Routing optimization on power packet dispatching system based on energy loss minimization

Electric energy management has been an essential technology in a variety of systems such as vehicles, robots, and microgrids [1, 2, 3]. Particularly in such systems, the strong demand for eco-friendliness has raised the importance of an efficient inclusion of distributed power sources. Then, the time-varying profile of the sources creates more complex power flow than in the conventional system based on a large stable power source. A key enabler for handling the complex power flow is the support of Information and Communication Technology (ICT) [4, 5, 6].

Power packet dispatching system has been proposed as a way of such smart power management [7, 8].

Figure 1 depicts the concept of the power packet dispatching system. A power packet is a unit of transfer of both power and information. A payload delivers a quantized unit of electric power. An information tag, attached to the payload in voltage waveform, represents information such as identification of the packet. The physical tag attachment enables the network to distinguish each packet and thus to process electric power in a fully digitized manner. This feature makes the power management highly compatible with the ICT.

A power packet is transferred from its source to a load through a network of apparatuses called power routers (hereafter we simply call them routers). Equipped with some temporary energy storage, the routers deliver a power packet in a store and forward manner [8]. Recent study [9] reported that the power packetization can be realized at a rating of several kW.

A remaining challenge is a routing optimization of a power packet. Similarly with the IP routing [10], in a power packet dispatching system of a number of routers, there are many possible paths between a pair of a source to a load. It then becomes necessary to set an appropriate strategy for selecting one. However, we cannot directly apply the routing problem for the IP routing. As we consider the flow of physical quantity, we need a new criterion for evaluating the power packet routing based on the dynamical behavior of the circuit.

In addition, a matching optimization of the distributed sources and loads is another important challenge. The network structure of a power packet dispatching system can vary dynamically. In a microgrid application, for example, electric vehicles (EVs) can connect to or disconnect from the system at an arbitrary time. Renewable sources such as photovoltaic cells and wind turbines can also have time-dependent output profiles. In this way, the sources are spatio-temporally distributed. Now suppose that a load does not have a specific demand for the origin of power but only for the amount of power, the system is requested to select one of the multiple sources available at the time.

In this paper, we address the above challenges by introducing an optimization of power packet routing based on a cost metric derived by circuit analysis. First, we propose a circuit model of power packet transfer. The model lets us derive the cost of power packet transfer between arbitrary node pairs on the dispatching network. Second, with the derived cost metric, we consider the path optimization problem. We develop an algorithm to solve the problem by reducing it to a general shortest path problem on a graph. Lastly, we verify the proposed algorithm through numerical simulations.

We assume a power packet dispatching system consisting of $n_{\mathrm{s}}$ sources, $n_{\mathrm{r}}$ routers, and $n_{\mathrm{l}}$ loads, where $n_{\mathrm{s}}$, $n_{\mathrm{r}}$, and $n_{\mathrm{l}}$ are positive integer. Figure 2 shows the circuit models for a source, a router, and a load.

A source consists of an ideal voltage source, an ideal switch, and series resistance. Turning on and off the switch, a source produces a power packet. The series resistance represents whole resistance between the voltage source and the output port, including internal resistance, switch’s conduction resistance, and line resistance. A router is equipped with multiple input/output ports that accept bidirectional power flow. The circuit consists of ideal switches, capacitors, and series resistance defined similarly to a source. We denote the number of capacitor in a router by $n_{\mathrm{c}}$. The capacitors are used as a temporary power storage in receiving a power packet and as a source in forwarding a power packet.

Now the dispatching system can be represented by a simple graph $G=G(\mathcal{N},\mathcal{E})$, where $\mathcal{N}$ and $\mathcal{E}$ represent sets of nodes (sources, routers, and loads) and edges (power lines between nodes), respectively. For each node $i\in\mathcal{N}$ we set an associated attribute $a_{i}$ that represents the role of the node, i.e. $a_{i}\in\{\mathrm{s},\mathrm{r},\mathrm{l}\}$, where the elements in the set represent “source,” “router,” and “load,” respectively. For each edge $e_{(i,j)}=(i,j)\in\mathcal{E}$, where $i,j\in\mathcal{N}$, we set an associated attribute $w_{(i,j)}(>0)$ that represents a cost for power delivery from $i$ to $j$. How to determine the cost is arbitrary at this stage; we will define one candidate later in this paper, with which we conduct numerical simulations.

Here we also define a state variable of the graph, namely voltage of each node:

where $N:=n_{\mathrm{s}}+n_{\mathrm{r}}+n_{\mathrm{l}}$ and the row vectors $\boldsymbol{v}_{i}\in\mathbb{R}^{m_{\mathrm{c}}}$ ($i=1,2,\dots,N$) represents voltage of node $i\in N$ in the following manner.

• •

  If $a_{i}=\mathrm{r}$, $v_{i}=[v_{(i,1)},\cdots,v_{(i,m_{\mathrm{c}})}]$ (voltage of the capacitors of the node).

• •

  If $a_{i}=\mathrm{s}$, $v_{i}=[V_{i},\cdots,V_{i}]$, where $V_{i}$ is voltage of the ideal source (filled with the voltage of the source).

• •

  If $a_{i}=\mathrm{l}$, $v_{i}=[0,\cdots,0]$ (filled with zero).

Note that the duplication of elements when $a_{i}=\mathrm{s}$ or $\mathrm{l}$ is just for keeping the size of the matrix.

