Towards negative cycle canceling in wind farm cable layout optimization

Since the integrated planning process which includes turbine and substation placement as well as internal and external cabling comes with a level of complexity that is hard to manage, often a single planning step at a time is considered for optimization in the literature.

For finding the optimal cable layout between turbines and substations with fixed positions—which is also the scope of this work—one of the first papers was by Berzan et al. (2016), in which they propose a hierarchical decomposition of the problem into several layers. They use well-studied graph problems to solve the so-called Circuit and Substation layers, in which only one substation is considered at a time, when there is only one cable type available.

Since then, various approaches have been taken for more elaborate problems considering different optimization functions and sets of constraints. With the high complexity of the problem in mind, metaheuristics, such as Genetic Algorithms (Zhao et al. 2004; Shirshak et al. 2017; Dahmani et al. 2015) or Simulated Annealing, (Lehmann et al. 2017) are very popular. While these approaches do not guarantee provably optimal solutions, they are able to provide good solutions within short running times. To the contrary, exact solutions can be provided by INTEGER LINEAR PROGRAM (ILP) or MIXED-INTEGER LINEAR PROGRAM (MILP) formulations, which need more time and therefore only work on small instances. Lumbreras and Ramos (2013), for example, consider losses along branches, stochasticity in wind inputs and component failures in an ILP and Cerveira et al. (2014) use a graph-theoretic flow model on wind farms with a single substation and use the resemblance to the CAPACITATED MINIMUM SPANNING TREE (CMST). Based on the flow model, they include constraints representing the CMST into an MILP formulation.

In our work, we use a flow model similar to the one presented by Cerveira et al. (2014) representing how turbine production is routed to one of multiple substations. We aim at finding a flow of minimum cabling cost and apply a well-known technique from network flow theory called negative cycle canceling. Negative cycle canceling was first proposed in the context of finding minimum cost circulations in flow networks (Klein 1967). Goldberg and Tarjan (1989) achieve a strongly polynomial running time for a cycle-canceling-based algorithm for the minimum cost flow by suitably choosing the cycles to cancel. The bound for the running time of this algorithm was later tightened by Radzik and Goldberg (1994).

Ouorou and Mahey (2000) employ negative cycle canceling to solve the Minimum Multicommodity Flow Problem with nonlinear cost functions. Negative cycle canceling is also used in combination with tabu search to tackle the Capacity Expansion Problem for multicommodity flow networks (de Souza et al. 2008), which can be modeled as a Multicommodity Flow Problem with non-convex and non-smooth cost functions.

Optimization problems that are similar to WCP appear for example in logistics. In the Single-Sink Edge Installation Problem introduced by Salman et al. (2001) the production of multiple sources must be transported to a single sink. On every connection a mixture of various cable types (including multiple copies of the same type) needs to be installed such that the cables provide sufficient capacity. Similarly, in the Buy-at-Bulk Problem (see (Gupta and Könemann 2011)), the cost of routing flow along a connection is given by a concave function representing economies of scale. In both cases, the amount of flow on a single connection is unlimited.

One of the main characteristics of our problem is the step cost function with an upper limit on every connection. Gabrel et al. (1999) consider similar step cost functions in a Multicommodity Flow Problem and provide a method for finding exact solutions using a specialized Bender’s Decomposition procedure. The exact solutions, however, come at the price of only being able to solve small instances with up to 20 vertices in reasonable time. Our approach, on the other hand, has been tested on instances with up to 500 vertices providing good solutions within 50 seconds on average and 7.5 minutes in the worst case.

In this section, we formalize the WIND FARM CABLING PROBLEM (WCP). We understand turbines and substations as vertices of a graph G=(V,E), i.e., if V_T and V_S denote the sets of turbines and substations, respectively, then V=V_T∪V_S and V_T∩V_S=∅. While the direction of a connection between a turbine and a substation or between two turbines does not matter in the real world, for the sake of modeling we impose an arbitrary direction on every connection. This implies that G is a directed graph, i.e., for every edge e there are vertices u and v such that e=(u,v) and we say e goes from u to v. We assume that turbine production that is transmitted to a substation is transmitted into the connection to the grid point. In particular, it is not routed to a second substation first. To simplify the description of our algorithm, we therefore assume that there are no edges connecting two substations. Moreover, we assume that each turbine has a standardized production of one unit and each substation has a capacity modeled by a function \(\text {cap}_{\text {sub}} \colon V_{S} \to \mathbb {N}\). Additionally, each edge is assigned a positive length by the function \(\text {len} \colon E \to \mathbb {R}_{>0}\), which represents the geographic distance between the endpoints of the edge.

i.e., we choose the cheapest cable type that has sufficient capacity to transport |x| units of flow.

