The National Academy of Engineering named the electric-power grid the greatest engineering achievement of the 20th century [1]. However, as recent large-scale power-grid failures illustrate, the (electro-mechanical) electric grid is being operated closer and closer to its limits. Specifically, the electric grid of the 20th century is aging and congested. Furthermore, it will not be able to meet future demands without operational changes and significant capital investments over the next decades [2]. Thus, the electric grid of the 21st century represents an open problem for the research community and industry. With the development of new technologies, such as, flexible AC transmission devices (FACTS), phasor measurement units (PMUs), renewable and distributed generation, flexible loads, and energy-storage solutions, the tools are available to enable a paradigm shift for the electric grid.

To overcome the limitations of today’s power grid, two main approaches are considered by engineers and scientists. The first approach investigates improvements to energy-delivery systems subject to boundary conditions given by today’s grid structures. The second approach seeks to develop and design a new paradigm for optimal future energy delivery systems, which takes into account novel emerging technologies. By treating the second approach (i.e., the greenfield approach) as the forecasted optimal “target” system, the first approach can be considered a coordinated effort to bridge today’s aging and congested system with the optimal future target system, as Figure 5.1 illustrates.

The design and development of the greenfield approach can be considered the long-term goal of power systems engineers and scientists, while the bridging approach can be considered a series of short-term projects. It is within this framework that the work is developed in this chapter. Namely, we build upon the ETH Zürich project “Vision for future energy networks,” which focused on a synergistic interconnected energy systems model as their green-field approach. To accomplish the interconnection of energy systems, the ETH project developed modeling tools such as the “energy hub” and multi-energy carriers and analyzed many scenarios under the new multi-energy context (see [3–5]).

This chapter expands upon the ETH project by applying the “energy-hub” approach to analyze the performance of energy delivery systems with energy storage in contingency scenarios (i.e, line outages). This is achieved by developing a novel model-predictive control (MPC) scheme that ensures efficient operation of energy systems and mitigates the effects of severe line-outage disturbances through feedback algorithms. Developed algorithms are illustrated through numerous simulation studies. The interdisciplinary research presented in this chapter lies at the intersection of power systems, optimization, and controls. Specifically, two model-predictive control-based cascade-mitigation approaches are analyzed herein. The first approach represents a multi-energy approach that considers a simplified system of interconnected natural gas pipelines and electric-transmission systems, which is then developed into a second practical, yet rigorously justified, cascade-mitigation scheme for just the electric bulk-power system.

1.2 Cascade mitigation

Currently, abnormal conditions are handled either through protection operation or operator intervention, depending on the severity of the abnormality. In the latter case, where conditions do not immediately threaten the integrity of plant or loads, operators institute corrective procedures that may include altering generation schedules, adjusting transformer tap positions, and switching capacitors/reactors. For more extreme abnormalities, the protection associated with vulnerable components will operate to ensure they do not suffer damage. This myopic response may, however, weaken the network, exacerbating the conditions experienced by other components. They may subsequently trip, initiating an uncontrolled cascade of outages. This pattern was exhibited during the blackout of the U.S. and Canada in August 2003 [6].

As the amount, type, and distribution of controllable resources increases, operators will find it ever more challenging to determine an appropriate response to unanticipated events. At a minimum, operators will require new tools to guide their decision-making. Given the increased complexity of response actions, a closed-loop feedback process will become indispensable. Furthermore, since power systems are suffused with constraints and limits on states and inputs, model predictive control (MPC) schemes can be particularly useful within the context of contingency management. For a general overview of MPC, see [7–9].

The first application of MPC to emergency control of power systems was [10], where voltage stability was achieved through optimal coordination of load shedding, capacitor switching, and tap-changer operation. A tree-based search method was employed to obtain optimal control actions from discrete switching events. To circumvent tree-based search methods, [11,12] employed trajectory sensitivities to develop MPC strategies. However, those methods focused on voltage stability and did not take into account energy storage nor thermal overloads of transmission lines. Distributed forms of MPC have also been proposed, with mitigation of line-outage cascades considered in [13].

The authors in [14,15] proposed a framework for electrothermal coordination in power systems, and developed temperature-based predictive algorithms that are amenable to energy markets and applicable within existing system controls. Other recent literature, cf. [16,17], focused on model-predictive control of electrical energy systems to alleviate line overloads within a standard DC power flow framework. Specifically, the authors in [17] extended the ideas of [16] to include a linearized current-based thermodynamical model of conductors and an auto-regressive model of the weather conditions (i.e., wind speed and ambient temperature) near transmission lines. This allowed [17] to set a hard upper limit on conductor temperature to ensure control objectives, and allowed MPC to operate the system closer to actual physical limits than if using standard (worst-case weather-based) thermal ratings. Furthermore, [17] illustrated that temperature-based control can outperform current control within a predictive framework.

The cascade-mitigation schemes described in this chapter examine a bi-level cascade-mitigation scheme that considers both the economic and static security objectives in operation of the system. The first level operates on a slow timescale (i.e., hourly) and determines a trajectory of optimal economical set points for generation, storage, load control, and wind curtailment. The second level responds to contingencies (i.e., line outages) on a much faster minute-by-minute timescale to ensure that the system is driven back to a secure operating regime and line overloads are alleviated. The second level does not consider fault conditions and automatic protection schemes, which operate on a sub-second timescale, and assumes transient short-term stability. The effect of energy storage on the performance of cascade mitigation is investigated through storage scenarios within a multi-energy “energy-hub” framework. A temperature-based cascade mitigation is described for the electric bulk-power system where the role of energy storage is highlighted. The work herein represents state of the art in model-predictive cascade mitigation.

1.3 Outline

The chapter is organized as follows. First, basic relevant concepts are described, including line-outage models. Then, relevant system models are introduced and the use of MPC is motivated for cascade mitigation. A bi-level cascade-mitigation scheme that considers economic and security objectives is then discussed. Two cascade-mitigation approaches utilizing model-predictive control (MPC) are discussed: shrinking-horizon and receding-horizon. The former employs a multi-energy “energy-hub” formulation within a simplified shrinking-horizon MPC framework to highlight the role of energy storage and represents a large-scale application of energy hubs within a cascade-mitigation framework. The second approach refines and improves upon the shrinking-horizon MPC scheme and focuses on receding-horizon MPC of electric bulk-power systems with energy storage and analysis of the feedback scheme is provided. The receding-horizon MPC scheme is complemented with a case study of the standard RTS-96 test system augmented with energy storage to highlight the practical, yet rigorously justified, cascade-mitigation scheme.

2.1 Line tripping

When lines exceed their limits, it is possible that these lines go out of service or trip. The term “line trip” refers to the event that causes a line to go from being in-service to out-of-service. When a line (i, j) is tripped (i.e., out of service) the following must hold:

• no flow (or losses) across (i, j): f_ij = 0.

• node i and j are decoupled in power flow equations. For example, the DC power flow equation, x_ij f_ij = θ_i – θ_j, that relates the voltage phase angles of nodes i and j to flow across line (i, j) no longer has to hold.

In general, one of the main goals of a system operator is to ensure that line flows stay within predefined flow limits, which represents a form of network reliability. Therefore, if f_ij (t) and u_ij are the (bidirectional) power flow and the power flow rating (i.e., limit) on arc (i, j) at time t, respectively, then, it is desirable to enforce the line flow limit:

$\left|f_{\mathit{ij}}\left(t\right)\right|\le u_{\mathit{ij}}\text{,}$

where (i, j) represents arc between nodes i and j. Thus, if system operations satisfy (1) at all times t, arc (i, j) will not be tripped (under normal operating circumstances).

While it is feasible to take inequality (1) into consideration upon determination of an hourly economic energy management schedule, it is unrealistic to expect such a constraint to be valid after the system undergoes a significant disturbance (e.g., multi-line outage). This is due to the fact that flows depend on the physics of the network and cannot be directly guided (e.g., FACTS devices¹ are not considered here), which means that the line flows may exceed their limit after a contingency has occurred.

There exists a myriad of approaches to model when an overloaded line should be tripped, ranging from deterministic hard constraints, as in [18], to soft-constrained probabilistic setups described in [19]. This section presents two models of line tripping: one is a simplified deterministic model and the other is probabilistic based on temperature overloads.

