Resource allocation problems in decentralized energy management
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Abstract
Changes in our electricity supply chain are causing a paradigm shift from centralized control towards decentralized energy management. Within the framework of decentralized energy management, devices that offer flexibility in their load profile play an important role. These devices schedule their flexible load profile based on steering signals received from centralized controllers. The problem of finding optimal device schedules based on the received steering signals falls into the framework of resource allocation problems. We study an extension of the traditional problems studied within resource allocation and prove that a divideandconquer strategy gives an optimal solution for the considered extension. This leads to an efficient recursive algorithm, with quadratic complexity in the practically relevant case of quadratic objective functions. Furthermore, we study discrete variants of two problems common in decentralized energy management. We show that these problems are NPhard and formulate natural relaxations of both considered discrete problems that we solve efficiently. Finally, we show that the solutions to the natural relaxations closely resemble solutions to the original, hard problems.
Keywords
Resource allocation Convex optimization Decentralized energy management1 Introduction
Our electricity supply chain is changing rapidly. Traditionally electricity is supplied by a small number of large producers that are centrally controlled. The goal of the current control methodology is to ensure the required balance between production and consumption at all times by letting production follow the demand. However, driven by environmental targets, a large number of smallscale generation units are being introduced in the system in recent years, many of which exploit renewable, uncontrollable sources such as wind and sun (Gönsch and Hassler 2016). To offset the loss of flexibility on the production side and avoid very high investment requirements in infrastructure, flexibility on the demand side is increasingly considered as a valuable alternative (Siano 2014; Vardakas et al. 2015). This flexibility on the demand side comes from appliances that are capable of changing their load profile without significantly decreasing user comfort. Examples are an electric vehicle (EV) shifting its charging from the evening to the night or a heat pump combined with heat storage which produces heat using electricity before it is needed and storing it in the heat storage. Many of these devices are, if not already present inside the homes of residential consumers, expected to be introduced in large numbers in the coming years.
To properly use the flexibility provided by electric appliances, specifically those used by residential customers, the traditional centralized control methodologies do not suffice as they do not scale to a large number of entities. Furthermore, it is hard to properly deal with the large diversity of devices. Thus, new approaches are required to schedule the use of flexibility of devices emerging on the customer side of the grid.
In recent years, several new approaches have been suggested in the literature. Some of these approaches require the devices to communicate their available flexibility to a central controller (e.g. the PowerMatcher (Kok 2013) and the Intelligator (Claessens et al. 2012)). However, such a centralized approach requires the disclosure of privacysensitive information. Furthermore, to be able to solve the central problem simplifying assumptions are generally made. For example, the PowerMatcher does not look ahead to potential future requirements of flexibility and can hence use up flexibility before the most opportune time. The Intelligator, on the other hand, aggregates the constraints for the individual devices into a single flexibility constraint on the central level, which can potentially cause the resulting solution to be infeasible on the device level. Furthermore, the approach requires the use of selflearning techniques to determine the relevant parameters. In case the parameters are hard to learn, the system operates suboptimally (Claessen et al. 2014).
Another approach is decentralized energy management (DEM). In such a decentralized approach, a centralized control steers the electricity consumption and production of devices through the use of steering signals. The flexible devices schedule their load profile, using their available flexibility, to best match the received signal. A popular steering signal is socalled timeofuse pricing, where the price of electricity varies over predetermined time intervals to incentivize customers to shift their consumption to times with abundant production (see, e.g.Telaretti et al. 2016; You et al. 2012). However, such (linear) pricing structures often suffer from drawbacks in that they often only shift the peak instead of lowering it or even cause new, higher peaks in case a large number of (automated) devices respond to the same price signal (Barbato and Capone 2014; McKenna and Keane 2014). Therefore, we believe more sophisticated steering signals are required to design a system that adequately uses the available flexibility from devices. Examples of approaches that use more sophisticated signals are Gerards et al. (2015), MohsenianRad et al. (2010) and Gan et al. (2013).
The approaches mentioned above, which use more general steering signals, typically deal with design of the approach on the level of the central controller (i.e. the decision on what the steering signal is going to be). However, the response of the devices to these steering signals is generally left as an open question. Furthermore, these responses are often assumed to be optimal with respect to the steering signal, to be able to prove certain properties of the approach (e.g. the convergence proved by Gan et al. (2013)). The devicelevel problems are generally formulated as nonlinear optimization problems that can be solved by generalpurpose nonlinear solvers. However, we note that the local hardware on which such a problem has to be solved is often limited (Molderink et al. 2010). Furthermore, the devicelevel problems have similar structures in many cases, which allows for specific solution methods that significantly reduce hardware requirements and increase the solution speed. The latter greatly benefits approaches where many devicelevel problems have to be solved (e.g. the approach by Gerards et al. (2015)).
For practical applications, it is not desirable that one has to design a new scheduling method for each new variant of a device. To avoid this, a higher abstraction of device classes is needed. One attempt to define a framework for DEM with abstract device classes in the EFPI platform developed by the Flexiblepower Alliance Network (2016). Therefore, we argue in this work that the devicelevel scheduling problems of many devices are in fact very similar and can be solved by the approaches we present.