In total, a wind farm is then modeled as a network \(\mathcal {N} = (G, V_{T}, V_{S}, \text {len}, \text {cap}_{\text {sub}}, \mathrm {c})\). The network incorporates turbines V_T, substations V_S with a capacity cap_sub each, and connections between turbines and substations described by the graph G, as well as the length of the connections len and costs per length c for using the connections. Note that we do not explicitly include the set of cable types K as all necessary information on them is incorporated in the function c.

The goal of WCP is to find a feasible flow f with minimum costs. Hence, it can be summarized as follows.

In this section, we describe an approach of finding and canceling negative cycles in order to solve WCP heuristically. The main idea of our heuristic is to repeatedly set up a residual graph from a flow, finding a negative cycle, and cancel negative cycles in the residual graph. Every cancellation yields a better solution to WCP. In the first part, we give an overview of our heuristic. Whereas in the second part, we describe the components used in the heuristic in more detail.

Algorithmic overview

Before we describe the algorithm, we introduce essential graph theoretical terms. We define a walk from u to w as a sequence of—not necessarily distinct—edges ((u,v₁)=:e₀,e₁,…,e_k:=(v_k,w)) such that the end vertex of e_i−1 is the same as the start vertex of e_i for i∈{1,…,k}. A walk is closed if u=w and it is side-trip free if e_i is not the reverse edge of e_i−1 for all i∈{1,…,k}, i.e., the walk does not contain a closed subwalk of length 2. Closed walks where the end vertices of all edges are distinct are called cycles.

Given a wind farm \(\mathcal {N}\) we first compute an initial feasible flow f (Lines 2–4; all line references in this section refer to Algorithm 1). For each turbine u∈V_T we perform a breadth-first search from u ignoring all edges and substations without free capacity. When the search finds a substation for the first time, the flow on the path from u to the substation is increased by 1. Starting with this initial flow, we iteratively identify simple changes of the flow that decrease the costs.

In each iteration of the heuristic, we set up the residual graph R (Line 8), which we define as follows. We denote the underlying directed graph of the wind farm \(\mathcal {N}\) by G as defined in the “Model” section. We set V(R)=V(G)∪{s} and E(R)=E(G)∪{(u,v):(v,u)∈E(G)}∪{(u,s),(s,u):u∈V_S}, where s is a special vertex representing a super substation.

Let f be the flow computed in the previous iteration and \(\Delta \in \mathbb {N}\) with initialization shown in Lines 5 and 7. In addition, we define the cost function \(\gamma \colon E (R) \to \mathbb {R}\) as explained below. We then search for a closed side-trip-free walk with negative total costs in R (Line 9). To this end, we use a slight adaptation of the Bellman-Ford Algorithm. If there is no such walk, we increment Δ and set up a new residual graph. Otherwise, if there is such a walk W in R, we split W into cycles C₁,…,C_l. Figure 1 shows an example of this decomposition into cycles. We check for each cycle C_i whether its total costs are negative and whether C_i has length at least 3 (Line 11). If both conditions hold, we cancel C_i (see Eq. 6 and Line 12). Note that the cost function γ is defined in such a way that cycles that decrease the cost have negative total costs. We cancel C by changing the flow f by Δ along C. More formally, we define a new flow f^′ for all (u,v)∈E(G) by

$$ f^{\prime} (u, v) = \left\{ \begin{array}{ll} f(u, v) + \Delta, & \text{if}\ (u, v) \in E(C)\text{,} \\ f(u, v) - \Delta, & \text{if}\ (v, u) \in E(C), \\ f(u, v), & \text{if}\ (u, v),(v, u) \not\in E(C). \end{array}\right. $$

(6)

Here, E(C) denotes the set of edges that form the cycle C; see Fig. 2 for an example of canceling a cycle.

Note that if a cycle has length exactly 2, it consists of an edge (u,v) and its reverse (v,u). Hence, canceling it means sending Δ units in both directions specified by (u,v) and (v,u), which does not change the flow. Canceling it would result in an infinite loop in Line 10.

Finally, if at least one cycle C_i in W was canceled, we reset Δ to 1 and build a new residual graph based on the new flow after the cycle cancellation in the current iteration. The heuristic terminates once it does not cancel a negative cycle in the residual graphs of the currently cheapest feasible flow for all values of Δ.