2.1.2 Probabilistic thermal outage model

The second model of line tripping is probabilistic and is utilized in Section 6 to more accurately represent the actual line-outage process for electrical transmission systems. For electrical systems, components, such as over-current relays, are often in place to protect the system against abnormal conditions by tripping lines (i.e., taking them out of service). However, these components operate for extreme over-current fault scenarios (e.g., 10,000A) and trigger automatically on a timescale of seconds and mili-seconds and are, therefore, not considered in this work. Instead, the line-tripping behavior of interest in this chapter occurs on a timescale of minutes, which shifts the focus from fault conditions to thermal conditions of transmission lines and sagging. Electrical transmission lines have prescribed power flow limits to prevent dangerous sagging and permanent damage. These limits are related to the thermal capacity of the conductor and the current flowing across the line. The method for calculating the current-temperature relationship of bare overhead transmission lines is described in detail by IEEE Standard 738 [20]. As the temperature of the conductor increases, the thermal expansion causes the line to sag. There is an inverse relationship between the overload on a line and the time it takes before the line sags excessively and must be taken out of service; however, there are no clear rules when a sagging line will be taken out of service (by operator or nature). For example, a human system operator may decide that the red flashing warning sign is sufficiently annoying and trip the line manually or an elephant may just walk into a sufficiently sagging line and cause an outage.² Excessive line temperature (and resulting sag or possible annealing) may eventuate in line tripping. In short: the higher the temperature, the more likely line tripping becomes. This inverse relationship between conductor temperature and mean time-to-trip (i.e., mean time-to-failure) is captured in the actual system by use of the exponential time-to-failure density parametrized by the temperature overload:

$P\left(\mathrm{line}\left(i,j\right)\mathrm{trips}\mathrm{at}\mathrm{time}t\right)=\mathrm{\lambda }\left(\mathrm{\Delta }{T}_{\mathit{ij}}\right)e^{-\mathrm{\lambda }\left(\mathrm{\Delta }{T}_{\mathit{ij}}\right)t}\text{,}$

  (3)

where rate parameter λ (ΔT_ij) > 0 is a nonlinear function of the temperature overload, defined such that the mean time-to-failure goes to zeros as temperature increases $\left(\mathrm{i}.\mathrm{e}.,\frac{1}{\mathrm{\lambda }\left(\mathrm{\Delta }{T}_{\mathit{ij}}\right)}\to 0\mathrm{as}\mathrm{\Delta }{T}_{\mathit{ij}}\to \infty \right)$. That is, for each temperature overload, there is different mean time-to-failure. Thus, the probability of line (i, j) tripping during time-interval T_s, given ΔT_ij, is defined by the cumulative density function (also called the unreliability function):

$P\left(\mathrm{line}\left(i,j\right)\mathrm{trips}\mathrm{during}\mathrm{time}k|\mathrm{\Delta }{T}_{\mathit{ij}}\left[k\right]\right)=1-e^{-\mathrm{\lambda }\left(\Delta T_{\mathit{ij}}\left[k\right]\right)T_s}\text{,}$

  (4)

where rate parameter λ(ΔT_ij[k]) > 0 can be based on the short-time (15-minute) emergency (STE) rating of a transmission line. It has been found experimentally that λ(ΔT_ij[k]) = (ΔT_ij[k]/15)⁶ gives reasonable line-tripping behavior, as shown in Figure 5.2. Notice how the mean time-to-trip decreases with increasing conductor temperature overload.

Furthermore, considering over-current protection on transmission lines (for large overloads), an additional condition can be added to the probabilistic line-tripping model:

$P\left(\left(i,j\right)\mathrm{trips}\mathrm{at}k\left|\frac{\left|f_{\mathit{ij}}^{\mathit{ac}}\left[k\right]\right|}{u_{\mathit{ij}}}\right.\ge \overline{\mathrm{\Omega }}\right)=1\text{,}$

  (5)

where $\overline{\mathrm{\Omega }}$ is an upper bound on allowable relative instantaneous overload. For example, if $\overline{\mathrm{\Omega }}=3\text{,}$ then a line flow of 300% of nominal thermal limit u_ij automatically trips line (i, j).

With this formulation for line tripping, if a line experiences an overload, the expected time to trip decreases as a function of the inverse of temperature-based rate parameter λ (ΔT_ij) and sampling time T_s.

Remark 1

(Implementation of line tripping) If line switching is an admissible action of the controller, a mixed-integer disjunctive line outage can be employed [22,23]. However, to be clear, line tripping is not considered as a decision available to the controller. That is, it is assumed that network topology is observable at all times and that line outages are known to the controller immediately after they occur, which means that line outages only represent a exogenous input for the system (i.e., a disturbance).

2.2 Cascade failures

Given a line-outage model, we can discuss cascade failures. From left to right, Figure 5.3 illustrates three stages of a cascading failure: initial disturbance (left side of figure), cycles of line outages and flow redistributions (center), and a terminal blackout (right). Cascade failures are initiated when a disturbance occurs that forces a redistribution of flows. When a line goes out of service, it can reduce the network’s overall capacity,³ which begets power overloads on the remaining lines as the power flows are redistributed according to Kirchhoff’s laws. If overloads are not alleviated in a timely manner, more lines may go out of service. The cycle of line outages and redistribution of flows, if left uncontrolled, is referred to as a cascade failure. A cascade failure generally terminates in a major blackout, with large areas of a network unable to supply demand.

Furthermore, for electric-power grids, the cascade is multi-scale in a somewhat unusual way. After the initial fault, the first stages of grid failure can proceed relatively slowly, on a scale of hours or minutes. As the cascade develops, the pace of failures can accelerate, with later waves happening on a scale of seconds. Figure 5.4 highlights the accelerating pace of major outage events during the 2003 blackout in the Northeastern U.S. and Canada. Note that the initial two outage events represent transmission lines overheating and sagging into trees. The multi-scale timing of outages has important consequences for any cascade-mitigation strategies. Since longer time scales during the initial cascade allow for significant computations to be performed, it provides an excellent opportunity for feedback control in mitigating the effect of cascading failures. This provides the motivation for the model-predictive control approach developed in later sections.

3.1 Multi-energy concept: the energy hub

Multi-carrier energy networks may be formulated in different ways. This section will focus the discussion on the “hybrid energy-hub” model initially developed by [25]. Under this formulation, the system operator of an energy-hub network can directly manipulate and control load, generator, and energy-hub (e.g., storage and converter) set points. The energy-hub model is linear, which makes it amenable for computation and optimization. For mathematical details on the energy-hub models, see [26].

Most common energy hubs consist of five simple interconnected elements: inputs, input-side energy storage, energy converters, output-side energy storage, and outputs. To properly describe the flow of power through the energy hub, we need to describe how power flows between these elements. Energy-carrier networks supply power to the hub at the input side, where storage may be available. The energy that was not utilized for storage is dispatched into converters that transform the energy accordingly. On the output side of the hub, converted energy may be utilized for storage or injected into the carrier network connected to that side.

One simple example of an energy hub is shown in Figure 5.5 where a campus energy plant is modeled as an energy hub. The chilled water plant (C), natural-gas steam boiler plant (B), gas turbine cogeneration plant (Cogen), electrical substation (T), and thermal-energy storage (TES) are considered part of the the same energy hub. The supply of electricity and natural gas represent the hub inputs while the outputs are cooling, heating, and electric loads (e.g., buildings and/or processes). The electrical energy is converted to cooling via the chilled water plant and to low-voltage electrical energy by the transformer at relevant efficiencies and injected into the output side. The natural gas consumed at the hub input can be dispatched to the steam boilers (for heating only) or to the Cogen plant where natural gas is converted into both electrical energy and heating. To illustrate the inclusion of storage devices within the energy-hub formulation, consider the chilled water-storage tanks in TES that can store cooling at some charge/discharge efficiency and standby losses. The chilled water plant can store cooling from the electric chillers while the boiler plant can store cooling via absorption processes. The TES can then charge or discharge to achieve objectives related to improved economics (e.g., reduce peak electric consumption) and/or increased reliability (e.g., provide backup when chilled water plant is out of service).

3.2 Energy hubs and cascade mitigation

Hub storage plays a significant role in cascade mitigation since it acts as a “buffer” against disturbances. That is, a system operator can employ stored energy to satisfy temporary energy shortages or overflows, while allowing time for conventional generators to effectively reconfigure their schedules.

Thus, the effectiveness of system operations in minimizing costs and rejecting disturbances depends on the available energy-hub storage infrastructure. Indeed, siting, sizing, and operational capability (e.g., power rating and standing losses) are salient parameters. Siting is important for reducing congestion during peak hours; however, the process for determining optimal location of energy hubs is non-trivial and is not considered here. Instead, this simulation will fix the location of hubs (and storage) within the system and study the effects of varying hub storage capacity and charge/discharge power limits on the cascade-mitigation process.

