thus, all the employees can be considered at one hierarchical level (see MI2-MI3).

Coordination and governance of Decentralized Corporation: Teams coordinate themselves by mutual adjustment (MI2), a form of horizontal coordination (MI3-PS3). Decision making power is distributed in the whole operative core (MI1-MI3-L3-PS1), according to Mintzberg [START_REF] Mintzberg | The structuring of organizations: A synthesis of the research[END_REF][Ch. 11 p. 210], this organization corresponds to the type E (Vertical and Horizontal Decentralization). The employees are very skilled and free, this allows the company to produce innovative services.

Form of Communication: No formal reporting is defined; the teams use direct communication. This type of communication is linked with the mutual adjustment; any difficulty in coordination will be corrected by immediate discussion between teams (see MI2-MI3-PS2).

Key part of the organization: The decentralized company mainly lies on its operative core. There is no need of technostructure or analysts for adapting the processes; a team can be modified any time for a better coordination between projects.

Enterprise Modeling for Decentralized Corporation: Consider the same project, integration of a new IS supporting design and development of games in Virtual Reality (VR) format. In the Decentralized Corporation, the project is launched by one or multiple teams. The new project team is formed and the roles are distributed by the team members.

Decentralization implies less formal relations and communication patterns, thus information about how the company is functioning and "who does what" is scattered. Distributing the roles within an EM project team will be different compared to a centralized company: The same group will also form the project team and share the role of the project leader. The EM facilitator and tool experts will be external consultants (the company maintains a focused set of skills and EM is not a field of expertise they want to develop in-house). These consultants will join the company for the project and will work on modeling in collaboration with the employees. In the Decentralized Corporation, voluntarily participation in a project is one of the main principles. Thus, the experts cannot be "assigned" and should nominate themselves for participation in the modeling session.

Implementation of the 4EM process (the 10 steps) will also vary compared to a centralized company: First of all, for the Decentralized Corporation, a process cannot be imposed (no manager to do so) but should be agreed upon. Conducting modeling sessions (step 5) and refining the models (step 9) will require participation from different employees and will rely strongly on their interest in the project and willingness to cooperate. Due to the distributed knowledge, gathering the background information (step 4) can take more time and efforts.

Creating the 4EM sub-models for Decentralized Corporation would also face some challenges: Since there is no executive level in the company, setting up the corporate goals, definition of the business rules, concepts and processes should be done through a collaborative process between the employees. Project-specific or team-specific concepts should be separated from the core concepts accepted by the company as a whole to avoid the information overload. Teams can have their local goals or priorities that are not always related to the corporate goals. Conflicts of interests need to be considered in such cases. Whereas the corporate processes are required to support the smooth collaboration between the teams and to ensure the corporate goals are met, each team can specify its own process(es) and working style. Since the actors and resources are identified by the project teams based on their current needs and can change all the time, the relations between actors and resources and the elements of a (corporate) goal model become difficult to capture within a single corporate Actors & Resources model.

Discussion

During this work, we identified some problems that need to be systematically resolved while applying the 4EM method in decentralized organizations. These problems are related to:

-Distributing the roles within an EM project -Implementation of the modeling process defined by 4EM (10 steps) -Creating the 4EM sub-models Depending on degree of decentralization5 , the described problems can be more or less apparent.

Distribution of roles and implementation of the modeling process defined by the original 4EM are challenging in decentralized organizations. Lack of explicit knowledge and use of informal communication can be overcome with advanced methods of knowledge management and communication around the project.

The employees can choose to participate or not in a project (compared to being assigned by a manager) -therefore, some efforts should be made to promote the EM project and to explain its importance to all the potential stakeholders.

Collaborative nature of work requires specific communication platforms and tools (social networks, groupware, etc.) in order to reduce the time needed to reach an agreement or to inform all interested participants.