In this work, we consider abstract models of several devices and argue that their load scheduling problem is in fact a resource allocation problem, where the resource to be allocated is the energy consumption and/or production of the device. Resource allocation problems are well studied in the literature and have applications in many fields; see for example the excellent surveys (Patriksson 2008; Patriksson and Strömberg 2015). One example of an application field is power dispatch, a problem studied in the traditional, centralized energy management field (see, e.g.Van Den Bosch 1985; Van Den Bosch and Lootsma 1987; Kleinmann and Schultz 1990). Another is that of green computing, where processor speed, and thus energy consumption, is dynamically changed to reduce the overall energy requirements. Within this technique of dynamic voltage and frequency scaling (DVFS) (see, e.g. Yao et al. 1995; Li et al. 2006), problems similar to those we encounter in decentralized energy management are commonly solved (Huang and Wang 2009; Tang et al. 2014). Another area of application is logistics (see, e.g. Norstad et al. 2011; Hvattum et al. 2013). This problem consists of finding optimal vessel route and speed schedules. Herein a resource allocation problem is used as a subroutine of the main solution method. Hence, also in this field, efficient solution methods are highly desirable.
In the area of decentralized energy management, one of the important device scheduling problems is that of finding optimal schedules for the charging of EVs. We show that this problem corresponds to the classical resource allocation problem, as surveyed by Patriksson. For this problem, many (efficient) solution methods exist. Next, we extend the EV problem to include the option to discharge electricity to the grid (vehicle to grid). In this way, the EV can function as a producer during times when this is beneficial. The resulting problem also models the behaviour of various other devices, e.g. a heat pump combined with a heat storage. We derive a property of any optimal solution to this more general problem and use this property to derive an efficient, recursive algorithm, which is based on a divideandconquer strategy. The resulting algorithm is also applicable to the aforementioned applications in DVFS and for the vessel route and speed scheduling problem.
In the final part of the paper, we consider variants of the previously studied problems where the decision variables no longer have a continuous range but are restricted to a discrete set, to more closely model the behaviour of several devices occurring in practice. We show these problems are NPhard in general. However, we derive efficient algorithms for natural relaxations of these problems. Furthermore, we show that these algorithms produce solutions that closely resemble feasible solutions to the original, hard problems.

An efficient solution method for an extension to the traditional resource allocation problem with applications in decentralized energy management and various other fields.

An NPhardness proof for the discrete variants of both the classical resource allocation problem and the extension discussed in this work.

Efficient algorithms for a natural relaxation of both discrete problems considered.
2 Background and definition of the EV charging problem
In this section, we consider the problem of scheduling the energy consumption (i.e. the load profile) of an EV under a received steering signal. We assume that the schedule is made for a time horizon discretized into a set of time intervals, i.e. the schedule details how much energy is charged for every time interval of, for example, 15 min. Let i denote the index of the time intervals, running from 1 to n, and \(x_i\) be the scheduled energy consumption of the EV for time interval i. We assume that the steering signals induce a cost function \(f_i(x_i)\) for every time interval i. Examples of cost functions are the price of the consumed energy (\(f_i(x_i) := a_i x_i\) with \(a_i\) the price of energy during interval i) or the squared deviation from a target consumption (\(f_i(x_i) = (x_i  d_i)^2\) with \(d_i\) the desired consumption for interval i). The total cost of a schedule is given by the sum \(\sum _{i=1}^n{f_i(x_i)}\) of the costs for the individual intervals. Note that by this cost definition we implicitly assume that the total cost is separable. Furthermore, we assume that the cost for an interval, given by \(f_i\), is convex and continuous in \(x_i\).
Many efficient methods to solve problem SRA exist (see, e.g. Patriksson 2008; Patriksson and Strömberg 2015). Most of these methods are based on optimality conditions tracing back to the nineteenthcentury scientist Gibbs (as noted by Patriksson 2008). These conditions were also used in previous work in the field of energy management to obtain solution methods to similar problems (Van Den Bosch 1985; Van Den Bosch and Lootsma 1987; Kleinmann and Schultz 1990). The optimality conditions can be shown to be necessary and sufficient using a generalization of the wellknown KKT conditions (Boyd and Vandenberghe 2004). We state them in the following lemma.
Lemma 1
Proof
follows directly from the generalized KKT conditions with subdifferentials (see, e.g. Rockafellar 1970). \(\square \)
These optimality conditions play a vital role when considering extensions of problem SRA. Furthermore, we note that it is possible to explicitly formulate the solution from the conditions in case the objective functions are quadratic (see Kleinmann and Schultz 1990). Example 1 gives some more insight in the conditions.
Example 1
Figure 1 depicts an example solution to an instance of Problem SRA with \(n=4\). In the first four plots of the figure, we plot the 4 objective functions \(f_1,\ldots ,f_4\) with the optimal solution \(x = (x_1,\ldots ,x_4)\). By Lemma 1, there exists a \(\lambda \) such that \(f'_i(x_i) = \lambda \) for i with \(0< x_i < u_i\). This is the case for \(i=2\) and \(i=4\) in our example. Furthermore, \(f'_1 (x_1) < \lambda \) and \(f'_3(x_3) > \lambda \) since \(x_1 = 0\) and \(x_4 = u_4\). The derivatives for this particular instance are given in the next 4 plots with the value of \(\lambda \) given by the grey line. In the last plot, the cumulative sum \(\sum _{i'=1}^i{x_{i'}}\) is shown. We note that the only restriction on this sum is that it should be equal to C for \(i=4\).
