2.1 Distribution network architecture

Urban electrical distribution networks are commonly designed according to a secured feeder architecture (see Figure 1). Primary substations act as the supply points of the medium-voltage distribution network, injecting power into the system through main feeders that interconnect the substations and supply multiple secondary substations. Along these feeders, secondary substations serve local areas and include distribution transformers that step down the medium-voltage level to low voltage for end users, as well as switching devices that control network operation. Each secondary substation is connected to a feeder through two switches, providing operational flexibility: normally closed switches (NCS) allow power flow under normal operating conditions, while normally open switches (NOS) are kept open to ensure radial operation so that each load is supplied by a single primary substation. In the event of a fault, the network can be reconfigured by operating the switches, enabling secondary substations to be resupplied from an alternative primary substation while repair actions are carried out.

2.2 Power grid planning

Distribution system operators (DSOs) are responsible for ensuring the safe, reliable, and cost-effective operation of power grids. This mandate includes maintaining existing infrastructures, guaranteeing continuity and quality of supply, and anticipating future developments in demand and distributed generation (Khator and Leung Reference Khator and Leung1997; Conejo et al. Reference Conejo, Carrion and Morales2010). In this context, long-term power grid planning is a core activity.

Long-term planning concerns the definition of how a distribution network should evolve over extended horizons, typically spanning multiple years. DSOs construct these plans considering multiple inputs, including consumption forecasts, expected production profiles – including the integration of renewable energy sources – and broader energy transition scenarios. The goal is to identify a future network architecture that remains technically feasible and operationally secure.

Infrastructure modifications in distribution grids are expensive and difficult to reverse. The cost of a single power line may range from 50 k $\unicode{x20AC}$ to 200 k $\unicode{x20AC}$ per kilometre, depending on technology (overhead vs. underground), cable type, and local conditions (rural, semi-urban, or urban). At the scale of a national distribution system – such as the French network, which extends over approximately 1.4 million kilometres – even minor changes can result in significant investments. Consequently, DSOs aim to minimise deviations from the existing infrastructure, favouring incremental and targeted interventions (Conejo et al. Reference Conejo, Carrion and Morales2010).

A distinctive feature of long-term planning is that it must consider not only the final target network but also the sequence of intermediate networks that will emerge as investment actions are gradually executed over many years. Each intermediate configuration must remain compliant with operational and topological constraints: such intermediate configurations correspond to networks that will be in service for extended periods. Ensuring their feasibility is therefore critical to maintaining security of supply and service quality over time.

According to the ERDF methodology (ERDF 2008) for the development of distribution network master plans in France, the long-term planning process can be represented as a workflow composed of six main stages: system diagnosis, load forecasting, definition of a long-term target network, identification of development strategies, techno-economic scheduling of investments, and assessment of the resulting quality of supply. Among these stages, the identification of development strategies is a key step, as it determines how the transition from the current network to the target configuration is achieved. More informally, given the current network configuration and the target configuration, this step generates the list of interventions to be performed and the corresponding resulting intermediate networks. The present work focuses on this stage, which remains largely manual and based on expert knowledge in current practice.

To ensure the operationality of intermediate networks during the development strategies step and for minimising the potential for wasting resources in the long-term power grid planning process, a number of implicit rules are usually followed by grid operators. Castellanos-Paez et al. (Reference Castellanos-Paez, Alvarez-Herault and Lalanda2023) extracted the following set of rules via in-depth interviews with experts from French DSOs that are in charge of the development strategies:

1. R1 There is no closed loop between two primary substations; that is, a normally open switch is required to ensure radial operation, meaning that each secondary substation in the network is supplied by exactly one primary substation.

2. R2 The network must tolerate the loss of any primary node, ensuring that a radial configuration, in which all secondary nodes remain supplied, can always be restored via switching operations (opening and closing lines).

3. R3 For urban areas, the number of connections of secondary substations are between 2 and 3.

4. R4 Only lines belonging to the target network can be created, and only lines not belonging to the target network can be removed. These operations are irreversible, for example a line cannot be removed and later re-added.

5. R5 There is only one connection between two nodes (distribution/source).

3.1 Acting upon the power grid

There are three possible kinds of actions available to reconfigure the power grid: edge addition, edge removal and edge switching. Each action changes the power grid graph. Whenever applying action $a$ over the power grid $G$ results in $G'$ , we denote this by $G \vdash _{a} G'$ . We describe effects of each possible action on $G$ .

