Fractal Grid – towards the future smart grid

Multiple research results have shown that built-up space is often structured according to fractal logic and recent investigations have compared the coherency between the spatial organisation of urban built-up spaces and street networks [10]. A similar approach is suitable for power systems. A given built-up area corresponds to an optimal topology of the power grid. This results from the spatial distribution of power sources and loads, the technical constraints, such as the maximal voltage variations, and the global cost. This latter cost is the sum of three components: the investment cost, the power loss cost and the non-supplied energy cost if the network is not reliable enough. Of course, the optimal solution is highly dependent on the spatial arrangement of the built-up area. Multi-scale analysis aims to characterise this dependency at different scales of observation and determines how it influences the optimisation of the power grid topology. For this purpose, the fractal dimensions of built-up areas are computed. These dimensions characterise how the built-up areas fill the space. Moreover, fractal analysis identifies scales in which spatial organisation changes. To identify ruptures or shifts in the scaling behaviour, a special method has been developed for urban pattern analysis [10]. A similar analysis can be performed for electrical distribution networks. This analysis can improve our understanding of the spatial organisation of the distribution network on inter-urban and intra-urban scales and identify hypertrophies or deficiencies in existing networks. For instance, it intends to study the link between spatial distribution of consumers or decentralised energy sources in urban areas and network morphology.

Beyond standard fractal analysis methods, a couple of specific methods adapted to networks and originally used in materials science, like percolation or multi-fractal formalisation [11], are applied. These methods allow us to e.g. explore the properties of the most direct path between two nodes and the distribution of meshes, which render the network more robust but also tend to increase the cost. Moreover, they can take into account simultaneously the topology of the network, its power flow and the multi-scale distribution of voltage drops. Then, taking voltage drops as a measure of power accessibility quality (the lower the voltage grid, the better the service provided by the electrical system), it is possible to assess accessibility to electrical distribution grids over a given area. All of these results are used to reshape the architecture of smart grids.

Fractality is used to design new smart grid architectures. This design is driven by the assumption that a self-similar topology can benefit the electrical system, from consumers to utilities. Indeed, nature shows that fractal networks emerge when a minimal guarantee of continuous supply is required while minimising the space covered by the network. On the other hand, minimising the coverage of a network decreases its resilience since the connectivity is reduced. However, fractal structures are often made up of multiple loops at different scales. This increases the network's resilience to link damage. Fractal structure can thus be the optimal design solution.

Hence, the objective is to design a fractal architecture for an urban distribution system. First, a built-up area is chosen from the open-access geographical information system database provided by IGN (IGN: National institute of geographical and forest information in France). Over this area, data concerning network geometry, substations at the HV/MV and MV/LV grid interfaces, cables, loads and dispersed generators are collected from the distribution system operator (DSO). Then, a number of fractal patterns (meshed or radial) are tested and adapted to interconnect substations, dispersed generators and loads and provide an optimal operation of the electrical system. The design aims at multi-scale coverage of the built-up area. This means that the spatial organisation of the electrical grid shall correspond to the needs and constraints of users located in the considered area. For this purpose, the results provided by the previous steps of the methodology are very useful. Finally, the cross-scale interactions between the designed Fractal Distribution Grid and the regional grids, which are connected at the HV/MV interfaces, are studied.

An important property of smart grids is their resilience, i.e. their ability to survive perturbations, repair themselves and adapt to new situations. In Fractal Grid, we address the robustness and adaptability of the network using tools from graph theory, optimisation and control theory. Random faults, broken links and power surges impact the network differently depending on its topology and the configuration of the generators and loads. This is mainly a static optimisation problem: in time and space. Important questions are: can a load-flow solution exist if a given link is broken? Which network topologies are most resistant? A number of studies in the complex network literature [12] link the graph topology of the network to its robustness and answer the main question, which nodes or links are most critical? We can apply these methods to power systems and compare their predictions to the load-flow calculations. Then, the results of the study guide the design of fractal architectures according to resilience criteria.

From another point of view, the adaptability of a network is a dynamical property because the network needs to shift from one state to another while it is in operation.

Using spectral properties of electrical networks, we have identified soft nodes for which no control or observation is possible [13]. Of course, future smart grids will need to minimise the number of soft nodes in order to increase their adaptability. Soft nodes for the designed fractal architecture are therefore identified by the same approach and the new issue is to show whether self-similarities reduce the number of soft nodes.

