4.2.1 Physical Dynamics

For simplicity, we consider only the dynamics of generators and ignore the details of the loads and transmission lines. We consider the power grid as a graph G in which each node represents a geographical site and each edge denotes a transmission line. For simplicity, we assume that each site has one synchronous generator and all generators are the same. If sites i and j are adjacent, we denote this by i ∼ j. It is challenging to incorporate the difference of generators into the model; however, it is complicated and beyond the scope of this section. Similarly to Ref. [56], we assume that each generator supplies a time-varying current, as well as time-varying power, using a constant voltage. The state of a single generator is represented by rotor rotation angle δ, which is described by the following swing equation [4]:

$\begin{array}{l}\hfill M\stackrel{..}{\delta }+D\stackrel{.}{\delta }=P_{\text{m}}-P_{\text{e}},\end{array}$

where P_m is the mechanical power, P_e is the electrical power, M is the rotor inertia constant, and D is the mechanical damping constant.

Fix an arbitrary node i shown in Fig. 4.1, where Z = R + jX is the impedance of the transmission line, E_i is the voltage of the generator, and Y is the shunt admittance. Then for any node k ∼ i, the current flowing through the transmission line between these two nodes is given by

$\begin{array}{l}\hfill I_{i,k}=\frac{E_i-E_k}{Z}.\end{array}$

Then the current flowing from the voltage source to the node, denoted by I_i, is given by

$\begin{array}{ll}\hfill I_i& =\sum _{k\sim i}{I}_{i,k}+Y{E}_i\hfill \\ \hfill & =\frac{1}{Z}\sum _{k\sim i}(E_i-E_k)+Y{E}_i,\hfill \end{array}$

due to Kirchoff’s current law. Since we assume that the voltage of the generator is constant in magnitude, we have $E_i=V{\text{e}}^{j{\delta }_i}$. Meanwhile, the electric power is given by

$\begin{array}{ll}\hfill P_{\text{e}}^i& =Re[E_i{I}_i^{*}]\hfill \\ \hfill & =\frac{V^2}{|Z{|}^2}\left[R\left(N_i-\sum _{k\sim i}cos({\delta }_i-{\delta }_k)\right)\right.\hfill \\ \hfill & \left.-X\sum _{k\sim i}sin({\delta }_i-{\delta }_k)\right]+V^{2}Re[Y],\hfill \end{array}$

which is obtained by substituting Eq. (4.3) into the expression of P_eⁱ.

Substituting Eq. (4.4) into the swing Eq. (4.1), we obtain the differential equation describing the dynamics of a power grid, which is given by

$\begin{array}{ll}\hfill M{\stackrel{..}{\delta }}_i+D{\stackrel{.}{\delta }}_i& =P_{\text{m}}^i-\frac{V^2}{|Z{|}^2}\left[R\left(N_i-\sum _{k\sim i}cos({\delta }_i-{\delta }_k)\right)\right.\hfill \\ \hfill & \left.-X\sum _{k\sim i}sin({\delta }_i-{\delta }_k)\right]+V^{2}Re[Y],\hfill \end{array}$

where N_i is the number of neighbors of node i. We notice that the dynamics of different nodes are coupled to the difference of rotation angles.

4.2.2 Protection

In power networks, protection devices are used to handle faults and also to prevent damage. For example, if the protection system detects a fault in a transmission line, the corresponding protective relays at both ends will be triggered to isolate the transmission line. The protection actions need to be very fast in order to prevent the propagation of the damage incurred by the fault. There are numerous types of protections such as overcurrent relays protecting the devices from large currents, and distance protection distinguishing the faults occurring in different parts of the power network. Note that both overcurrent relay and distance protection do not need communications, since local measurements (e.g., the local current) are used to make the decision.

In a sharp contest, communications are needed for differential protection, whose mechanism is illustrated in Fig. 4.2 and explained as follows. Suppose that two relays are installed at the two ends of a power transmission line. Both relays have communication transceivers; they can communicate with each other. They measure the currents flowing through the two ends, denoted by I₁ and I₂, simultaneously. We define the directions of the currents as flowing into the power line. When there is no fault in the transmission line, we have

$\begin{array}{l}\hfill I_1+I_2=0,\end{array}$

according to Kirchhoff’s current law.

When a fault occurs (e.g., the transmission line breaks and falls to the ground), there is usually a significant current flowing into the ground since the ground usually has a low impedance. It is possible that the ground has a high impedance. In this situation, the leaking current is very weak and is very difficult to detect. For simplicity, we assume that the ground has a low impedance; thus Eq. (4.6) is broken due to the leaking current. Therefore when the two relays find that I₁ + I₂ is significantly different from zero, they trigger the breakers to take action such that the fault is isolated. If the two relays are not located at the same place (e.g., at the two ends of a transmission line), telecommunication is needed across the two relays. In a more generalized scenario [57, 58], the measurements are propagated through a communication network (no longer point-to-point communications) in order to provide backup protections. The corresponding communication network for differential protection has been optimized to maximize the overall reliability in Ref. [58].