Here we set some assumptions on power delivery within the dispatching system. The first one is on the time synchronization of the system. We fix the time duration of one power packet¹¹1Note that we do not fix the time duration of payload. at $T$, and denote the $k$-th time slot, $(k-1)T\leq t<kT$, simply by $k$ ($k=1,2,\dots$), where $t$ represents continuous time. Second, we fix the amount of energy transferred in a time slot on each edge and denote it by $U$. This is achieved by changing the length of the payload. This restriction is to minimize the energy change in the network. Third, when a router is connected to another node, no overlap of capacitor is allowed. For example, suppose that a router node $i$ receives a power packet from $j$ and forwards it to $k$, the capacitors used for delivery on $(j,i)$ and on $(i,k)$ must be different. This restriction is to ensure a one-to-one delivery of unit energy.

Under the assumptions, the connection between any nodes can be classified into three types: router-router, source-router, and router-load connections. Figure 3 shows the circuit model for the three types of connection.

The detailed operation of each connection is explained in the following subsections.

Based on the circuit models derived above, we consider flow control on the power packet dispatching system. A general router network contains multiple distributed sources available. Thus, considering a power supply to a specific load, there are multiple candidates of the power source to be used. In this section, we develop an algorithm to find the optimal choice of the source and the path from the source to the load on the router network.

Apparently there are arbitrary definition for being the optimal. In this paper, we define it as “the energy loss becomes the lowest among the possible paths.” This is a natural way of defining the cost because energy loss is one of the most critical criteria in a power distribution network.

We define the cost by the proportion of the energy loss to the total energy received, namely $\Delta E_{\mathrm{loss}}/\Delta E_{\mathrm{receive}}$. With Eqs. (5) and (6), we can calculate the cost for router-router connection as follows:

where $v_{0}(k)=v_{1}(k)/v_{2}(k)$. The definitions of $v_{1}$, $v_{2}$, $T$, $C$, and $R$ are same as introduced in Eqs. (5) and (6). Similarly, for source-router connection, we have

and for router-load connection, we have

Here we define the path. Given a pair of a source node $n_{\mathrm{s}}\in\mathcal{N}$ and a load node $n_{\mathrm{d}}\in\mathcal{N}$ of a graph $G=G(\mathcal{N},\mathcal{E})$, a path from $n_{\mathrm{s}}$ to $n_{\mathrm{d}}$, $p(n_{\mathrm{s}},n_{\mathrm{d}})$, is a sequence of edges that connects the source and load without loop. That is, $p(n_{\mathrm{s}},n_{\mathrm{d}})=((n_{1},n_{2}),(n_{2},n_{3}),\dots,(n_{k-1},n_{k}))$ s.t. $n_{i}\in\mathcal{N}$ ($i=1,2,\dots,k$), $(n_{j},n_{j+1})\in\mathcal{E}$ ($j=1,2,\dots,k-1$), $n_{1}=n_{\mathrm{s}}$, $n_{k}=n_{\mathrm{d}}$, and $n_{i}\neq n_{j}$ ($i\neq j$).

Based on the definitions above, the problem to be solved is expressed as follows. Given a source node $n_{\mathrm{s}}$ and a load node $n_{\mathrm{d}}$, find $p(n_{\mathrm{s}},n_{\mathrm{d}})$ along which the sum of energy loss $\sum_{i}w_{(n_{i},n_{i+1})}$ becomes the lowest.

The algorithm including path finding and update of voltage variables is shown in Algorithm 1. As the first step, it updates the cost between all possible connections of nodes (Lines 2–12). At the beginning of this step, we choose one capacitor of those in each router to avoid duplicated use (Line 2). Then, the cost on all the edges is calculated. Next, the second step is the shortest path finding (Line 13). This can be achieved by any algorithm for all pair shortest path problem. We apply Floyd–Warshall algorithm [11] in the following numerical simulation. The third step is calculation of payload length $T_{0}$ (Lines 14–15). This step ensures the unit energy transfer along the path. The last step is to update voltage of capacitors used in the path (Lines 16–22).

Here is given a comment on the step of capacitor selection (Line 2). The selection of capacitors affects the result of the path finding. In other words, the algorithm gives the optimal path under a particular selection of the capacitor indexes. In this study we avoid its arbitrariness by setting the initial voltage of all capacitors within a router at a uniform value. If they are at different voltage levels, one way is to select the pair with lowest ratio of voltage difference.

In this paper, we first proposed the circuit models for connection of network nodes in a power packet dispatching system. Using the model, we derived the cost of power packet delivery as a function of voltage distribution. Then we considered the optimal routing problem based on the models and developed the algorithm to solve it. Through the numerical simulations we confirmed that the proposed algorithm can find the optimal path for the power delivery.

The optimality is not measured in the sense of geometric distance but of energy loss along the path. Thus, the optimal path varies when the voltage distribution of the sources and the router nodes. As is mentioned in Introduction, the inclusion of EVs and renewable sources causes dynamic change of the voltage distribution. Even under this condition, the proposed method creates efficient allocation of distributed power sources to the demand of loads.

This work was partly supported by JSPS KAKENHI Grant Numbers 20K14732 and 20H02151, Cross-ministerial Strategic Innovation Promotion Program (SIP) 18088028, and by Program on Open Innovation Platform with Enterprises, Research Institute and Academia (OPERA) JPMJOP1841 from Japan Science and Technology Agency.