To overcome the challenges related to the creation of the main sub-models defined by 4EM, we propose the following extensions illustrated in Figure 1: 1 We propose to decouple the local (created by each team) and the corporate (created for the whole organization) models; 2 We propose to create a "goal model loop" that specifies the dependencies between local and corporate models; The idea behind the separation between local and corporate models has been explored by the FEA with the segments (core mission-area and businessservices) [START_REF] Roger | A comparison of the top four enterprise-architecture methodologies[END_REF] [Ch. FEA, The FEA Perspective on EA], where some segments may be local or shared according to their scope. Two types of corporate models are possible : a concrete model containing various local models parts, and an implicit model based on lateral communications between peers (using only local models). In this work, we explore the first one, where a model is built upon local models to maintain a corporate vision about the full organization. This choice is close to a federated model, but it matches a compromise between the two cases presented (the centralized case can use the corporate model, and the decentralized case has more flexibility than before for modeling). Goal model, Business Rule model, Business Process model and other models can be seen on the corporate level and on the business unit (BU) or team level for the decentralized organizations (yellow rectangles in Figure 1). The goal model allows sharing of common values across the company: for example,increasing the number of players can be considered as corporate goals and shared between the game development team, the cloud client team, and the new console development team. The teams can use the local Goal model to define their internal goals. These goals can support the corporate goals, can complement them or can be defined independently. The In order to identify how the local (internal) goals of each team are related to the shared (corporate) goals, we add the cyclic relation (a loop) between local and corporate goal models. This loop specifies that the local goals can be defined by the corporate goals as well as the corporate goals can be defined by the local goals. Indeed, compared to the centralized organization, where the corporate goals define the local goals of each team, in decentralized organizations, we need to consider the situations when the team goals can be leveraged to the corporate level: for example, an innovative solution for a new game console can set up the goals for the game developers. Distributed decision making (L3-PS1) is acknowledged since each team decides locally on its goal model.

In a decentralized organization, resources and actors are shared between teams. Compared to traditional (centralized) organizations, it is the resources and actors of all the teams that define the resources and actors of a corporation and not the other way round. This is reflected by a relation between the corresponding models in Figure 1. We also add a relationship between Actors & Resources and Concepts Models since the concepts should also be attached to their owners (or creators) in a company. This relationship reflects the concept of direct communication defined in MI3 and PS2.

To create each local model, internal meetings has to be made regularly to ask questions and change their processes : what are the new objectives ? where could the work be optimized ? Thereby, each team decides internally its way of working. In the decentralized context, representatives of each team meet occasionally together to establish corporate goals and acknowledge the change in their services interfaces (services they offer to other teams, like an API for development, a version control system, etc.). Thus, if a team needs to use or create a service, it may check if another is already serving it (no duplication). To extend this way of working, we may assume that a subscription to external services implies subscription about their updates. Then, at least one component must be shared in the whole organization : the services catalog (supported by centralized or decentralized technologies). This process assumes teams are collaborating and have the capacities to make meetings (physically or not), as collaboration and communication are important in decentralized organizations.

This discussion is limited to the presented cases; more empirical work and experimentation is required.

Conclusion and Perspectives

In this paper, we study how decentralization impacts the enterprise modeling methods in the organizations. We gather centralization and decentralization criteria in Table 2, then we propose four extensions to the EM methods in order to better capture the specifics of decentralized organizations. We illustrate our proposal on the example of 4EM in Figure 1. We described a corporate model updated by a meeting of representatives of each team composing the organization, but other ways may be explored where peers review their productions and decisions continuously. This work is a step in a larger research project study-

  [START_REF] Gedin | des Rochettes, d'Erceville, and Poujois. Rapport de synthèse du groupe "organisation[END_REF]:[START_REF] Bērziša | Capability as a service in digital enterprises[END_REF] Concentration that is characterized by grouping of action and control capabilities in one location, under one top position; (2) centralization characterized by grouping of action and control capabilities in one or more locations, under one top position; (3) deconcentration characterized by scattering of action and control capabilities in one or more locations, where local directors make decisions, but they still follow a central top position; (4) decentralization that is similar to deconcentration, but without the central top position. The following characteristics can be identified from those 4 levels: CIG1 Concentration/Dispersion of the actions (IT production, development team) and control (quality management, organization analysts) capabilities. CIG2 Allocation of decision rights (top/middle layer in large corporations context).Criteria according toPearlson & Saunders: They propose 3 criteria [14][8]. PS1 Allocation of decision rights (one top position, or every employees).