Edge addition. Edge addition consists of adding a new edge to the open network. Let $x, y \in V$ with $(x,y) \not \in E$ . We denote the action of adding $(x,y)$ to $G$ by $add(x,y)$ . It results in the power grid $G(V, E', \lambda ')$ , where $E' = E \cup \{(x,y)\}$ and $\lambda '(e) = \lambda (e)$ for all $e \in E$ , and $\lambda '(x,y) = open$ .

Edge removal. Edge removal consists of removing an edge from the open network. Let $(x,y) \in E$ such that $\lambda (x,y) = open$ . We denote the action of removing $(x,y)$ as $remove(x,y)$ . It results in the power grid $G(V, E', \lambda ')$ , where $E' = E \setminus \{(x,y)\}$ and $\lambda '(e) = \lambda (e)$ for all $e \in E'$ .

Edge switching. Edge switching consists of swapping the state of two edges that are adjacent to the same secondary station. Let $x \in S$ , $y, z \in V$ such that $\lambda (x,y) = close$ and $\lambda (x,z) = open$ . We denote the switching action by $switch(x,y,z)$ . It results in the power grid $G(V, E, \lambda ')$ , with $\lambda (e) = \lambda '(e)$ for all $e \in E \setminus \{(x,y), (x,z)\}$ and $\lambda '(x,y) = open$ , $\lambda '(x,z) = close$ .

Non-interfering actions. Executing an action causes a change in the power grid graph. Intuitively, concurrent execution of actions that involve the same edges leads to ambiguities during plans’ execution. We formalise the notion of “allowed concurrency” in our domain, inspired by the notion of non-interference in planning (see, e.g., (Rintanen et al. Reference Rintanen, Heljanko and Niemelä2006)). We denote by $\operatorname {Aff}(a)$ the set of edges that is involved in the execution of action $a$ , and in particular we have that $\operatorname {Aff}(p(x,y)) = \{(x,y)\}$ if $p \in \{add, remove\}$ and $\operatorname {Aff}(switch(x,y,z)) = \{(x,y), (x,z)\}$ .

We say two actions $a_1$ and $a_2$ are non-interfering (or compatible) if $\operatorname {Aff}(a_1)$ and $\operatorname {Aff}(a_2)$ are disjoint sets. Intuitively, non-interfering actions are “compatible”, in the sense that they can be carried out on the graph “at the same time” without ambiguities. A set of actions $A$ is non-interfering if all its elements are pairwise non-interfering. Thus, we extend the notion of affected edges to sets of actions, with $\operatorname {Aff}(A) = \bigcup _{a \in A} \operatorname {Aff}(a)$ . We say that $G \vdash _A G'$ if we obtain $G'$ by applying all the actions $a \in A$ ; if $A$ is non-interfering, there is no ambiguity in the resulting graph $G'$ (as no edge is affected by multiple actions).

3.2 Compliant networks

The DSOs’ rules described in Section 2.2 provide properties that power grids must comply with to be considered valid. These can be succinctly expressed in terms of graph-theoretic properties. We say a power grid $G(P \cup S, E, \lambda )$ is compliant whenever the following properties are satisfied.

3.3 Long-term planning problem

Finally, given an initial power grid $G(P \cup S, E, \lambda )$ and a target one $G_T(P \cup S, E_T, \lambda _T)$ , we define the long-term planning problem as computing a sequence of sets of actions $(A_0, \ldots , A_n)$ such that $G_i \vdash _{A_i} G_{i+1}$ , with $G_0 = G$ , $G_n = G_T$ , and each intermediate network $G_i$ satisfying the radiality, reconfigurability and degree-compliant properties. We denote by $B = E_T \setminus E$ the set of buildable edges, that is edges present in the target network but absent from the initial one, and by $R = E \setminus E_T$ the set of removable edges, that is edges present in the initial network but not in the target.

Relationship with DSOs rules. C1 requires that the network evolves from $G$ to $G_T$ by the end of the horizon. C2 requires every intermediate network $G_i$ to satisfy radiality (R1), fault-tolerance under the loss of any primary node (R2), and the degree compliance (R3). R4 is captured by C3–C5, which constrain admissible line additions and removals and enforce their irreversibility.

In the remainder, we refer to the bounded version of LPP whenever we assume an integer upper bound $k \in \mathbb{N}$ on the length of the solutions. We refer to the sequential version of the problem whenever we add the requirement that $|A_i| = 1$ , and parallel otherwise (i.e.,, $|A_i| \ge 1$ ).