Besides differential protection, communications may be needed in other protection schemes. It is well known that phasor measurement units (PMUs) are very useful for protection. For example, the reports from PMUs can be used to detect and locate transmission line faults [59]. They can also be used in adaptive out-of-step control and distance relaying of multiterminal transmission lines [1]. Since the action of protection should be very fast (thus very short delays), communication could be a bottleneck of PMU-based protection schemes.

4.2.3 Smart Metering

Communications are needed in advanced metering infrastructure (AMI) [60], which is a critical mechanism in smart grids [61, 62]. The AMI is needed by the demand response mechanism in future power grids. Demand response is defined as “changes in electric usage by end-use customers from their normal consumption patterns in response to changes in the price of electricity over time, or to incentive payments designed to induce lower electricity use at times of high wholesale market prices or when system reliability is jeopardized” [63]. It has been intensively studied, for example the demand response for areas with large populations [64] and the charging of electric vehicles [65].

In the AMI system, each power consumer (e.g., factory, household, workshop, building, etc.) is equipped with smart meters capable of two-way communications. A smart meter can monitor the activities of the power consumer. It reports the most recent power consumption to a control center; meanwhile it receives a power price from the power market. These reports can be used to achieve the balance of power demand and supply, and avoid power consumption peaks (e.g., the dinner time or summer noons), by setting the correct power price for demand response and adjusting the power generations [66, 67]. Although the communication requirements, such as throughput and delay, are quite loose for AMI (e.g., a report with at most tens of bytes per few minutes) compared with the capability of modern data communications, the aggregated data traffic can be large in densely populated areas (consider an area with thousands of smart meters). Therefore an event-triggered communication mechanism has been proposed in Ref. [68] such that the throughput requirement is substantially relaxed. Among many modern communication technologies, wireless communication has great potential for AMI since it is inexpensive and can be deployed quickly.

4.2.4 Communication Systems in Industrial Control and Smart Grids

Compared with modern data communication systems such as 4G cellular systems, WiFi and Internet, the communication networks in industrial control systems have special requirements, e.g., predictable throughput, very high reliability, robustness in hostile environments, scalability, and the maintenance capability [27]. Here we explain two typical types of communication systems for industrial controls. Readers can consult Ref. [28] for a comprehensive introduction.

• Foundation Fieldbus [29]: Fieldbus is a family of industrial control communication systems. It has been widely used in numerous industrial control applications. It corresponds to IEC61158. Foundation Fieldbus is a Type 1 Fieldbus. In sharp contrast to modern communication systems, the physical layer of Fieldbus is based on current modulation, in which each device senses the voltage drops at the terminating resistors, thus still being analog. The access control is managed by the Datalink layer using tokens. Fieldbus is real-time and reliable, which is important for control systems. However, its data rate is around 30 kbps, thus being too low for applications with fast dynamics.

• Supervisory Control and Data Acquisition (SCADA) [30]: SCADA has been used to monitor or control the plant dynamics of industrial infrastructures such as power grids. It consists of a human-machine interface, supervisory computer system, remote terminal units (RTUs) connected to sensors, and communication infrastructure. Either wired or wireless communications are used. For large area networks, SONET/SDH optical communication is an option. A major disadvantage of SCADA is the slow speed, which may take minutes to collect data from the power grid [31]. Hence it cannot be used in fast data communications.

4.3.1 Physical Dynamics

The physical dynamics mainly describes the locations and motions of the robots. They can be modeled in either a deterministic or probabilistic manner.

Probabilistic model

The motion of a robot can also be modeled in a probabilistic manner. Denote by x(t) and u(t) the state and control of a robot, similarly to the deterministic model. Then the dynamics can be modeled by a Markov chain, whose transition probability is given by

$\begin{array}{l}\hfill p(x(t)|u(t),x(t-1)).\end{array}$

  (4.11)

In probabilistic models, the modeling of the robot motion is of key importance. Two models are possible for robot motion:

• Velocity motion model: In this model, a robot can be controlled by manipulating its rotational and translational velocities (denoted by v(t) and w(t)). Hence the control action is given by

$\begin{array}{l}\hfill \mathbf{u}(t)=(v(t),w(t)).\end{array}$

  (4.12)

To evaluate the transition probability p(x(t)|u(t), x(t − 1)), one first assumes that the control u(t) is exact and calculates the expressions for the location after a small time of motion. Then we relax the assumption that the control is exact and assume that the actual control action is contaminated by random perturbations, which is given by

$\begin{array}{l}\hfill (\widehat{v},ŵ)=(v,w)+({ϵ}_v,{ϵ}_w),\end{array}$

  (4.13)

where (v, w) is the expected control and (ϵ_v, ϵ_w) is the random perturbation, which is assumed to be centered at zero and has a bounded variance. Typical distributions of (ϵ_v, ϵ_w) include Gaussian distribution and triangular distribution. From these distributions of random perturbation, p(x(t)|u(t), x(t − 1)) can be evaluated. The detailed algorithm can be found in Ref. [69].