3

 3 We propose to add relations between Actors & Resources and Concepts models 4 We propose to add relations between Business Processes, Actors & Resources, and Technical Components & Requirements models

Fig. 1 .

 1 Fig. 1. The extended 4EM for decentralized organizations

Table 1 .

 1 Actor involvement in the EM process steps (R-responsible, P-participates)

	Process step (1-10)	Problem owner	Domain expert	EM project leader	EM facilitator	Tool expert
	1) Define scope and objectives of the project	R	-	P	-	-
	2) Plan for project activities and resources	R	-	P	P	-
	3) Plan for modeling session	P	-	R	P	-
	4) Gather and analyse background information	-	-	P	R	-
	5) Interview modeling participants	-	P	-	R	-
	6) Prepare modeling session	P	-	P	R	-
	7) Conduct modeling session	-	P	-	R	P
	8) Write meeting minutes	-	-	P	R	P
	9) Analyse and refine models	P	-	P	R	P
	10) Present the results to stakeholders	R	P	P	P	-

Table 2 .

 2 Criteria of organizational centralization/decentralization

		Characteristic	Decentralized	Centralized	Criteria in the literature
	1	Concentration of operations	Dispersed operations in multiple different processes) locations (Multiple facilities containing	Concentration in the HQ (One facility processes) containing the whole production	L1, CIG1
	2	Concentration of functions	Dispersed functions in different locations (Multiple facilities containing one or more functions)	Concentration in the HQ (One facility with all the functions)	L2, CIG1
	3	Forms of coordination	Horizontal coordination; Mutual adjustment (peer-to-peer)	Vertical coordination; Direct supervision (hierarchy); Standardization of work process, output, and/or skills	MI2, MI3, CIG1, PS3
	4	Allocation of decision rights	Distributed in the whole organization (democratic); Vertical and Horizontal decentralization; Selective shared within the organization) decentralization (decision making power	Top position / Hierarchy (decision making power in the upper hierarchy); centralization Vertical centralization or Horizontal	MI1, MI3, M3, L3, CIG2, PS1
	5	Structure of communication	Direct communication (informal communication between peers)	Hierarchy / Formal communication (strict processes of communication)	MI3, PS2
	6	Number of hierarchical levels	Few levels (Flat organizations, managers of ad-hoc teams)	Multiple levels (Top hierarchy, middle various ramifications) line managers, team managers, with	M1
	7	Flexibility of hierarchical levels	Regular reorganization (Changing roles/shape of organization, adaptation to the environment)	Fixed hierarchies (Fixed roles and/or peoples, slow change management)	M2
	8	Key Part of the organization	Technostructure (Structure and Processes of the organization)

Electronic Arts (www.ea.com), Take 2 (www.take2games.com), and many others for centralized case. Valve Corporation[START_REF]Valve handbook for new employees[END_REF] for decentralized case (known for its ease in adjustment and creation of internal teams without hierarchy).

As in many organizations, some characteristics of the Decentralized Corporation correspond to centralization and some to decentralization according to Table2.

Evaluation of "degree of decentralization" for a given company remains beyond the scope of this work. For simplicity, one can consider a company "more decentralized" if it exhibits more "Decentralized" characteristics from Table2.

ing decentralization and its impact on the organizational IT. More studies on post-modern organizations and their business models will be done to better understand the possibilities offered by modern technologies (P2P, mobile, cloud, virtualization, etc.).