Example. Figure 2 illustrates the constraints considered in this work. Figure 2a shows a compliant configuration, where radiality, redundant connectivity, and degree constraints are all satisfied. Figure 2b shows the reconfigurability of the graph from Figure 2a under the failure of $P_{11}$ ; in particular, radiality can be restored by closing edges $(S_2,S_9)$ and $(S_1,S_4)$ . Figure 2c shows a configuration that violates radiality for two distinct reasons: nodes $S_6$ and $S_9$ are isolated, and the feeder associated with $P_{12}$ contains a cycle. Figure 2d shows a graph that is radial but not redundantly connected. Figure 2e shows how the violation of connectivity in the graph provided in 2d implies non-reconfigurability: in particular, in case of failure of $P_{11}$ , node $S_1$ can be resupplied by closing $(S_1,S_4)$ , whereas $S_2$ , $S_5$ , and $S_{10}$ remain isolated.

Fig. 2.

Example graph configurations illustrating the role of constraints.

4.1 Data model

Let $G(P \cup S, E, \lambda )$ and $G_T(P \cup S, E_T, \lambda _T)$ be a pair of power grids. We describe the predicates and facts we use to represent inputs and outputs of the $LPP(G,G_T)$ problem.

Input. To describe the available stations in the power grid, we use atoms of the form $node(x)$ for $x \in P \cup S$ . The atom $node\_attr(x, primary)$ models that node $x$ is a primary node, $x \in P$ . Edges that belong to the start configuration are described by the predicate $start/3$ . In particular, if $\lambda (x,y) = o$ , we have the atom $start(x,y,o)$ , with $o \in \{open, close\}$ . Similarly, edges in the target configuration are described by the predicate $target/3$ , with the atom $target(x,y,o)$ whenever $\lambda _T(x,y) = o$ . We state that an edge $(x,y)$ may be built or removed by using the atoms $buildable(x,y)$ and $must\_remove(x,y)$ . As input, we expect an undirected representation of the graph, and in particular we assume the fact $start(x,y,o)$ with $x \lt y$ .

Output. A solution $(A_0, \ldots , A_k)$ to the $LPP(G, G_T)$ problem is described by means of the $action/2$ predicate, where the atom $action(t,a)$ models that $a \in A_t$ . In particular, $a$ will be function terms of the form $add/2$ , $remove/2$ and $switch/3$ , mirroring how possible actions are defined in Section 3.1. As an example, $action(3,switch(x,y,z))$ means that the solution includes the action $switch(x,y,z)$ at time $3$ , that is $switch(x,y,z) \in A_3$ .

4.2 Encoding of the problem

We present the ASP encoding used to model the LPP problem. We describe it incrementally, with each paragraph assuming all the rules that precede it. We refer to the $i$ -th line in the encoding by $r_i$ . We present the encoding to compute a bounded solution of length at most $k$ .

Initial definitions. We start with definitions that make the encoding more readable. Let $G(P \cup S, E, \lambda )$ be a power grid. We denote which nodes of our network are respectively primary stations or secondary stations by the $primary/1$ and $secondary/1$ predicates, respectively. We will have the atom $primary(x)$ if $x \in P$ , and $secondary(x)$ if $x \in S$ . The $opposite/2$ predicate expresses the notion that open line and closed lines are opposite concepts. The $network/4$ predicate stores the “state”, with $network(i,x,y,o)$ stating that $\lambda _i(x,y) = o$ in the $i$ -th power grid. We assume $t=-1$ corresponds to the initial state of the network.Footnote ¹ The $edge/4$ predicate is a “symmetric closure” of $network/4$ over time (that also keeps track of each edge’s state) that will be useful to define the effects of the $switch$ action ( $r_{13}$ – $r_{14}$ ).

Generating action sequences. The following rules generate the search space of possible action sequences. Informally, we first guess “the plan’s length” $(r_8)$ by guessing a $time/1$ atom (and propagating backwards existence of previous time-points, up to zero $(r_9)$ :

Next, the following choice rules impose that edges to be built and removed will be subject to an $add$ ( $r_{12}$ ) and $remove$ action ( $r_{10}$ ), respectively, exactly once within the whole solution. We prohibit the removal of closed edges ( $r_{11}$ ). The third choice rule ( $r_{13}$ – $r_{14}$ ) guesses edge switching operations. The last constraint $r_{15}$ imposes that we perform at least one action in each time-point.

The atom $action(t,a)$ encodes that at time $t$ the action $a$ is performed, where the action $a$ is a function term as described in Section 4.1. We assume that actions taken at time $t$ act upon the network as available in the previous time-step $t-1$ , and take effect at time $t$ .