• Odometry motion model: In this model, odometry measurements are used. Usually odometry is more accurate than velocity. At time t, the odometry of a robot estimates the relative location change; i.e., it reports the advance from x(t − 1) to x(t). The location change has three components, namely δ_rot1, δ_trans, and δ_rot2, as illustrated in Fig. 4.3. In the odometry motion model, it is assumed that these three components experience independent random perturbations. Given the distributions of these random perturbations, the conditional probability p(x(t)|u(t), x(t − 1)) can be obtained, whose details can be found in Ref. [69].

4.3.2 Communications and Control

The procedure of communication and control of a robot is illustrated in Fig. 4.4. In each cycle, a robot exchanges information with neighbors through its communication module. Then it updates its own processor state and calculates the control action, which will be carried out subsequently. Hence the communication and control law of each robot (say robot i ∈ I, where I is the index set of robots) consists of the following elements:

• Communication alphabet A, which consists of the symbols for communications.

• Processor state sets Wⁱ, which describe the local state of computing.

• Allowable initial values $W_0^i$, which denote the initial state of computing.

• Message generation law: $ms{g}^i:X^i\times W^i\times I\to A$, which means the message generated according to the physical state, computing state, and robot index.

• State transition law: $st{l}^i:X^i\times W^i\times A^{|I|}\to W^i$, which makes the robot transit to a new state of computing according to the current physical state, computing state and all messages.

• Control law: $ct{l}^i:X^i\times X^i\times W^i\times A^n\to U^i$, which calculates the control action at time t ∈ [l, l + 1), where l is the time instant of the previous update of the computing state; hence the control action u(t) is a function of x(l), x(t), current local computing state, and all received messages.

Several simplifications can be made to the above rather general model:

• The communication and control law can be static, which does not change with time and thus means that Wⁱ is a singleton.

• The communication and control law can be data sampled, if the control action u(t) is independent of x(t) and is dependent on only x(l) (the physical state of the last sample time).

4.3.3 Coordination of Robots

The robots in a robotic network need coordination in order to accomplish their tasks. In the subsequent discussion, we introduce two major types of coordination tasks, namely rendezvous and connectivity maintenance.

Connectivity maintenance

The other task of coordination is the maintenance of network connectivity. First we define the r-disk graph of geographical nodes, where two nodes are connected when their distance is no more than r. When the r-graph is connected (i.e., for any pair of nodes in the graph we can always find a path between them), then the connectivity maintenance problem is to enable the robots to form a connected r-disk graph.

We first consider the maintenance of the connectivity of two robots. Suppose that two robots i and j have positions p^[i] and p^[j] with ∥p^[i] − p^[j]∥ < r. Then we define the connectivity constraint set as

$\begin{array}{l}\hfill \chi (p^{[i]},p^{[j]})=B\left(\frac{p^{[i]}+p^{[j]}}{2},\frac{r}{2}\right),\end{array}$

  (4.15)

where B(c, a) denotes a ball with center c and radius a.

Assume that robots i and j are within distance r at time t. Then it can be shown that, if the control actions uⁱ(t) and u^j(t) satisfy

$\begin{array}{l}\hfill u^i(t)\in \chi \left(p^{[i]}(t),p^{[j]}(t)\right)-p^{[i]}(t)\end{array}$

  (4.16)

and

$\begin{array}{l}\hfill u^j(t)\in \chi \left(p^{[i]}(t),p^{[j]}(t)\right)-p^{[j]}(t),\end{array}$

  (4.17)

as illustrated in Fig. 4.5, then the following two conclusions hold:

• Robots i and j will still be in the connectivity constraint set $\chi \left(p^{[i]},p^{[j]}\right)$.

• Their positions are still within distance r.

Then we can extend the two-robot case to the generic case with arbitrarily many robots. When the locations of the robots are p^[1], p^[2], …, p^[n], the corresponding connectivity constraint set is defined as (as illustrated in Fig. 4.6)

$\begin{array}{l}\hfill {\chi }_i\left(p^{[1]},\dots ,p^{[n]}\right)=\left\{x\in {\chi }_{p^{[i]},p^{[j]}}|j\ne i,\text{s.t.},‖p^{[i]}-p^{[j]}‖\le r\right\},\end{array}$

  (4.18)

which means the joint set of the connectivity constraint set of robot i with all other robots that are within distance r of robot i. Then it is shown in Ref. [5] that, if the robot network keeps the control actions at time l to satisfy the following condition:

$\begin{array}{l}\hfill u^{[i]}(l)\in {\chi }_i\left(p^{[1]}(l),\dots ,p^{[n]}(l)\right)-p^{[i]},\end{array}$

  (4.19)

then the following goals can be achieved:

• At the next time slot l + 1, each robot will still be in its connectivity constraint set of time slot l.

• All the communication links are still maintained.

• The connectivity of the robot network in time slot l + 1 is maintained, if the network is connected in time slot l.

• The number of connected components in time slot l + 1 is not larger than that of time slot l.

The concept of the connectivity constraint set provides a sufficient condition for the control actions of the robots to maintain the connectivity of the network. It can be relaxed and applied in more complicated situations. The details can be found in Chapter 4 of Ref. [5].