3 Problem SRA with cumulative bounds
Problem CRA allows modelling of the charging and discharging of an electric vehicle. Furthermore, the same formulation also models a standalone battery used inside a house to, for example, store solar energy for later use or a heat producer (e.g. a heat pump or combined heat and power unit) together with a heat vessel to store the heat for usage later on. In the latter case, \(B_j\) represents the cumulative heat demand up to time j, which must be met by the device, and \(C_j\) represents this demand plus the storage capacity.
Problem CRA has many applications outside of decentralized energy management. For example, it is a subroutine in the vessel routing and scheduling problem where an optimal route for a fleet of vessels has to be determined together with the speed for the vessels on the legs of their route (Norstad et al. 2011; Hvattum et al. 2013). Also, problem CRA models the problem of finding a tradeoff between processor speed and energy consumption and determining a schedule of tasks for such a processor that satisfies arrivals and deadlines while minimizing energy consumption in the field of DVFS with agreeable deadlines (Huang and Wang 2009). Commonly the assumption that the objective functions are equal for all times intervals is made in both aforementioned fields. We note that we do not make this assumption.
When considering problem CRA, we note that we no longer have the constraint that a specific amount of the resource must be used over the given set of intervals, i.e. we can have that \(B_n < C_n\). However, if this is the case, we can add an additional variable \(x_{n+1}\) with \(f_{n+1}(x_{n+1}) = 0\), \(B_{n+1} = C_{n+1} = C_n\) and \(l_{n+1} = 0, u_{n+1} = C_n  B_n\). This essentially allows us to ensure that the total usage of the resource over the set of intervals sums up to \(C_n\) without incurring additional cost, hence preserving optimality. An optimal solution to the original problem now follows by discarding the variable \(x_{n+1}\). Hence, in the following, we assume that \(B_n = C_n\).
The base of our solution approach is that we first generate a candidate solution by solving problem SRA with \(C = C_n\) and the given bounds on the resource usage in the different intervals. This solution may be infeasible for several of the cumulative resource constraints, i.e. \(\sum _{i=1}^j{x_i}\) might be smaller than \(B_j\) or larger than \(C_j\) for several j. However, since the objective functions are convex, intuitively it seems that in an optimal solution the constraint that is violated most in the (infeasible) candidate solution should be tight in an optimal solution to problem CRA. This property is in fact proven and used in both the applications of DVFS (Huang and Wang 2009) and vessel speed optimization (Hvattum et al. 2013) to construct efficient algorithms to compute a solution. However, in those settings, the objective function is the same for every time interval, whereas in decentralized energy management this does not need to be the case. In the following, we show that this property still holds when we assume each \(f_i\) to be an arbitrary continuous and convex function.
Lemma 2
Consider an instance of CRA with \(C_n = B_n\) and let y be an optimal solution to the instance of SRA obtained by ignoring the cumulative bounds for all indices except the last. Assume that y is not feasible for the considered instance of CRA and let k be the index of the variable that maximally violates the cumulative bounds, i.e. \(k = {{\mathrm{arg\,max}}}_j \{ \sum _{i=1}^j{y_i}  C_j, B_j  \sum _{i=1}^j{y_i} \}\). Then, there is an optimal solution x to the considered instance of CRA such that, if \(\sum _{i=1}^k{y_i} > C_k\), then \(\sum _{i=1}^k{x_i} = C_k\) and, on the other hand, if \(\sum _{i=1}^k{y_i} < B_k\), then \(\sum _{i=1}^k{x_i} = B_k\).
Proof
Let x be an optimal solution to the considered instance of CRA and assume that \(\sum _{i=1}^k{y_i} > C_k\) and \(\sum _{i=1}^k{x_i} \ne C_k\). Since x is feasible, it follows that \(\sum _{i=1}^k{x_i} < C_k\). Let l be the last index before k for which the upper cumulative bound is met by x with equality, i.e. \(l := \max \{ j < k  \sum _{i=1}^j{x_i} = C_j \}\) and \(l := 0\) if this set is empty. Furthermore, let m be the first index after k for which the upper cumulative bound is met by x with equality, i.e. \(m := \min \{ j > k  \sum _{i=1}^j{x_i} = C_j \}\). Note that \(\sum _{i=1}^n{x_i} = C_n\) by assumption; hence, m is well defined and \(m \le n\).