The $affected\_by/4$ predicate ( $r_{16}$ – $r_{21}$ ) marks networks’ edges that are affected by an action, according to the definition of $\operatorname {Aff}(\cdot )$ . In particular, the atom $affected\_by(t,x,y,a)$ states that $(x,y) \in \operatorname {Aff}(a)$ , and that $a$ is performed at time $t$ . Affected edges will either change their state (i.e., whether they are open or close), or be added/removed at the next time-point.

Notice that (see $r_{10}$ – $r_{12}$ ) we can add an edge $(x,y)$ only if it we have the $buildable(x,y)$ fact, and similarly we can remove it only if we have the $must\_remove(x,y)$ fact. Thus, for add and remove actions, we know precisely the affected edge in the ASP representation. On the other hand, the available switch actions are defined on the previous network state ( $r_{13}$ ), and an edge may be affected by switch actions from different endpoints. As an example, edge $(S_6, S_9)$ in Figure 2 may be affected by $switch(S_6, S_8, S_9)$ but also $switch(S_9, S_6, S_2)$ . Thus, the definition of $affected\_by/4$ takes this into account ( $r_{18}$ – $r_{21}$ ). To ensure our solutions consist of compatible action sets, we forbid to guess non-compatible actions (i.e., actions affecting the same edge).

Actions’ effect. The following rules apply the effect of actions taken at time $t$ on the network at time $t-1$ , according to the formalisation of Section 3.1. They share the use of atoms $network(t-1,\_,\_,\_)$ and $action(t,\_)$ – past network state and action to execute – in the body, and have $network(t,\_,\_,\_)$ – how the network is affected by the action.

Actions of the form $add(x,y)$ simply define that the edge $(x,y)$ will exist at the next time-step:

The edge switching $switch(x,y,z)$ operation first identifies the correct pair of edges that is involved in the action, then flips their state in the next time-step:

All edges that are not affected by an action are implicitly carried over to the next time-step, with the same state as before. This is a form of inertia. The combination of $remove(x,y)$ affecting $(x,y)$ and intertia implicitly removes edges.

Encoding radiality I (Connectedness). To enforce the radiality property, as by Definition1, we require the closed network to be a forest with roots in the primary nodes $P$ . First, we ensure that $(i)$ all secondary $s \in S$ reach at least one primary $p \in P$ in the closed network, and $(ii)$ there are no loops between primaries over the closed network. We start by defining the standard notion of graph reachability over the closed network by means of the predicate $reach/3$ ( $r_{27}$ – $r_{30})$ :

Then, we use the atom $reaches\_primary(t,x,y)$ to model that node $x$ is connected to the primary station $y$ at time $t$ . The constraints model, respectively, that two distinct primaries should not reach each other ( $r_{32}$ ), and that all secondary stations should be connected to at least one primary station ( $r_{33}$ ).

Encoding radiality II (Acyclicity). Since the problem deals with undirected graphs and the previous rules ( $r_{32}$ – $r_{33}$ ) enforce that each secondary station is connected to exactly one primary station in the closed network (recall that if it were connected to two there would be a loop between primaries, which we forbid by means of a constraint ( $r_{32}$ ), it is sufficient to enforce that the network at time $t$ has the correct number of edges to enforce acyclicity. Since we are looking for $|P|$ disjoint trees and the number of edges in a tree of $q$ nodes is $q-1$ , we want our network to have exactly $|V| - |P|$ edges. The predicates $num\_primaries/1$ and $num\_nodes/1$ store the number of primary nodes and overall number of nodes in the network, while the constraint ( $r_{36}$ – $r_{37}$ ) enforces the correct number of overall edges.

Encoding reconfigurability. Secondary stations must be redundantly connected (Definition2) to an extra primary substation over the open network, distinct from the one they are connected to in the closed network. Recall that a secondary station cannot be connected to two primary stations in the closed network, as this would create a loop between primaries and violate radiality.

The predicate $redundancy\_reach/3$ models reachability over the open network ( $r_{38}$ – $r_{41}$ ), with the atom $redundancy\_reach(t,x,y)$ modelling that node $x$ is connected to node $y$ over the open network with a path that does not traverse any primary substation – notice the not primary(Y) atom in the body of the recursive definition of $redudancy\_reach/3$ ( $r_{41}$ ). This ensures the only way networks’ edges involving primaries are taken into account in computing $redundancy\_reach/3$ is via direct edges. Finally, $redundancy(t,s)$ denotes that at time $t$ the secondary station $s$ reaches a primary substation over the open network, and we prevent a secondary station that does not ( $r_{44}$ ), ensuring reconfigurability as by Definition3, that is that all secondary nodes are redundantly connected.