In this paper, we introduce a heuristic that is able to calculate feasible solutions for WCP in milliseconds and thus, it provides an alternative to MILPs, even though our algorithm may not solve the problem to optimality. In this section, we evaluate the solution quality and running time of our heuristic and compare it to the baseline MILP which minimizes the step cost function in Eq. 5 subject to the constraints given by Eqs. 2–4. We analyze the solution quality and performance on the benchmark sets published by Lehmann et al. (2017) using different criteria namely the number of turbines |V_T| (Fig. 3a), and the benchmark sets \(\mathcal {N}_{i}\) with 1≤i≤4 (Fig. 3b). The benchmark sets include data on cables and their characteristics.

We calculate the baseline by the MILP using Gurobi 7.0.2 (Gurobi optimizer reference manual 2018). Our code is written in C++14; compiled with GCC 7.3.1 using the -O3 -march=native flags. The simulations run on a 64-bit architecture with four 12-core CPUs of AMD 6172 clocked at 2.1 GHz with 256 GB RAM running OpenSUSE 42.3. Though we have a multi-core machine, we run all simulations—including the MILP—in single-threaded mode to ensure comparability. In addition, for the simulations with regards to the MILP we opt for a quantity measurement (similar to (Lehmann et al. 2017)). That means we run a large fraction of benchmark instances with a time limit of one hour for every instance instead of running a few instances for a longer time and solving them—if possible—to optimality. This provides us with a broader range of results.

Summarizing the evaluation, the Negative Cycle Algorithm is a good alternative especially when it comes to large instances. For small instances there is room for improvement, e.g., by analyzing cases where the MILP is better.

Based on canceling negative cycles we present a heuristic for the WIND FARM CABLING PROBLEM (WCP). It runs very quickly—in the order of milliseconds to a few minutes depending on the size of the wind farm given as input—and provides very good results. Our comparison to the solutions of an MILP solver after one hour indicates that our heuristic often produces better solutions even though it takes only a fraction of the time.

Moving forward in this research in progress, we want to continue identifying strengths and weaknesses of our heuristic by running further analysis on the data provided by our simulations and by elaborating on the correctness of our algorithm. We plan to conduct further simulations on the MILP-side with longer running times of several days or even weeks. Furthermore, we hope we are able to compare our heuristic to other (meta-)heuristics tackling WCP. All of these will help us improve our algorithmic approach to solving WCP by canceling negative cycles.

So far, in all our simulations we only considered one set of cable types. It would be interesting to run our algorithm for different cable types and compare the findings depending on the characteristics of various cable type sets. We also assume standardized turbine production throughout the wind farm. We hope we are able to account for non-uniform productions by adjusting the residual graph so that the non-uniform net-flow at turbines is maintained.

Those instances, for which the MILP provides better solutions, show that there are non-optimal feasible flows that are not further improvable by our algorithm. Such a flow can be seen as a local minimum for our heuristic. To escape these local minima there are multiple viable strategies. One way could be to allow changes that temporarily increase the total costs, e.g., by canceling cycles with small positive total costs if no negative cycle is found—similar to metaheuristics like Simulated Annealing. As another approach we could search for more complex circulations with negative total costs in the residual graph, e.g., two cycles that share an edge, and cancel those. From a more theoretical standpoint, it would be very interesting to see if optimality can be achieved by identifying only a small set of more complex circulations.

So far, we restricted our heuristic to a single initialization strategy, namely breadth-first search. Other techniques might influence the trajectory of canceling negative cycles and therefore our heuristic might converge to other local minima.

Since our heuristic finds good solution within a short period of time, it might be interesting to see how those solutions can help the MILP to solve WCP. More specifically, a solution given by our algorithm can be given to the solver as an initial feasible solution from which the optimization procedure can be started (warm start). Then, the performances of the MILP with warm and with cold start can be compared in further simulations.

In our model, it is not required that every turbine has only one edge with outgoing flow. When applying AC-flow or its DC-approximation including phase angles at vertices, it might be desirable to prohibit splitting flow at vertices. In the existing literature, requiring unsplittable flow is often neglected to reduce the complexity of the problem. In terms of future work, allowing only one edge per turbine with outgoing flow in our heuristic seems to be possible by suitably modifying the residual graph and the residual costs. With that, we hope that our model represents real-world wind farms more realistically.