Since \(\sum _{i=1}^l{x_i} = C_l\) and \(\sum _{i=1}^k{x_i} < C_k\), it follows that \(\sum _{i=l+1}^k{x_i} < C_k  C_l\). Also, since \(\sum _{i=1}^l{y_i}  C_l \le \sum _{i=1}^k{y_i}  C_k\) by construction of k, it follows that \(\sum _{i=l+1}^k{y_i} \ge C_k  C_l\). Combining this, we obtain that \(\sum _{i=l+1}^k{y_i} >\sum _{i=l+1}^k{x_i}\), and hence, there exists an index s with \(l < s \le k\) such that \(y_s > x_s\). Similarly, since \(\sum _{i=1}^m{x_i} = C_m\) and \(\sum _{i=1}^k{x_i} < C_k\), it follows that \(\sum _{i=k+1}^m{x_i} > C_m  C_k\). Also, since \(\sum _{i=1}^m{y_i}  C_m \le \sum _{i=1}^k{y_i}  C_k\), it follows that \(\sum _{i=k+1}^m{y_i} \le C_m  C_k\). Combining this, we obtain that \(\sum _{i=k+1}^m{y_i} < \sum _{i=k+1}^m{x_i}\), and hence, there exists an index t with \(k < t \le m\) such that \(y_t < x_t\).
The proof for the case that \(\sum _{i=1}^k{y_i} < B_k\) and \(\sum _{i=1}^k{x_i} > B_k\) is symmetric. \(\square \)
Lemma 2 is the base of a solution approach for CRA. For this approach, we first ignore the intermediate cumulative bounds of CRA, and afterwards, we iteratively satisfy the ignored constraints using a divideandconquer approach. We start by calculating an optimal solution for the instance of SRA, which we get by setting \(C = C_n\). For this optimal solution, we determine the index k where this solution maximally violates the cumulative bounds. By Lemma 2 we know that there exists an optimal solution for which the corresponding bound is tight for index k, meaning that we can set both \(B_k\) and \(C_k\) to the value of the violated bound (i.e. either to \(B_k\) or to \(C_k\)). Note that this splits the original instance of CRA into two independent instances of CRA: one for the indices up to and including k and one for the indices after k. These problems can be solved separately following the same procedure. Hence we can recursively solve problems of the form SRA, until we no longer have violations of the intermediate cumulative bounds. Combining the individual solutions of these instances then gives a solution to CRA that is optimal by Lemma 2 and the fact that the individual solutions are optimal for their respective time intervals.
Example 2
Figure 2 depicts an application of Algorithm 1. The problem instance is the same as in Example 1 except that we added lower and upper cumulative bounds. These bounds are depicted in middle plot showing the cumulative sum. Above the cumulative sum, the objective functions and derivates are plotted together with the original solution x to Problem SRA. This solution serves as a candidate solution to Problem CRA, which does not consider the cumulative bounds. This candidate solution is not feasible, since \(\sum _{i=1}^2{x_i} > C_2\). Based on this, we split the problem into two subproblems. In the first subproblem, we have to decrease \(x_1\) and/or \(x_2\). Note that by doing this we obtain \(x'_1 < u_1\), and thus, we find \(\lambda _1\) with \(f'_1(x'_1) = f'_2(x'_2) = \lambda _1\). In the second subproblem, we increase \(x_3\) and \(x_4\). Doing this gives us \(x'_4 = u_4\); thus, we find \(\lambda _2\) such that \(f'_3(x'_3) = \lambda _2 > f'_4(x'_4)\). Combining the solutions to the subproblems, we obtain the optimal solution \(x'\) to Problem CRA. This solution is given in the bottom eight plots.
4 Resource allocation over a discrete set
In the previous sections, we assumed that the scheduled load of the device in an interval can be any value between the given lower and upper bound for that interval. However, this might not be the case in practice for several devices. For example, a heat pump generally runs at a limited number of predefined levels. Also, current charging stations for EVs only allow charging at specific amperages, i.e. at specific power levels (Kesler et al. 2014). To model this, we have to replace the continuous range for \(x_i\) by a finite set. We show that this makes problem SRA NPhard in general. However, we show that there is a natural relaxation in the setting of decentralized energy management that leads to efficient solution methods which give solutions that are quite close and similar to solutions to the original discrete version of the problem.
4.1 Discrete SRA
Lemma 3
The decision problem of determining whether a feasible solution to dSRA exists is NPcomplete, even if all sets \(Z_i\) are the same.
Proof
Clearly, the problem of existence of a feasible solution is in NP. The NPcompleteness if the sets \(Z_i\) may differ follows from a reduction from the partition problem. For this we define \(Z_i := \{0,p_i\}\) and \(C = \frac{1}{2} \sum _{i}{p_i}\), where \(p_1,p_2,\ldots ,p_n\) are the integers from the set to be partitioned.
For the case that all \(Z_i\) are equal, we assume that \(Z_i = \{z_0,z_1,\ldots ,z_m\}\) for every i and use a reduction from even/odd partition. In the even/odd partition problem, a set \(P = \{p_1,p_2,\ldots ,p_{2l} \}\) of 2l nonnegative integers is given with total sum 2B. The problem asks whether there is a subset \(A \subset \{1,2,\ldots ,2l\}\) such that \(\sum _{i \in A}{p_i} = B\) and for \(i=1,2,\ldots ,l\) we have: \(2i1 \in A \Leftrightarrow 2i \notin A\).