Under the energy-hub paradigm, we are able to combine multiple types of energy systems and study their combined performance. Therefore, we need to consider multiple types of energy storage, namely, natural gas storage, electrical storage, and thermal storage. Note that, in this work, we consider energy storage to have no standing losses, constant charging and discharging efficiencies⁴ (see Table 5.1), and we neglect the economics of construction, operation, and maintenance of storage facilities. That is, energy storage represents a cost-free service available to the system operator.

3.3 Energy-Storage model

The systems considered in this chapter will have energy storage located at various nodes throughout the network. During normal operation of the power system, energy storage plays a significant role in minimizing generation costs from conventional generators, as it allows the system operator to preposition energy in storage during off-peak hours to satisfy demand in the presence of intermittent generation (e.g., wind power) or take advantage of arbitrage. However, during contingency operation, economics become a secondary concern, but energy storage can still play a significant role in cascade mitigation as it acts as a “buffer” against disturbances. That is, a system operator can employ stored energy to satisfy temporary energy shortages or overflows, while allowing time for conventional generators to effectively reconfigure their schedules against ramp-rate limits.

Energy storage is available in many forms (e.g., hydrogen fuel cells, grid-scale battery systems, hydro, gas-filled caverns, thermal energy-storage, molten salts, etc.) and energy-storage devices can be located at various nodes throughout a network. Let $n\in {\mathrm{\Omega }}_i^E\subset \mathcal{Q}$ be an energy-storage device at node i, where $\mathcal{Q}$ is the set of storage devices in the system. Assume steady-state storage-power values, a constant slope for ${\dot{E}}_n\left(t\right)=d{E}_n\left(t\right)/\mathit{dt}\text{,}$ and treat the storage interface as a conversion process with charging and discharging efficiencies η_c,_n and η_d,_n, then the relationship between storage state of charge (SOC) and power injected/consumed by device n is:

${\dot{E}}_n\left(t\right)=\frac{d{E}_n\left(t\right)}{\mathit{dt}}\approx e_n\left(t,f_{\mathit{Qn}}\right)\times f_{\mathit{Qn}}\left(t\right)\text{.}$

where the SOC switching mechanism e_n is defined as:

$e_n\left(t,f_{\mathit{Qn}}\right)=\left\{\begin{array}{cc}\hfill {\eta }_{\mathrm{c},n},\hfill & \hfill \mathrm{if}{f}_{\mathit{Qn}}\left(t\right)\ge 0\left(\mathrm{charge}/\mathrm{standby}\right)\hfill \\ \hfill \frac{1}{{\eta }_{\mathrm{d},n}},\hfill & \hfill \mathrm{if}{f}_{\mathit{Qn}}\left(t\right)<0\left(\mathrm{discharge}\right)\hfill \end{array}\right.\text{.}$

Since energy-storage devices have two distinct states of operation, charging and discharging, that achieve different efficiencies, energy-storage devices introduce switches in the SOC formulation. The following reformulation of the SOC makes this non-convex nonlinearity more apparent:

${\dot{E}}_n\left(t\right)={\eta }_{\mathrm{c},n}{f}_{Q\mathrm{c},n}\left(t\right)+\frac{1}{{\eta }_{\mathrm{d},n}}{f}_{Q\mathrm{d},n}\left(t\right)\text{,}$

$f_{\mathit{Qn}}\left(t\right)=f_{Q\mathrm{c},n}\left(t\right)+f_{Q\mathrm{d},n}\left(t\right)\text{,}$

$0=f_{Q\mathrm{c},n}\left(t\right)f_{Q\mathrm{d},n}\left(t\right)\text{,}$

where the rate-limited charging (c) and discharging variables (d), f_Qc,_n ∈ $\left[0,{\overline{f}}_{Q\mathrm{c},\mathrm{n}}\right]$ and f_Qd,_n ∈ $\left[-{\overline{f}}_{Q\mathrm{d},\mathrm{n}},0\right]\text{,}$ model the switching mechanism explicitly as a complimentarity condition in (8c). The nonlinear complimentarity condition ensures that q can either charge or discharge, but not both simultaneously. To circumvent the nonlinearity, a mixed-integer linear (MIL) formulation can be employed:

$\mathbf{MIL}\mathbf{:}{f}_{Q\mathrm{c},n}\left(t\right)f_{Q\mathrm{d},n}\left(t\right)=0\iff \left\{\begin{array}{c}\hfill 0⩽f_{Q\mathrm{c},n}⩽{\overline{f}}_{Q\mathrm{c},n}\left(1-z_n\right)\hfill \\ \hfill -{\overline{f}}_{Q\mathrm{d},n}{z}_n⩽f_{Q\mathrm{d},n}⩽0\hfill \\ \hfill z_n\in \left\{0,1\right\}\hfill \end{array}\right.\text{.}$

where z_n is a binary integer. For example, if z_n[k] = 1, then f_Qc,_n[k] ≡ 0 and device n is consequently operating in discharging mode at time step k. While the above linear formulation is equivalent to the nonlinear complimentarity condition, the use of integers is generally not desired, as it greatly increases computational complexity.

To avoid utilizing integers in the linear model, one can ignore the complimentary condition in (8c). This implies that simultaneous charging and discharging is now feasible and is equivalent to a convex relaxation of the original SOC model. Replace (8) with the strictly linear and continuous formulation:

${\dot{E}}_n\left(t\right)={\eta }_{\mathrm{c},n}{f}_{Q\mathrm{c},n}\left(t\right)+\frac{1}{{\eta }_{\mathrm{d},n}}{f}_{Q\mathrm{d},n}\left(t\right)\text{,}$

$f_{\mathit{Qn}}\left(t\right)=f_{Q\mathrm{c},n}\left(t\right)+f_{Q\mathrm{d},n}\left(t\right)\text{.}$

Employing a forward Euler discretization to (10) with sample time of T_s seconds admits linear continuous first-order discrete SOC dynamics that represents the full linear energy-storage model:

$E_n\left[k+1\right]=E_n\left[k\right]+T_s{\eta }_{\mathrm{c},n}{f}_{Q\mathrm{c},n}\left[k\right]+\frac{T_s}{{\eta }_{\mathrm{d},n}}{f}_{Q\mathrm{d},n}\left[k\right]\text{,}$

$f_{\mathit{Qn}}\left[k\right]=f_{Q\mathrm{c},n}\left[k\right]+f_{Q\mathrm{d},n}\left[k\right]\text{,}$

$f_{Q\mathrm{c},n}\left[k\right]\in \left[0,{\overline{f}}_{Q\mathrm{c},n}\right],f_{Q\mathrm{d},n}\left[k\right]\in \left[-{\overline{f}}_{Q\mathrm{d},n},0\right]\text{.}$

3.4 Power flows

Energy hubs are interconnected via multiple energy-supply networks. The previous section defined how power flows through an energy hub, however, to describe the flow between energy hubs, we need to consider power networks. A power network is a collection of a set of nodes i ∈ $\mathcal{N}$ (e.g., buses) and a set of arcs (i, j) ∈ $\mathcal{A}$ (e.g., transmission lines and gas pipelines) that define a simple graph, as shown in Figure 5.6. The nodes either consume power from the network (i.e., loads), inject power into the network (i.e., generators), or act as throughput nodes that neither inject nor consume power. The sum of power flows into the network (e.g., hub outputs, generators) must equal the sum of flows out of the network (e.g., hub inputs, loads, and losses). In fact, any network must satisfy Kirchhoff’s first law (also called the “power balance”). That is, the net flow into a node must equal the net flow out of the node. Generally, a node may have generators (f_Gn) and/or loads (f_Dn) available and, in a system with energy-storage devices, the charging (discharging) corresponds to additional demands (injections) (f_Qn). Therefore, the power-balance equation is formulated as:

${\displaystyle \sum _{n\in {\mathrm{\Omega }}_i^D}{f}_{\mathit{Dn}}}\left[k\right]-{\displaystyle \sum _{n\in {\mathrm{\Omega }}_i^{\mathrm{G}}}{f}_{\mathit{Gn}}}\left[k\right]+{\displaystyle \sum _{j\in {\mathrm{\Omega }}_i^{\mathrm{N}}}{f}_{\mathit{ij}}^{\mathrm{total}}\left[k\right]+{\displaystyle \sum _{n\in {\mathrm{\Omega }}_i^{\mathrm{E}}}{f}_{\mathit{Qn}}}\left[k\right]=0}\forall i\in \mathcal{N}\text{,}$

where $f_{\mathit{ij}}^{\mathrm{total}}=f_{\mathit{ij}}+\frac{1}{2}{f}_{\mathit{ij}}^{\mathrm{loss}}$ is the total flow across line (i,j) and:

• Ω_i^G – Set of generators at node i (hub outputs and conventional)

• Ω_i^N – Set of nodes adjacent to node i

• Ω_i^D – Set of demands (loads and hub inputs) at node i

• Ω_i^E – Set of energy-storage devices at node i

The power-balance equation in (12) determines the net power generated or consumed at each node. As well as contributing to the power balance, line losses f_ij^loss also drive the temperature dynamics associated with line overloads. This dual role will be carefully studied in Section 6.