In [14] the challenges facing electric power systems management are identified. Planning and operating the next-generation mix involves making decisions on time scales ranging from several years (concerning new grid expansion) down to time scale scales of fractions of a second. The spatial scales of power systems are related to the sizes of the transmission, distribution, and customer (industrial, residential) systems [15]. It is essential to coordinate distributed energy resources, storage units and active loads in future power systems in order to ensure flexibilities at every level of the system and high reliability. For this purpose, in recent years, several works have proposed hierarchical optimisation [16]. This considers decomposition into sub-systems, which are locally optimised, and the interactions between them [17]. Methods like multi-agent system modelling are seen as a promising technology to perform this kind of optimisation. Although several research works deal with multi-agent systems in the power engineering field, as shown in [18], most only focus on operation at micro-grid and distribution network scales [19–21] or on market modelling and simulation [22–24].

In Fractal Grid, a methodology and a simulation tool based on the multi-agent approach to manage a fractal power system are developed. This considers the major actors in a power system, from the regional system operator to the customer, in a day-ahead and intraday timeframe (see Fig. 2). In addition, the role of electricity markets at the different scales is analysed. Moreover, this simulation tool aims to cover networks with different topologies [15].

We performed fractal analysis of two distribution power grids, located in the French administrative areas of Jura and Haute-Saône, which feature both intra- and inter-urban zones.

We compared the power grid with the built-up spatial organisations. To compute the fractal dimension, the results were obtained using the box-counting method, which is a way of determining the fractal dimension by computing the number of boxes used to cover an area. The box size varies to cover different scales. In this case, boxes went from 3 to 27,300 m for Jura and 32,800 m for Haute-Saône. For very small boxes, the fractal dimensions of both Jura and Haute-Saône's distribution power grids were very similar and close to 1. For box sizes up to 2000 m, fractal dimensions increased to reach 1.75. Indeed, when growing up, the boxes include more and more network bifurcations and the resulting fractal dimension increases. Above 6000 m, they varied in an oscillating fashion. This remains unexplained but it seems to be a very typical feature of distribution networks. Fractal dimensions of HV and LV grids are lower when they are considered separately. This is a strongly expected result, as the combined LV and HV cover space more homogeneously than they do separately. The same kind of analysis was finally carried out on different towns in Jura and Haute-Saône to compare built-up patterns and the spatial arrangement of power grids. We identified three different types of area as shown in Fig. 3. For St Claude, Champagnole, the grid is hypertrophied in comparison with the built-up surface. On the other hand, for Lure, Luxeuil, Héricourt and Vesoul, a dense area is covered by a relatively small grid. This is the most efficient of the three types in terms of grid proportion versus built-up surface. Dole, Lons and Gray are some intermediate cases.

Several Use Cases are considered in order to study alternative multi-scale organisation forms and their impact on the power system. As an example, a first use case, named ‘DSO constrained control’, considers a strategy in which the DSO manages system operation using a constrained control grid tariff. In a day-ahead timeframe the DSO has to guarantee safe operation of the grid. After forecasting possible constraint violation, the DSO calculates a ‘grid tariff’ for retailers. Then, the retailer establishes a flexibility plan that is proposed to the customer in order to reduce the number of constraints violated. Consequently, the next day, customers try to implement this plan. If constraints are violated, some load automatically disconnects from the grid and DSO charges the retailers at the Grid Tariff.

An alternative Use Case is that of an ‘aggregator for load flexibility’. The DSO can operate the flexibilities offered by the different grid stakeholders in order to ensure safe operation. In this context, a local flexibility market is established. Flexibilities are offered mainly by aggregators, each of which manages a portfolio of customers. In the day-ahead timeframe, the DSO requests flexibilities by setting bids in a flexibility market and the aggregators set flexibility offers. The results of this market are the flexibilities obtained by the DSO. In real time, the aggregator implements the flexibility plan established and the DSO evaluates the technical feasibility of applying the proposed flexibility and pays the aggregator. This Use Case puts the emphasis on a new smart grid actor, i.e. the aggregator, and evaluates the possibility of establishing local markets to trade flexibilities. In this scheme, flexibility requests may also come from the TSO, while in a more general case, an aggregator may tender its available flexibilities on different markets.