Consider an instance I of even/odd partition and let \(k := \lfloor \log _2 (2B) \rfloor \), i.e. k is the unique integer such that \(2^k \le 2B < 2^{k+1}\). To transform this instance of even/odd partition to an instance \(I'\) of dSRA, we take \(n = m = 2l\). Furthermore, we choose \(z_{2i1} = p_{2i1} + 2^{k+i}\) and \(z_{2i} = p_{2i} + 2^{k+i}\) for \(i=1,2,\ldots ,l\) and \(C = B + \sum _{i=1}^{l}{2^{k+i}}\). In the following, we show that I is a yesinstance iff \(I'\) is a yesinstance.
First assume that I is a yesinstance. Thus there exists a subset A with \(\sum _{i \in A}{p_i} = B\) and \(2i \in A \Leftrightarrow 2i1 \notin A\). By defining \(x_{2i} = p_{2i} + 2^{k+i}\) if \(2i \in A\) and \(x_{2i} = 0\) otherwise, and \(x_{2i1} = p_{2i1} + 2^{k+i}\) if \(2i1 \in A\) and \(x_{2i1} = 0\) otherwise, it now follows that \(\sum _{i=1}^n{x_i} = \sum _{i \in A}{p_i} + \sum _{i=1}^{l}{2^{k+i}} = C\).
On the other hand, assume that \(I'\) is a yesinstance for the dSRA problem and x a corresponding solution. Note that the l most significant bits of C in binary representation are 1 and correspond to \(2^{k+1},2^{k+2},\ldots ,2^{k+l}\). Since \(\sum _{t=1}^n{x_t} = C\), it follows that for \(i=1,2,\ldots ,l\) there is exactly one of the values \(p_{2i} + 2^{k+i}\) or \(p_{2i1} + 2^{k+i}\) contained in \(\sum _{i=1}^n{x_i}\). If we now define A as the subset of \(\{1,2,\ldots ,n\}\) of the indices of these values that are contained in \(\sum _{i=1}^{n}{x_i}\), it follows that \(2i \in A \Leftrightarrow 2i1 \notin A\). Furthermore, by construction we have \(\sum _{i \in A}{p_i} = \sum _{t=1}^n{x_t}  \sum _{t=1}^l{2^{k+t}} = B\).
This shows that checking whether a feasible solution to dSRA exists is NPcomplete, even if we assume that the \(Z_i\) are the same for every index i. \(\square \)
The above lemma shows that checking whether a feasible solution exists for dSRA is NPcomplete. However, in the case of energy management, the variable \(x_i\) expresses the power produced or consumed by the device over a time interval. Restricting this value to be constant over an entire time interval does not reflect all operational possibilities as many flexible devices have the option to shift between different power levels at any moment. We may model this by modifying problem dSRA to allow convex combinations of the points in each set \(Z_i\), i.e. we introduce multipliers \(y^0_i,y^1_i,\ldots ,y^{m_i}_i \in [0,1]\) with \(\sum _{j}{y^j_i} = 1\) and \(\sum _j{y^j_i z^j_i} = x_i\). We note that this is also common practice in DVFS (see, e.g. Kwon and Kim 2005; Li et al. 2006).
Considering problem rdSRA, we note that, by the convexity of \(f_i\), we can assume that, for any i, only at most two consecutive multipliers \(y^j_i, y^{j+1}_i\) are nonzero. Hence we can replace \(F_i\) by the piecewise linear function obtained from \(f_i\) by linearizing it on each of the intervals \((z^0_i,z^1_i),(z^1_i,z^2_i),\ldots ,(z^{m_i1}_i,z^{m_i}_i)\). By doing this we obtain an instance of problem SRA with piecewise linear objective functions.
As mentioned before, we are interested in very fast and efficient solution methods, since the considered problems generally have to be solved often and on embedded platforms. To this end, we study Problem SRA with piecewise linear objective in more detail. The breakpoints of the piecewise linear function \(f_i\) are given by the set \(Z_i\). Furthermore, we use \(s_i^j\) to denote the slope of the piece between breakpoints \(z_i^{j1}\) and \(z_i^j\). We first observe that pieces with a smaller slope will always be preferred by an optimal solution.
Lemma 4
Consider an instance of Problem SRA where each \(f_i\) is a piecewise linear function. Furthermore, assume that a piece with slope s is used in an optimal solution (i.e. the corresponding \(x_i\) is larger than the left breakpoint of this piece). Then, any piece with a slope smaller than s is also used by the optimal solution.
Proof
Consider an optimal solution x and assume there is a piece with slope \(s' < s\), which is not used, with corresponding objective function \(f_{i'}\). Let \(s^{\star }\) be the slope of the first piece of \(f_{i'}\) that is not completely used by x. By the convexity of \(f_{i'}\), it follows that \(s^{\star } \le s' < s\). This is a contradiction with Lemma 1. \(\square \)
Example 3
Figure 3 depicts an application of Algorithm 2. In this example, piecewise linear approximations of the objective functions used in Examples 1 and 2 are used. These approximations are given in the upper four plots with the derivatives, i.e. the slopes of the pieces, given in the four plots below that and the cumulative sum given in the last plot. The particular solution depicted in the plots is the solution after 2 steps in the while loop of the algorithm, where the first piece of both \(f_2\) and \(f_4\) (both in blue) has already been used. In the next step, the first piece of \(f_1\) (red dotted) is used since it has the lowest slope, i.e. it is the first slope in S. After this, the second piece of \(f_1\) (green dashed) is added. The ordered set S over the different steps of the algorithm is given below the plots in the lower part of the figure, with the piece used and removed marked in red and crossed out (note that this is always the first piece of S) and the new piece added marked in green. Note that in the last step the piece \(s^2_2\) is only partially used.