Besides interconnecting energy hubs, the main difference between a simple graph and a power network lies with the physics of the particular power flows. That is, there exists a physical relationship between the flow across an arc and the connected nodes. For a simple example, consider an electrical power network and the linear DC flow model:

$x_{\mathit{ij}}{f}_{\mathit{ij}}={\theta }_{\mathit{ij}}\text{,}$

where x_ij represents the reactance of arc (i,j), f_ij is the power flow across said arc, and θ_ij := θ_i – θ_j represents the voltage phase angle difference between nodes i and j. In (13), we see that the electrical power flowing across any arc depends on the difference in the phase angle between connected nodes and the reactance of the arc. This physical relationship between nodes and arcs manifests itself differently depending on the nature of the power network model and may be linear (e.g., DC power flow) or nonlinear (e.g., AC power flow). A commonly cited nonlinear power flow constraint is exhibited in natural gas networks, where the flow of natural gas through pipelines depends in a nonlinear manner on the pressure p_i at the nodes [28]:

$f_{\mathit{ij}}^{\mathrm{gas}}=\left\{\begin{array}{cc}\hfill k_{\mathit{ij}}\sqrt{p_i-p_j},\hfill & \hfill \mathrm{if}{p}_i⩾p_j\hfill \\ \hfill -k_{\mathit{ij}}\sqrt{p_j-p_i},\hfill & \hfill \mathrm{if}{p}_i<p_j\text{,}\hfill \end{array}\right.$

where k_ij is a constant pertaining to the specific gas and pipeline properties. In addition, power is necessary to maintain pressure at the nodes, which introduces the nonlinear compressor constraints:

$f_{\mathit{com},\mathit{ij}}^{\mathrm{gas}}=c_{\mathit{ij}}{f}_{\mathit{ij}}^{\mathrm{gas}}\left(p_i-p_j\right)\text{,}$

with c_ij a constant describing the properties of the compressor along arc (i,j).

Regardless of the type of power network, a set of equations describing the appropriate power flow and power balance can be formulated. In this work, the cascade-mitigation schemes employ linear models of power flows (i.e., linearized gas flow and DC power flow). The simulation results will illustrate the value of such simplified models for cascade mitigation.

3.5 Transmission line losses

The probabilistic temperature-based line-outage model described in Section 2.1.2 requires consideration of conductor temperature, which depends on ohmic I²R losses, local ambient conditions (wind speed, insolation), and conductive and radiative conductor heat losses. Thus, to alleviate temperature overloads caused by ohmic heating in transmission lines, it is necessary for a controller to model and manipulate I²R line losses. However, the DC power flow model in (13) ignores active line losses. To establish a relationship for losses on branch (i, j), the AC expression for active power flow across a transmission line can be manipulated to give:

$f_{\mathit{ij}}^{\mathrm{loss}}=f_{\mathit{ij}}+f_{\mathit{ji}}=g_{\mathit{ij}}\left(U_i^2+U_j^2-2U_i{U}_{j}cos{\theta }_{\mathit{ij}}\right)\text{,}$

where g_ij,U_i,θ_ij are line’s (i,j) conductance, nodal voltage magnitude, and voltage phase angle difference, respectively. Assuming voltage magnitudes are close to 1.0 pu and approximating cos θ_ij by a second-order Taylor series expansion gives:

$f_{\mathit{ij}}^{\mathrm{loss}}\approx 2g_{\mathit{ij}}\left(1-cos{\theta }_{\mathit{ij}}\right)\approx g_{\mathit{ij}}{\theta }_{\mathit{ij}}^2\text{,}$

$\Rightarrow f_{\mathit{ij}}^{\mathrm{loss}}\approx \frac{r_{\mathit{ij}}}{r_{\mathit{ij}}^2+x_{\mathit{ij}}^2}{\theta }_{\mathit{ij}}^2\approx \frac{r_{\mathit{ij}}}{x_{\mathit{ij}}^2}{\theta }_{\mathit{ij}}^2\text{,}$

where the final step follows because x_ij ≥ 4r_ij for most transmission lines. Thus, the “DC” line losses can be written as:

$f_{\mathit{ij}}^{\mathrm{loss}}\approx \frac{r_{\mathit{ij}}{\theta }_{\mathit{ij}}^2}{x_{\mathit{ij}}^2}=r_{\mathit{ij}}{f}_{\mathit{ij}}^2\text{,}$

with the DC flow f_ij defined in (13). Note that the loss term f_ij^loss is quadratic in θ_ij and is therefore not suitable for a strictly linear constraint formulation.

A computationally amenable model of the quadratic losses can be incorporated into a linear optimization formulation by applying a (piece-wise) linear approximation of losses that circumvents the need for integer optimization (see [29–31]). PWL[f_ij^loss] In fact, you can employ PWL[·] to line losses and develop a strictly (piece-wise) linear model of active line losses in (19) with the following relations:

$f_{\mathit{ij}}^{\mathrm{loss}}:=\frac{r_{\mathit{ij}}}{x_{\mathit{ij}}^2}\mathrm{\Delta }\mathit{\theta }{\displaystyle \sum _{s=1}^S\left(2s-1\right)}{\theta }_{\mathit{ij}}^{\mathrm{PW}}\left(s\right)\text{,}$

${\displaystyle \sum _{s=1}^S{\theta }_{\mathit{ij}}^{\mathrm{PW}}\left(s\right)={\theta }_{\mathit{ij}}^{+}+{\theta }_{\mathit{ij}}^{-}}\text{,}$

${\theta }_{\mathit{ij}}={\theta }_{\mathit{ij}}^{+}-{\theta }_{\mathit{ij}}^{-}\text{,}$

${\theta }_{\mathit{ij}}\in \left(-{\theta }_{\text{max}},{\theta }_{\text{max}}\right)\text{,}$

$0\le {\theta }_{\mathit{ij}}^{+},{\theta }_{\mathit{ij}}^{-}\text{,}$

${\theta }_{\mathit{ij}}^{\mathrm{PW}}\left(s\right)\in \left[0,\Delta ,\theta \right]\text{.}$

The linear constraint formulation presented in (20) is, in fact, a convex relaxation of the “DC” line losses. Figure 5.7 illustrates the relaxation of PWL line losses. Note that the relaxation given by (20) yields a value for f_ij^loss that is greater than or equal to the piece-wise linear approximation. PWL[f_ij^loss] Equality occurs only when both |θ_ij| ≡ θ_ij⁺ + θ_ij⁻ and θ_ij^PW(s) > 0 ⇒ θ_ij^PW(n) = Δθ, ∀ n < s (i.e., adjacency conditions). Under such conditions, the relaxation is considered “tight” and the model is exact with respect to PWL, and the convex relaxation of line losses provides a more accurate method for estimating line losses than standard linearization. When the losses are relaxed (i.e., not tight), overestimated losses are denoted “fictitious losses,” as they exist only as an artifact of the MPC controller model and not in the actual system.

3.6 Model-predictive control

Literature provides two generic approaches for mitigating cascade failures in power networks. The first method predicts disturbances a priori and is based on an off-line computation of all possible or likely failures in the network – the so-called N – k problem, e.g., [32]. With such an approach, control policies are devised to deal with each possible disturbance. A major drawback of this approach is that it does not scale well, since the number of salient contingencies to consider increases exponentially with network size. A second method is based on retroactive control, whereby the uncertainty surrounding the disturbance has been revealed and one can utilize the knowledge available about the disturbance to determine control responses in real-time to mitigate the effects of the disturbance. In the latter approach, the multi-timescale nature of cascading failures provides sufficient time for post-contingency computations. In addition, power/energy systems are suffused with constraints on control inputs and states, which makes model predictive control (MPC) particularly useful in a cascade-mitigation scheme.

MPC is an advanced method of process/batch control that has gained prominence over the last 30 years from its extensive deployment in the chemical industry. For a thorough technical discussion of predictive control in linear systems (see [8]). Basically, MPC provides a method for controlling dynamic systems with constraints on inputs and/or states using tools from optimization. MPC implementations solve on-line, at each sampling instant, a finite-horizon optimal-control problem based on a dynamic model of the plant. Most MPC approaches can be described by the following algorithm:

1. Determine a control sequence that optimizes an objective function over a prediction window, where the measured state at time step k is the initial state.