Encoding degree-compliance. We use aggregates to count the number of adjacent edges to each node, with the atom $degree(t,x,d)$ meaning that node $x$ has degree $d$ in the network at time $t$ . The constraints $(r_{46}$ – $r_{47})$ enforce that $d \in [2,3]$ , as by Definition4:

Final configuration. We encode that the final network obtained by applying the guessed actions sequence is exactly the target network $G_T$ , as in Definition5: it contains the same edges as $G_T$ and in the same state ( $r_{49}$ – $r_{50}$ ), and all edges to be removed have been removed ( $r_{51}$ ).

Preferred solutions. Answer sets of the above logic programme encode sequences of (sets of) actions that reconfigure $G$ into $G_T$ . As customary in ASP applications, multiple answer sets may exist, that is multiple ways to reconfigure $G$ into $G_T$ . ASP optimisation (via weak constraints) enables us to single out preferred solutions, according to different criteria. For the purposes of this paper, we are interested in reducing the overall number of actions executed in the plan, as well as its length (lexicographically). We can express this by the constraints:

The weak constraint $r_{52}$ minimises the total number of actions,Footnote ² while the weak constraint $r_{53}$ favours shorter plans.

5.1 Results

Below we present the results of our experiments. All solutions have been validated using an external purposely-designed tool to assess the satisfaction of the DSO rules.

Fig. 3.

(a) A point $(i,t)$ indicates that $i$ instances are solved in $t$ seconds. (b) A point $(x,y)$ denotes a synthetic instance solved in $y$ seconds by exponential and in $x$ by bounded. Points below the bisector indicate faster exponential search.

Synthetic instances. In order to select the best approaches to be employed on real data, we first test our encoding in the bounded and exponential search version, searching for sequential and parallel plans, on the synthetic instances. We report our results in terms of the cactus plot and scatterplot in Figure 3. First, we can observe in Figure 3a that we are able to solve more problem instances dealing with parallel plans wrt sequential plans. Intuitively, this is due to non-interfering actions “compressing” the solutions’ length, which translates to dealing with smaller logic programmes. In ASP planning, this typically correlates with easier problems, given that long plans are a known pain point also for ASP-based planning (see, e.g., (Son et al. Reference Son, Pontelli, Balduccini and Schaub2023)). Exponential search performs worse than bounded search over sequential plans, as it incurs in overhead due to multiple solver calls that are required to expand the search horizon, while the bounded encoding operates directly on it. In the parallel setting, the exponential search approach dominates, solving 54 more instances wrt the bounded at horizon $\frac {1}{2}g$ , 78 wrt $g$ and 93 wrt $2g$ , with all instances being mostly trivial for the parallel approach. The scatterplot in Figure 3b shows that for parallel plans, all instances slow down as bounds increase. In particular, we can observe (top-right corner of the scatterplot) that these synthetic instances – despite being trivial whenever they admit a short solution – are challenging if we try to solve at a much higher bound: we observe several instances that can be solved within seconds by the exponential search approach but timeout with larger bounds. Table 1 shows that all techniques solve fewer instances as $|V|$ increases. Parallel exponential search scales better as $|V|$ increases. This is confirmed by runtimes distributions in Figure 4. The largest instances are comparable in size to those handled by human experts, both for training and for the actual long-term planning of power grids. Larger grids are generally decomposed into networks of between 20-25 nodes. In that sense, our approach can readily support the work of experts. Overall, these results on synthetic instances motivate our focusing on parallel plans and exponential search approaches to solve real-world scenarios.

Table 1.

Number of solved instances by graph size and method. Cell content $x;y$ denotes that we solve $x$ instances with sequential planning, and $y$ instances with parallel planning, respectively

Fig. 4.

Parallel plans runtime comparison across graph sizes. Red bar is the median runtime.

Real instances. Table 2 reports runtime of our exponential search approach, with and without optimisation statements (as discussed in Section 4). We are able to solve instances of up to 42 nodes, proving the approach is useful for operation support in realistic scenarios, and can solve instances of size larger than those experts deal in daily operations. Runtimes are comparable with the ones we obtained over synthetic instances of the same size. On some solved instances, we are also able to prove optimality of parallel plans. In real-world scenarios, this translate in lower costs and simpler-to-put-in-place operations. These solutions were validated by an automated tool and visually reviewed by domain experts, who confirmed their consistency with DSO planning rules and their comparability, at the topological level, to solutions typically produced by experts.

Table 2.

Runtime ( $r$ ) on real instances of a given size ( $\#$ Nodes), with total number of actions ( $\Sigma |A_i|$ ) and maximum number of concurrent actions ( $\max |A_i|$ ). Dashes (–) denote timeout. The “Opt?” column marks instances where we are able to prove optimality of the parallel plan