Lemma 5
Algorithm 2 solves SRA with piecewise linear convex objective functions to optimality in \(O(M \log n)\) time, where M is the sum of the number of pieces of the objective functions.
Proof
The feasibility of the algorithm follows from the convexity of the \(f_i\)’s, and the optimality follows directly from Lemma 4.
When considering the time complexity, note that the first sorting in Line 2 can be done in time \(O(n \log n)\). Furthermore, the heaviest operation in the while loop is the insertion of the slope of a piece in the ordered vector of at most \(n1\) other slopes. This can be done in time \(O(\log n)\) when using an appropriate data structure. Since the while loop runs for at most M iterations and \(M \ge n\), it follows that the total complexity is \(O(M \log n)\). \(\square \)
Hochbaum and Hong (1995) exploit the linear complexity of median find to solve problem SRA with quadratic objective functions in linear time. Based on their findings, we may formulate an approach for problem rdSRA with complexity O(M). Instead of sorting (part of) the pieces such that they have nondecreasing slopes, we iteratively guess the slope of the last piece used by an optimal solution. For this guess, we use the piece with the median value of the slopes, which we can find in linear time (Blum et al. 1973). In case of an even number of slopes, we pick either of the two middle values. After this we split the pieces in sets \(S_1 := \{ s_i^j  s_i^j \le m \}\) and \(S_2 := \{ s_i^j  s_i^j > m \}\), where m is the slope of the found median piece, and determine a solution x with \(\hat{C} := \sum _i{x_i}\) using the pieces in \(S_1\).

If \(\hat{C} = C\), we have found an optimal solution.

If \(\hat{C} > C\), we check whether we can obtain a feasible, and hence optimal, solution, by using the piece with slope m only partially. If this is not the case, we know that an optimal solution cannot use pieces from \(S_2\) nor piece m. Thus, we recursively call the algorithm using only the pieces in \(S_1 \setminus \{ m \}\)

If \(\hat{C} < C\), we know that an optimal solution must use all the pieces in \(S_1\). Thus, we mark these pieces as used and recursively call the algorithm on the pieces in \(S_2\) with \(C := C  \hat{C}\), since the remaining pieces must be used for this amount.
Example 4
Figure 4 depicts an application of Algorithm 3. The same instance is used as in Example 3. Here we depict the first application of the algorithm. Since there are a total of 8 pieces, the algorithm picks \(S_1 = \{s^1_1,s_1^2,s_4^1,s_2^1 \}, S_2 = \{s_2^2,s_3^1,s_3^2,s_4^2 \}\), and \(m = s_1^2\) in the top plots of the figure in blue, grey and dotted green, respectively. The guessed solution has a cumulative sum \(\hat{C}\) lower than C, as shown. For clarity, the cumulative sum of the solution without m is also plotted. This would be used in case \(\hat{C} > C\). In the depicted case, the algorithm concludes that each of the currently used pieces, i.e. those in the set \(S_1\), must be used in an optimal solution and it is recursively called on the set \(S_2\) with \(C := C  \hat{C}\).
Corollary 1
Algorithm 3 solves rdSRA to optimality in time O(M), with \(M = \sum _i{m_i}\), i.e. the total number of pieces.
Proof
The optimality of the algorithm follows immediately from the fact that pieces are used in order; hence, the algorithm finds the same solution as Algorithm 2 which is optimal by Lemma 5. The complexity of O(M) follows from the fact that all the steps before a potential recursive call in the algorithm can be solved in time linear in the number of pieces formed by the breakpoints in Z. For the first call this number of pieces is exactly M, and for subsequent, recursive calls the number is halved each time. Hence the complexity is \(O(M + M/2 + M/4 + \cdots ) = O(M)\). \(\square \)
We note that the actual running time of the median find algorithm with linear asymptotic complexity is high for small to mediumsized sets (Blum et al. 1973). Thus, while Algorithm 3 has a lower asymptotic complexity than Algorithm 2, the latter might in practice be more efficient for decentralized energy management applications, where the number of intervals is typically low.
As mentioned before, for problem rdSRA, which we can solve efficiently using either Algorithm 2 or 3, we assumed that devices can switch between different consumption levels or states somewhere during a time interval. However, excessive switching may cause undesirable wearing of the device, e.g. the lifetime reduction that may occur for heat pumps and compressors in airconditioning systems due to frequent on/off cycling or increased degradation of the internal battery within EVs from excessive switching between different charging and discharging currents (Kesler et al. 2014). Thus, solutions that avoid excessive switching are desirable. However, if we look at the solutions produced by Algorithms 2 and 3, we see that they have \(x_i \notin Z_i\) for at most one index i. This leads to the following corollary.