2. Apply the computed control profile until new process measurements become available.

3. When new measurements are available, set k = k + 1 and repeat step 1.

MPC is most often formulated in the state space by linear discrete-time difference equations. The mathematical formulation is given below.

MPC Formulation The objective of MPC is to drive the system from its current state to some reference state, given by a set point, x^sp, in the “best possible” way. In addition, power systems are often modeled with a mix of differential and algebraic states, which beget a set of differential-algebraic equations (i.e., equality constraints). In this work, the constraints are assumed linear (i.e., nonlinear MPC is not discussed) and an l₂-norm describes the objective function, therefore, MPC can be formulated as a quadratic programming (QP) problem over a finite prediction horizon⁵ of length M:

$\begin{array}{c}{U}^{*}\left[k\right]=\underset{u\left[l|k\right]}{min}\left|\right|x\left[M|k\right]-x_{k+M}^{\mathrm{sp}}\left|\right|{}_{S_M}+{\displaystyle \sum _{l=0}^{M-1}L\left(x\right[l\left|k\right],u\left[l|k\right])}\end{array}$

$\begin{array}{c}\mathrm{s}.\mathrm{t}.x\left[l+1|k\right]=\mathit{Ax}\left[l|k\right]+\mathit{Bu}\left[l|k\right]+\mathit{Fz}\left[l|k\right]\end{array}$

$\begin{array}{c}0=\hat{A}x\left[l|k\right]+\hat{B}u\left[l|k\right]+\hat{F}z\left[l|k\right]\end{array}$

$\begin{array}{c}\mathit{Cx}\left[l|k\right]+\mathit{Du}\left[l|k\right]+\mathit{Gz}\left[l|k\right]\le d\end{array}$

$\begin{array}{c}x\left[l|k\right]\in \chi ,u\left[l|k\right]\in \mathcal{U},z\left[l|k\right]\in \mathcal{Z}\end{array}$

$\begin{array}{c}x\left[M|k\right]\in {\mathcal{T}}_x\end{array}$

$\begin{array}{c}x\left[0|k\right]=x_k^{\mathrm{meas}}\text{,}\end{array}$

where x[l|k], u[l|k], and z[l|k] represent the dynamic state, control input, and algebraic state variables, respectively, at predicted time 0 ≤ l < M, given initial measured state $x_k^{{{}^{\mathrm{m}}}^{\mathrm{eas}}}$ at time k. The optimizer,

$U^{*}\left[k\right]=\left\{u^{*}\left[0|k\right],u^{*}\left[1|k\right],\dots ,u^{*}\left[M-1|k\right]\right\}\text{,}$

represents the open-loop optimal control sequence over the prediction horizon at time k. The appropriately-sized matrices A,B,F and $Â,\hat{B},\hat{F},C,D,G$ describe dynamic and algebraic constraints, respectively. The objective function in (21a) is defined by:

$L\left(x\left[l|k\right],u\left[l|k\right]\right)=\left|\right|x\left[l|k\right]-x_{k+l}^{\mathrm{sp}}\left|\right|{}_Q+\left|\right|u\left[l|k\right]-u_{k+l}^{\mathrm{sp}}\left|\right|{}_R\text{,}$

where $x_{{{}_k}_{+l}}^{\mathrm{sp}}$ and u_k + l^sp refer to a reference trajectory at time k + l, the norms are defined by ||y||_B ≡ y^TBy, and weighting matrices S_M$\underset{¯}{\succ }$ 0 and Q$\underset{¯}{\succ }$ 0 are non-negative definite while R 0 is positive definite. Expressions (21b) and (21c) describe the differential-algebraic (DAE) dynamics. Expressions (21d), (21e), and (21f) define static inequality constraints, bounds on states and inputs, and a terminal state constraint set, respectively. Equation (21g) establishes the initial state for MPC.

In this chapter, two specific MPC techniques are employed: shrinking-horizon and receding-horizon MPC. These two approaches are described in detail within the context of cascade mitigation in Sections 5 and 6, respectively.

4.1 Level 1: economically optimal energy schedule

Over a 24-hour period, Level 1 computes an optimal energy schedule that determines how to best operate energy storage, conventional generation, flexible loads, and available renewable energy based on forecasts. The Level 1 schedule is, therefore, similar to standard economic dispatch [33], except that the temporal coupling introduced by energy storage implies optimization over a horizon rather than a single time step. In addition, any quadratic line losses can be included with a standard piece-wise linear (PWL) DC approximation as presented in [29].

The Level 1 model enforces line-flow limits to ensure that, under accurate model and forecast scenarios, no lines are overloaded (i.e., the system is “safe” and economical). The dispatch schedule is computed as a multi-period quadratic programming (QP) problem whose objective is to minimize energy (fuel) costs of conventional generators:

$\mathrm{Cost}\left(f_{\mathit{Gn}}\left[k\right]\right)=a_n{\left(f_{\mathit{Gn}}\left[k\right]\right)}^2+b_n{f}_{\mathit{Gn}}\left[k\right]\text{,}$

where a_n [$/h-pu²] and b_n[$/h-pu] are constant parameters for generator n and f_Gn[k] is its output power at time step k.

The Level 1 schedule establishes a reference signal over a multi-hour horizon, consisting of the economically optimal system set points x^sp, and the operator control actions u^sp required to achieve those optimal set points. The schedule is submitted to the operator and recomputed every hour. For details on Level 1 formulation, see [22,26].

4.2 Level 2: corrective controller

The Level 2 controller operates in the background to track the reference trajectories computed from Level 1 (i.e., the economic set-point values). The corrective controller employs a linear model of the actual system and operates on a minute-by-minute timescale.⁶ If a disturbance takes place (e.g., a line outage), Level 2 computes corrective control actions u[k] in a MPC fashion that steers the system towards a safe and economically optimal state as provided by a Level 1 reference.

Level 2 considers ramp-rate limits on conventional generators, dynamics and power ratings of grid storage devices, and can incorporate the thermal response of overloaded lines. Note that, in Level 2, lines are no longer subject to a hard flow limit constraint and, instead, the controller seeks to drive overloaded lines below respective limits. The Level 2 MPC-based cascade-mitigation scheme is formulated as a quadratic programming problem (QP) over a finite prediction horizon M.

The Level 2 cascade mitigation discussed in Section 5 employs a shrinking-horizon (i.e., M → 0) MPC-based scheme within a simplified framework of energy hubs. In Section 6, receding-horizon MPC (i.e., M fixed) is described for a temperature-based cascade-mitigation scheme within a bulk electric-power system setting.

5.1 Shrinking-horizon MPC (SHMPC)

In this MPC-based cascade-mitigation scheme, if lines are overloaded after M_T minutes, they automatically trip (i.e., a sensor may measure overloads every M_T minutes and trip lines based on simple overload criteria). That is, a deterministic line-tripping model is employed in the multi-energy setting. Therefore, it is sensible to utilize a prediction horizon of M_T minutes and assume new measurements are available every minute (and that computation of open-loop control sequences is instantaneous). Furthermore, in attempting to halt the cascade, load-shedding is only allowed in the final time step M_T, as a last resort to bring line flows within their limits. Accordingly, there is no reason for our prediction window to extend beyond minute M_T and, therefore, the prediction window shrinks by one minute after every measurement update. This method of MPC is often referred to as “fixed-point” or “shrinking-horizon” MPC (SHMPC) [34]. SHMPC considers a prediction horizon that is fixed in time as illustrated in Figure 5.9.

Consider an energy system at time k over M = M_T minutes (i.e., initial prediction horizon) with sample time T_s = 1 minute. Then, the SHMPC algorithm is given by:

Shrinking-Horizon MPC Algorithm:

1. Given initial state x[k], solving an optimal control problem over horizon [k, k + M] yields the open-loop control sequence {u[l|k]}_l = 0^M − 1.

2. For time [k, k + 1), apply the first instance of control sequence u[0|k] to the system with a zero-order hold.

3. Measure new system states x[k + 1], set k := k + 1.

4. Set M : = M – 1 and, if M > 0, repeat step (1). Else, STOP.

As can be realized from the above SHMPC process, with each successive control action, the horizon shrinks by 1 minute until horizon has length 0. At the end of the horizon some physical process may have reached a critical juncture, such as presented with the simple deterministic line tripping model: lines may trip.

If the MPC model has no errors and matches the real system, no overloads will remain at time k + M_T, and the cascade will be halted. On the other hand, if the MPC model is imperfect, some overloads may remain. If more lines do trip, one would re-run SHMPC with redrawn horizon M = M_T and this shrinking/redrawing horizon cycle continues until SHMPC relieves all line overloads. At that point, the slow-timescale Level 1 is updated with a new post-disturbance network topology, and the latest values for generation, storage and load, and the optimal schedule over the remainder of the 24-hour period are determined. An overview of the energy-hub system operation under closed-loop control is illustrated in Figure 5.8. The cascade-mitigation scheme interfaces with the “real system” as well but on a faster timescale. Note that without the look-ahead feature of MPC, a closed-loop controller acting only at minute k + M_T would shed more load, as it would not be able to properly allocate storage utilization to overcome possible future generator power limits.