Corollary 2
There always exists an optimal solution to problem rdSRA such that \(x_i \notin Z_i\) for at most one i.
As a consequence, the optimal solutions we obtained for Problem rdSRA do not cause much extra wearing of the devices over any feasible solution to Problem dSRA.
4.2 Discrete CRA
Similarly to the greedy approach for Problem SRA with piecewise linear objective, presented in Sect. 4.1, we now construct a greedy approach to solve CRA with piecewise linear objective that prefers pieces with a smaller slope. However, we also need to satisfy the cumulative bounds \(B_i\) and \(C_i\) for every i. We note that to satisfy the lower bound \(B_i\), we can only use pieces of the functions \(f_{i'}\) with \(i' \le i\). Therefore, we iteratively build up a solution that satisfies the lower bounds \(B_1,B_2,\ldots ,B_k\) in iteration k. To do this, during iteration k, we only consider pieces of the functions \(f_1,f_2,\ldots ,f_k\) and increase the use of these pieces until \(\sum _{i=1}^k{x_i} = B_k\).
Next we consider the upper cumulative bound \(C_i\). We note that we cannot increase the use of the pieces of function \(f_i\) by more than \(C_j  \sum _{i'=1}^j{x_{i'}}\) for every \(j \ge i\), lest we violate the upper bound \(C_j\). Thus, to ensure feasibility, we introduce a variable \(V_i := \min _{j \ge i} \{ C_j  \sum _{i'=1}^j{x_{i'}} \}\) to track how much we can increase \(x_i\), using the pieces of \(f_i\), without violating the upper bound \(C_i\). Note that \(V_i\) depends on all \(x_j\); thus, after we increased the use of the pieces of some \(f_j\) by \(\delta \), we need to update \(V_i\) for every i. Furthermore, note that whenever we increase \(x_j\) by \(\delta \), \(\sum _{i'=1}^i{x_{i'}}\) is also increased by \(\delta \) for every \(i \ge j\). This implies that \(V_i\) decreases by exactly \(\delta \) for every \(i \ge j\). On the other hand, for \(i < j\), we note that \(V_{i1} = \min \{ V_i,C_i  \sum _{i'=1}^{i1}{x_{i'}} \}\). This can be used to iteratively update \(V_{j1},V_{j2},\ldots ,V_1\).
Example 5
Figure 5 depicts an application of Algorithm 4. For this instance, we added the cumulative bounds used in Example 2 to the instance used in Examples 3 and 4. A colour coding is used to denote (parts of) the pieces that are used in each iteration of the main for loop of the algorithm (brown dashed, blue dotted, red and green dashdotted, respectively). The same colour coding is used in the plot that depicts the cumulative sum till the kth interval for each of the partial solutions constructed. Furthermore, the solution marked in the top plots is the optimal solution the algorithm produces after completion. In the table, the values of various variables used inside the algorithm are denoted for each iteration of the while loop in the algorithm. These iterations are denoted in the column Iteration. Here, \(S_{\mathrm{start}}\) denotes the set S at the start of the considered iteration of the while loop, and \(S_{\mathrm{end}}\) the set S at the end of the iteration. New additions to S are depicted in green. In particular we note that \(s_2^2\) is deleted from S at the end of iteration 4, since \(V_2 = 0\). Thus the algorithm must use \(s_3^1\) instead to ensure that the bound \(B_3\) is met.
Lemma 6
The greedy approach for CRA with piecewise linear convex objective, given in Algorithm 4, gives an optimal solution to CRA with piecewise linear convex objective functions. The algorithm runs in time O(nM).
Proof
The feasibility of the algorithm follows from the fact that the pieces are added in increasing order and that any new piece is either the first piece of an objective function (Line 4) or a piece for which the previous piece is already fully used (Line 22). Also, note that the lower cumulative bounds are satisfied since the constructed solution enforces this for every index in the main for loop of the algorithm (Lines 3–28). Finally, note that in every step of the main for loop, \(V_i \le C_j  \sum _{k=1}^{j}{x_k}\) for \(j \ge i\). Hence, in Line 7, we ensure that the upper cumulative bounds are satisfied.
To prove optimality of the algorithm, consider an instance of CRA with piecewise linear convex objective functions. Let y be an optimal solution, and let x be the solution produced by Algorithm 4. Furthermore, let i be the smallest index for which \(x_i \ne y_i\). We consider two cases.
Case 1 \(x_i > y_i\): Let j be the smallest index for which \(x_j < y_j\). This index must exist since \(\sum _i{x_i} = B_n = \sum _i{y_i}\). Furthermore, let \(s_i\) and \(s_j\) be the slopes of the pieces on which \(x_i\) and \(x_j\) lie, respectively. In case \(x_i \in Z_i\), we pick the piece with \(x_i\) as endpoint, and in case \(x_j \in Z_j\), we pick the piece with \(x_j\) as begin point. By the fact that the right and left derivatives of \(f_i\) and \(f_j\) are nondecreasing, it follows that \(s_i \ge f^+_i(y_i)\) and \(f^_j (y_j) \ge s_j\). Also, by the optimality of y it follows that \(f^+_i(y_i) \ge f^_j(y_j)\). Finally note that, since both x and y are feasible, we can decrease \(x_i\) and increase \(x_j\), until either \(x_i\) is equal to \(y_i\) or \(x_j\) is equal to \(y_j\), without violating feasibility. Finally, note that doing so does not increase the objective value.