To contrast the performance of the shrinking-horizon MPC scheme, we consider the base case of a “dumb” controller. The “dumb” controller seeks only to satisfy demand (i.e., avoid load shedding) and line flows are given by the power flow solution with no regard for line flow limits. Therefore, the “dumb” controller will be unlikely to alleviate overload and undergo significant cascading failures.

5.2 Linear SHMPC model outline

The energy-hub model from [26] is employed in the SHMPC scheme, which is detailed in [22] and outlined below.

5.3 Simulations of SHMPC-based cascade mitigation

The formulation of energy hubs described in [26] permits the construction of arbitrarily large interconnected energy-hub networks. In this section, the effects of disturbances (i.e., line outages) under different energy-storage scenarios for small and large energy-hub systems are investigated with simulations. Because current power-grid operating and planning standards ensure power systems are in a reliable condition even if one contingency occurs (i.e., N – 1), the initial disturbances will consist of multiple simultaneous line outages. Each system consists of an electrical network, a natural gas network, district heat loads, wind turbines, and multiple energy hubs that couple the four different energy types.⁷ The smaller system is useful in describing how the SHMPC approach mitigates a cascade failure, while the larger system allows us to better showcase the cascading effects and role of mitigation.

Employing the SHMPC cascade-mitigation scheme, the following simulations investigate the performance of the scheme under different energy-storage scenarios. That is, simulations are used to investigate how the availability (i.e., capacity) and performance (i.e., charge/discharge power limits) of energy-storage devices impact the level of load control.

The random grid-generating techniques proposed in [35] are employed again to construct two multi-energy systems – a small 11-hub system and a larger 69-hub system and subject each to a multi-line outage. The small energy-hub system is shown in Figure 5.10. Due to the size of electrical and natural gas networks and the random interconnectivity with energy hubs, a meaningful visualization of the larger energy-hub system is not clear and is excluded here. The smaller, simpler system enables a clear understanding of how storage is utilized to mitigate cascade failure, whereas the larger system becomes useful in the discussion as it exhibits meaningful cascade behavior.

The topological characteristics of the two systems are given in Tables 5.2a and 5.2b where N, E, G, and D represents the number of nodes, lines, generators, and loads for each network, respectively. There are 11 energy hubs in the small system and 69 energy hubs in the large system to couple the four different energy networks. The energy conversion efficiencies are given in Table 5.2c. All hubs connecting the wind network to the electrical network have output storage, while applicable input and output hub storage is added randomly to all of the remaining hubs.

The energy-storage scenarios are considered by varying storage capacity and storage power limits from the nominal values by a given factor. For example, for scenario (capacity, power) = (0.25, 0.10), energy-storage capacity is at 25% of nominal and power limits are 10% of nominal. The idea is to compare the effect of different energy-storage scenarios on the average load shed over the 24-hour period. In particular, the scenarios include systems with “no” (0), “some” (≈ 0.25), “nominal” (1.0), and “a lot” (10) of storage capacity, while the system storage devices are subject to “small” (0.10), “nominal” (1.0), and “large” (5 ≥) power limits.

• Small 11-hub system:
In the nominal (1,1) SHMPC case, the 11-hub system from Figure 5.10 undergoes a disturbance that results in the outage of lines 2,5, and 8. The loss of the three lines leaves line 9 (natural gas) with a significant overload, which must be cleared to avoid tripping the line. Under the SHMPC scheme, generators are reconfigured (considering power limits), storage is utilized, and minimal load is shed over the 5-minute interval to avoid tripping line 9. The Level 1 schedule is updated to reflect the multi-line outage and some load (≈ 20%) must be shed until wind power becomes available towards the end of the day. In the nominal base case with the “dumb” controller, line 9 is not protected and trips after 5 minutes, which leaves the system in a weak state and results in heavy load-shedding (≈ 50%).
The results of the 11-hub system are shown in Tables 5.3 and 5.4 Increasing storage capacity generally reduces the amount of load shed and improves performance of cascade mitigation, which is expected, since more energy can be stored and is available to inject into the system upon the disturbance. However, the general trend suggests that lowering power limits improves performance. The reason behind this trend is that the MPC scheme does not care about the state of the system beyond the halting of the cascade and will maximally utilize (free) stored energy to satisfy demand and avoid shedding load. However, with small power limits, the amount of stored energy is effectively rationed during the cascade-mitigation process and some conventional generation is needed to satisfy demand and halt the cascade. This balance between generation and energy utilization allows MPC to halt the cascade and best positions the multi-energy system to satisfy load demand over the subsequent period (following the cascade).
In the base case, the amount of load shed under scenario (10, 5.0) is relatively low, because line 9 is “accidentally” not overloaded at minute 5. The term “accidentally” is used here since the base-case controller attempts to satisfy all demand at the lowest cost with no regard for flow limits. Furthermore, for the (10, 5.0) scenario, the base case sheds marginally less load than the MPC scheme over the 24-hour period. This is because the optimization does not penalize the spilling of wind power, which means that the MPC scheme can use available energy storage and wind power at the same cost. Thus, stored energy in the future (i.e., beyond the 5-minute prediction window) has no value to the MPC controller. MPC employs available energy storage and spills wind power, while the base case utilizes available wind power. The MPC response is optimal over the 5-minute period, but with the cascade halted, it leaves the system with less energy stored. This results in a marginal increase in load-shedding over the remaining hours.

• Large 69-hub system:
The trends from the 11-hub system apply to the 69-hub system as well, and the results are provided in Tables 5.5 and 5.6. As storage capacity increases and power limits decrease, the performance of MPC increases for the same reasons mentioned above.
In the base case, the trend is not as strong, but nonetheless discernible. While the complexity of the network obfuscates the subtleties of the result, it is worth noting that for the scenario without storage (i.e., zero capacity), the amount of load shed is similar to low- and medium-capacity scenarios (0.2, *) and (1.0, *).
The behavior of both the base-case system and the MPC cascade-mitigation scheme are depicted in Figures 5.11 and 5.12, for energy scenarios (0, 10), (0.20, 10), (1.0, 10), and (10,10). (Note that thicker lines represent higher capacity scenarios.) In fact, for scenarios (*, 10), the base-case system generally sheds more and more load as storage capacity increases. This is because the base case is uncontrollable in the sense that it does not regard line-flow limits and, for larger systems, this results in significant line tripping as seen in Figure 5.11a. At high power limits and with increasing storage capacity, the base-case controller will inject more stored energy into the system (Figure 5.11c) which, coupled with non-trivial generation levels (Figure 5.12), results in line tripping (Figure 5.11a) and fragmentation of the system. The fragmentation leads renewables to become isolated from loads, which is evident in the base case in Figure 5.11b, since the increase in available wind power (in evening and at night) is unable to recover shed load. The fragmentation is most severe in the highest capacity case, where available wind power is separated from loads and can only be utilized to aimlessly increase energy-storage levels.

Note that “No Disturbance” cases represent economically optimal energy management without any line outages. However, in the absence of a disturbance, there is still a need to shed some nominal load when wind power reaches its nadir and no more power can be injected by conventional generation without exceeding flow limits. This is a remnant of the fact that the random networks generated do not properly capture the design and planning of real transmission power networks.

Furthermore, it is worth noticing that the zero-capacity MPC scheme (0, *) sheds only 1.1% of total load while the optimal MPC storage configuration (10, 0.1) sheds 0.4% of total load. Therefore, one should ask if the investment made in storage devices is worth the marginally improved cascade-mitigation performance (a 0.7% reduction in load shed). Indeed, with the base-case minimal load shed of 14.9%, the MPC scheme without storage may provide a sufficient cascade-mitigation solution. However, other factors, such as intermittency and congestion reduction, suggests the need for considering energy-storage devices in the investment process.

6.1 Receding-Horizon MPC

The SHMPC approach suffers from two major drawbacks:

• An unpredicted event could take place towards the end of the shrinking-horizon period, which leaves the system unable to recover in the remaining time.

• As the horizon shrinks and approaches the final time, the control law typically “gives up trying” since there is too little time to go to achieve anything useful in terms of objective function reduction.