Case 2 \(x_i < y_i\): This case can be treated completely symmetrical to the previous case.
The above process can be repeated until \(x = y\) without increasing the objective value. This shows that x is indeed optimal.
Finally, we show that the time complexity of the algorithm is O(nM) with M the sum of the number of pieces of each \(f_i\), i.e. \(M = \sum _i{m_i}\). Note that a slope \(s_i^j\) can only be added to and subsequently removed from S once. Furthermore, if a slope is picked to be used in the while loop and not removed, it follows that \(R = 0\) after this iteration, and hence, the while loop is finished. Finally, note that the steps inside the while loop can all be executed in time O(n), since the size of S is never more than n. Hence, the total complexity of the for loop in the algorithm is O(nM), which clearly dominates the complexity of the other steps. \(\square \)
We note that the asymptotic complexity of Algorithm 4 is slightly lower than the complexity when we recursively apply Algorithm 1 using Algorithm 2 to obtain solutions for the resulting SRA problems. However, the latter method might be more favourable in practice, specifically if the number of intervals for which the cumulative bounds are tight is low, as can be expected in many applications (Yao et al. 1995; Vidal et al. 2014). Furthermore, the asymptotic complexity of Algorithm 4 is the same as the complexity when we recursively apply Algorithm 1 using Algorithm 3. As noted before, however, the latter method is probably only efficient for large instances, i.e. instances where the number of breakpoints is very large.
From a practical point of view, it is again important to consider how often solutions to rdCRA switch between the different values in the feasible set \(Z_i\) for each i. We obtain the following result, which is similar to Corollary for Problem rdSRA.
Corollary 3
There exists an optimal solution to rdCRA such that for any two indices \(i < i'\) with \(x_i \notin Z_i\), \(x_{i'} \notin Z_{i'}\) there exists an index \(i \le k < i'\) with \(\sum _{i''=1}^k{x_{i''}} \in \{B_k,C_k\}\), i.e. solution x meets either the lower or upper cumulative bound tightly for an index in between i and \(i'\).
This result follows immediately from the fact that we can recursively apply Algorithm 2 to Problem rdCRA by Lemma 2. In practice we do not expect these cumulative bounds to be met often. Therefore, the above shows that, in most practical cases, the expected number of time intervals for which two operational values are chosen instead of one is low.
5 Conclusion
In this work, we considered scheduling problems motivated by the domain of decentralized energy management. Within this domain, flexible appliances have to schedule their load profile based on steering signals received from a central controller. We showed that many of these problems can be modelled as resource allocation problems. Furthermore, several variants of these scheduling problems extend the traditional resource allocation problem by adding cumulative lower and upper bounds on the resource usage. This problem has applications in various fields, and we showed that a simple, recursive algorithm can be used to solve this problem in a very general form. The asymptotic complexity of the algorithm is low and is quadratic in the case of quadratic objective functions.
Furthermore, we considered discrete versions of the two studied variants of resource allocation problems. While traditionally discrete resource allocations only add the restriction that the decision variable \(x_i\) should lie in \(\mathbb {Z}\), the application area of DEM leads to a more restrictive variant where the feasible set for each \(x_i\) is a finite subset of \(\mathbb {R}\). This makes the problem NPhard, even if each of these sets \(Z_i\) is the same. However, in the domain of decentralized energy management, there is a natural relaxation for this problem, similar to one found in the area of DVFS (also known as speed scaling). These relaxations allow efficient solution methods while producing solutions that are very similar to solutions of the discrete problems we originally introduce. More precisely, in most practical cases we expect that the extra switching between operational values in the solutions to the relaxations is minimal, causing only minimal extra wearing of the considered devices in practice.
The models herein do not capture all (common) constraints and objectives encountered in the area of DEM. For example, another important constraint often seen is that of minimum run times. Such a constraint forces a device, for example a heat pump, to stay on for a minimum amount of time when it is switched on. This is commonly done to minimize device wearing and increase efficiency. An example of an important factor for the local objective of a device is the degradation of the device. Such constraints and objectives often add a dependency between the time intervals considered in the scheduling problem. This dependency causes the problem to become more difficult, which is left for future work.
A challenge in energy management is the unpredictability of several aspects, particularly the production from new, renewable sources and the availability of flexible devices. As the devicelevel problems covered herein can be solved very efficiently, we believe such uncertainty can be accounted for, by rescheduling (subsets of) devices. In case the central controller, responsible for sending steering signals, notices a (large) mismatch between a prediction and the corresponding observation (e.g. an EV arrives later or needs to depart sooner, or the solar production is more/less than expected), it can update the steering signals sent to the devices to compensate. The central controller then requests the device to reschedule based on the updated steering signal. We believe the efficient solution methods developed herein assist in ensuring that such an approach can be used in practice to also address observed mismatches between predictions and reality.
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