The above two shortcomings surrounding SHMPC can be overcome with the notion of receding-horizon MPC, which considers a prediction horizon that does not shrink but, instead, remains fixed in length and moves with time. This makes receding-horizon MPC immune against the above above drawbacks and, therefore, offers a more robust control paradigm than SHMPC. That is, consider a system with a prediction horizon of M_T minutes and sample time T_s = 1 minute and assume the initial time step is k, then the receding-horizon MPC is summarized by the following algorithm.

Receding-horizon MPC algorithm:

1. Given x[0|k] := x[k], solve an optimal control problem over horizon [k, k + M_T] to retrieve an open-loop control sequence ${\left\{u\left[l|k\right]\right\}}_{l=0}^{M_T-1}$.

2. Apply the first instance of control sequence to the system: u[k] := u[0|k].

3. Measure the new system state x[k + 1] := f (x[k],u[k]).

4. Set k = k + 1 and repeat step (1).

As the above MPC process illustrates, with each successive control action, the horizon recedes, as the name implies. Note that in this work, the prediction and control horizons are assumed equal to M_T.

The Level 2 model for the receding-horizon MPC cascade-mitigation scheme is similar to the SHMPC scheme, where we considered ramp-rate limits on conventional generators and the dynamics and power limits of grid-storage devices. However, with the temperature-based cascade-mitigation scheme, receding-horizon MPC also incorporates the thermal response of overloaded lines. Note that in Level 2, lines are no longer subject to a hard flow-limit constraint. Rather, the controller seeks to drive conductor temperatures below their respective limits. Note that the receding-horizon MPC optimization is still formulated over a finite prediction horizon as described in (21).

The details of the Level 2 receding-horizon MPC model, system states, and controls are developed and discussed in [31]; however, for sake of completeness, a summary of the model is provided below. The states and inputs associated with the proposed formulation of an MPC cascade-mitigation scheme for an electric-power system are outlined here.

Dynamic states (x): there are three types of dynamic states:

• $\mathrm{\Delta }{\hat{T}}_{\mathit{ij}}\text{,}$ line (i, j) conductor temperature overload w.r.t. limit T_ij^lim.

• E_n, state of charge (SOC) for energy-storage (ES) device n.

• f_Gn, power output level for generator n.

Control inputs (u): the formulation employs five types of control inputs:

• Δf_Gn, change to conventional generator n output level.

• $f_{G_{W}n}^{\mathrm{spill}}\text{,}$ wind spilled from nominal, for wind turbine n.

• f_Dn^red, demand response (reduction) from nominal, for load n.

• f_Qc,_n , f_Qd,_n, charge (c) and discharge (d) rates for ES n.

Uncontrollable inputs: there are three types of forecast (uncontrollable) inputs (i.e., exogenous disturbances):

• f_Dn^nom, nominal available power from wind turbine n.

• f_Dn^nom, nominal demand, for load n.

• d_ij, ambient temperature and solar gain, for line (i, j).

Algebraic states (z): models require nine types of algebraic states:

• f_ij, real power flowing through line (i, j).

• f_ij^loss real power losses for line (i, j).

• θ_ij, phase-angle difference between nodes i and j.

• θ_ij⁺, θ_ij⁻, absolute value approximation of |θ_ij |.

• {θ_ij^PW(s)}_s = 1^SS-segment PWL approximation of |θ_ij|².

• $f_{{G_W}^{{}_n}}\text{,}$ real power injected by wind turbine n.

• f_Dn, real power consumed by load n.

• f_Qn, total power injected or consumed by ES n.

From the above descriptions, state and input vectors are defined by:

$x=\mathrm{col}\left\{\mathrm{\Delta }\hat{T},E,f_G\right\}\text{,}$

$u=\mathrm{col}\left\{\mathrm{\Delta }{f}_G,f_{G_W}^{\mathrm{spill}},f_D^{\mathrm{red}},f_{\mathit{Qc}},f_{\mathit{Qd}},\mathit{\psi }\right\}\text{,}$

$z=\mathrm{col}\left\{\theta ,{\theta }^{+},{\theta }^{-},{\theta }^{\mathit{PW}},f,f^{\mathrm{loss}},f_D,f_{G_W},f_Q\right\}\text{.}$

6.1.4 Mitigating simultaneous charging and discharging

This section compares the simple linear “simul-charge” energy-storage model used in the receding-horizon MPC to the more accurate but (non-convex) complimentarity-based model. First, define the models’ respective actions by the superscripts (.)^S and (.)^C. Then, for a given optimal storage device action, f_Q*, the following holds:

$f_{\mathit{Qc}}^C\left[l\right]-f_{\mathit{Qd}}^C\left[l\right]=:f_Q^{*}:=f_{\mathit{Qc}}^S\left[l\right]-f_{\mathit{Qd}}^S\left[l\right]\text{.}$

  (33)

Note that there exists only one unique complementarity-based control action (due to the condition f_Qc^C[l]f_Qd^C[l] = 0). However, without complementarity (i.e., under the simultaneous charge/discharge formulation), multiple solutions may exist. One side effect of allowing simultaneous charge/discharge events is identified by the following:

Theorem 5

For a given optimal storage flow f_Q*[l], the simultaneous charge/discharge model (compared with the complementarity-based model) underestimates SOC (i.e., ΔE[l + 1] := E^C[l + 1] − E^S[l + 1]) by

$\mathrm{\Delta }\mathit{E}\left[l+1\right]=T_s\frac{1-{\eta }_c{\eta }_d}{{\eta }_d}{\displaystyle \sum _{m=0}^{l}min\left\{f_{\mathit{Qc}}^S\left[m\right],f_{\mathit{Qd}}^S\left[m\right]\right\}\text{.}}$

  (34)

Proof. The proof follows directly from considering the two cases: f_Qc^C = 0 and f_Qd^C = 0.

From the theorem, it is straightforward to see that the simultaneous charge/discharge model exactly matches the complementarity-based model when one of the following holds:

• η_c = η_d = 1 (perfect efficiency),

• min{f_Qc^S[l], f_Qd^S[l]} = 0 (complementarity is satisfied),

• $f_Q^{*}\left[l\right]=\overline{f_Q}$,

where $\overline{f_Q}=\overline{f_{\mathit{Qc}}}=\overline{f_{\mathit{Qd}}}$ has been assumed for presentation clarity. (Generalization to $\overline{f_Q}\ne \overline{f_{\mathit{Qd}}}$ is straightforward.) The last condition stems from

$min\left\{f_{\mathit{Qc}}^S\left[l\right],f_{\mathit{Qd}}^S\left[l\right]\in [0,\overline{f_Q}-f_Q^{*}\left[l\right]\right\}\text{.}$

  (35)

This means that the controller can (erroneously) employ simultaneous charge/discharge to achieve a lower-than-actual SOC, which could be advantageous to reduce the cost of SOC deviations from the Level 1 reference. Furthermore, the controller can utilize simultaneous charge/discharge to reduce line overloads by fictitiously “burning” excess power through energy-storage inefficiencies (η_c, η_d < 1).

To reduce the effect and occurrence of simultaneous charge/discharge events, two steps have been implemented. Firstly, to reduce the worst-case behavior of the simultaneous charge/discharge formulation, the following constraint is utilized:

$\frac{f_{\mathit{Qc},n}\left[l\right]}{{\overline{f_Q}}_{c,n}}+\frac{f_{\mathit{Qd},n}\left[l\right]}{{\overline{f_Q}}_{d,n}}\le 1\forall l,n\text{,}$

  (36)

where ${\overline{f_Q}}_{c,n},{\overline{f_Q}}_{d,n}$ are the rate limits on charging and discharging, respectively.

Secondly, most devices at most time steps will satisfy f_Qn[l|k] ≠ 0. This knowledge can be used to enforce complementarity-like constraints, and limit occurrences of simultaneous charge/discharge events. When MPC first runs, the charge/discharge status of storage devices over the prediction horizon is most likely unknown. In order to initialize the status, simultaneous charging/discharging is permitted for that first prediction trajectory. When MPC next runs, at time k, the charge/discharge status of each storage device over the prediction horizon is determined from its status at the corresponding time step in the previous prediction trajectory (i.e., k – 1). It should be noted that the prediction horizon at time step k – 1 only extends to f_Qc,n[M − 1|k], so no prior value is available for initializing the status of f_Qc,n[M − 1|k]. Therefore, the Level 1 status at the corresponding time can be used to establish the charging state for all devices at this terminal time step. Figure 5.13 outlines the algorithm employed in MPC.

Remark 6

This algorithm introduces a delay of one time step in the transition of storage devices from charging to discharging, or vice-versa. To address this issue, computation of the MPC trajectory for time k can be repeated using the latest status information. At this re-run, storage devices with f_Qn[l|k] ∈ [− α_tol,α_tol] are handled in accordance with line 13 in Figure 5.13.

To summarize, constraint (36) limits the worst-case behavior of simultaneous charge/discharge, and the algorithm in Figure 5.13 reduces the frequency of simultaneous charge/discharge events. The methodology of the proposed simul-charge algorithm is illustrated in Figure 5.14. The red arrows represent the original complementarity-based (non-convex) model, while the green region represents the effect of constraint (36). The total impact of the heuristic is given by the union of red arrows and the dashed red line, which represents the admissible set of storage behaviors. Thus, by shrinking the simul-charge area from the green-blue rectangle to just the red dashed line, the heuristic makes the storage model more representative of reality, but at a slightly increased computational cost.

6.2 Base-case Controller

To benchmark the performance of the proposed MPC scheme, a base-case controller was developed. This base-case was meant to provide an indication of human-operator behavior during a system emergency (disturbance). Clearly, modeling a human operator is non-trivial as standard emergency procedures vary broadly across utilities. Furthermore, the experience of a human operator is not amenable to an implementable (and repeatable) algorithmic framework and, as far as the authors are aware, no data sets exist that capture operator behavior during (simulated) contingencies. However, the formulation presented here captures the underlying goals of the operator:

1. alleviate thermal overloads by rescheduling or curtailing generation, while considering ramp-rate limits and incremental generator cost curves;

2. employ sensitivity-based methods, such as power transmission distribution factors (PTDFs), generation shift factors (GSFs), and transmission loading relief (TLR) procedures to make quick control decisions to relieve thermal overloads [37];

3. shed load as an absolute last resort; and

4. ignore energy storage.

Thus, mapping the above operator traits into an MPC-based framework serves as the base case:

• Base-case implementation

– Replace $\mathrm{\Delta }{\hat{T}}_{\mathit{ij}}\left[l\right]$ with a relative overload metric:

${\hat{o}}_{\mathit{ij}}\left[l\right]=10max\left\{0,\left(\left|f_{\mathit{ij}}\left[l\right]\right|+0.5f_{\mathit{ij}}^{\mathit{loss}}\left[l\right]\right)/f_{\mathit{ij}}^{\text{lim}}-1\right\}\text{.}$

That is, if a line is 25% overloaded, ${\hat{o}}_{\mathit{ij}}=2.5\text{.}$

– Consider PTDF, GSF, and TLR implicitly as onestep linear MPC process akin to Level 2 (i.e., set M = 1) and include overloads ${\hat{o}}_{\mathit{ij}}\left[0|k\right],{\hat{o}}_{\mathit{ij}}\left[1|k\right]$ in the objective and terminal costs.

– Heavily penalize load control and adjustment of SOC.

– Remove terminal constraints on overloads, ${\mathcal{T}}_x\text{.}$

– Set weighting matrices R_base = R, Q_base = S_M, and S_M,base = S_M.

6.3 Actual system model (plant)

The AC power flow is generally accepted as a valid representation of the actual physical power system (i.e., the plant). Therefore, the control actions recommended by the MPC, which utilizes the strictly linear model described in Section 6.1.2, is applied to an AC model of the system at each time step. This interaction between predictive DC controller and actual AC plant is illustrated in Figure 5.15. Furthermore, the resulting losses from the AC power flow are utilized in the nonlinear IEEE Standard 738 conductor temperature model to better capture the effects of MPC recommendations on the actual system. Finally, the actual energy-storage model does not allow for simultaneous charging and discharging in the same time step and instead employs the projected control action f_Qn[k] such that f_Qc,n[k]f_Qd,n[k] = 0.

The higher the temperature, the more likely line tripping becomes. To capture the inverse relationship between temperature and expected time to trip in the actual system, the probabilistic thermal line-outage model from Section 2.1.2 is employed.

6.4 Case Study: IEEE RTS-96

The bi-level control scheme is applied to an augmented version of the IEEE RTS-96 power system test case, which is described in full details in [38]. A brief overview of this test case is included here.

6.4.1 Overview

The RTS-96 system consists of 138 kV and 230 kV subsystems. The network is organized into three interconnected physical regions, as illustrated in Figure 5.16. It consists of 73 nodes and 120 branches, of which 15 branches are in-phase transformer (IPTs), one is a phase-shifting transformer (PST), and the remainder are overhead transmission lines (138 and 230 kV). Buses are denoted with three digits: the first digit indicates the area while the latter two are intra-area designators. Bus types are indicated by color: generator (blue), load (yellow), and zero-injection (white). Edges represent transmission lines (black) and transformers (aqua/gray). The disturbance is displayed with stars: lines 113-215 and 123-217 were tripped. Note that the three underground cables in the original RTS-96 system have been replaced by equivalent overhead lines to enable application of a single thermodynamic model. Transformer temperature overloading is not considered in this case study, as their thermodynamic models differ from those presented here.

The aim of this case study is to explore the contingency management achievable with the proposed hierarchical control scheme. Unfortunately, the RTS-96 system is designed as a highly reliable system, with unusually high thermal ratings for lines. To bring the system closer to its limits and engender worthwhile scenarios, thermal ratings f_ij^lim were reduced by 40%, yielding line temperature limits in the range of 60 to 70 °C. Furthermore, ramp rates have been reduced by 82.5% to highlight Level 2 performance and enhance the role of storage in congestion management. For the temperature dynamics, the RTS-96 system data only specifies per-unit resistance, reactance, and line length, but not the conductor types (i.e., diameter, heat capacity). Therefore, this case study employed ACSR conductors, 18/1 Waxwing (138 kV) and 26/7 Dove (230 kV), which represent reasonable choices given the reduced line ratings. The parameter values for Dove and Waxwing conductors, along with other system parameters, are provided in Table 5.7. The values in brackets represent ranges.

Table 5.7

Network model parameters used in case study

Model Parameter	Value		Units
Sampling Time, Ts	60		s
3-phase power base, Sb	100		MVA
Energy-storage base, Eb	100		MWh
Monetary unit base, Mb	10,000		$
Storage SOC limits, $\overline{E_i}$	2		pu
Storage power limits, $\overline{f_{\mathit{Qc}}},\overline{f_{\mathit{Qd}}}$	0.25		pu
Nominal wind power, $f_{G_W}^{\mathrm{nom}}$	Fig. 5.17		pu
Nominal loads, $f_{{}_{\mathrm{D}}}^{\mathrm{nom}}$	Week 1, day 1 in [38]		pu
Overcurrent protection limit, $\overline{\mathrm{\Omega }}$	3		-
Ambient Temperature, Tamb	35		°C
Wind speed, angle, υw ∠ θw	0.61, π/2		m/s, rads
Line-to-line base voltage, Vb	138	230	kV
Thermal rating, fijlim	1.05	3.00	pu
Conductor diameter, Dij	15.5	23.5	mm
Heat capacity, mCp, ij	383	916	J/m-°C
Ampacity, Iijlim	439	753	A
Resistance per unit length, Rij	[103,118]	[55,66]	μΩ/m
Temperature limit, Tijlim	[62,64]	[67,71]	°C
Temperature coefficient, τij	0.796	0.888	-
Loss coefficient, ρij	0.157	0.066	°C-m/W
Ambient coefficient, γij	0.193	0.104	-
Solar heat gain rate, qs,ij	14.4	21.9	W/m

6.4.3 Simulation results

The case study described in Section 6.4.1 is simulated in MATLAB according to Level 1, Level 2, and base-case implementations. Initially, the system is operated economically according to Level 1. However, at hour 18 (low wind, high demand), a two-line outage (i.e., the disturbance) trips lines 113-215 and 123-217. Transient (short-term) stability was assumed. Performance and behavior of the Level 2 MPC (with horizon lengths of M = 5, 10, 20, 30, and 45) and the base case are discussed below.

The double-line outage caused the remaining inter-area transmission line 107-203 to become severely overloaded (greater than 1.25f_ij^lim). The Level 2 MPC scheme alleviated the temperature overloads and brought the system safely to the updated economic set points provided by Level 1. In contrast, the base case underwent a cascading failure, with line tripping bringing the system to a voltage collapse after 29 minutes, as exemplified by non-convergence of the AC power flow. The base-case cascading failure evolved as follows:

• k = 3: Line 107-203 tripped at ΔT_ij[3] = 13.5° C.

• k = 16: Line 114-116 tripped at 8.14° C.

• k = 26: Line 113-123 tripped at 11.4° C.

• k = 28: Lines 103-109 and 112-123 tripped at 16.8° C and 20.7° C.

• k = 29: Voltage collapse “blackout.”

This process is illustrated in Figure 5.18, where it can be seen that the minimum voltage magnitude fell below 0.87 pu.

Figure 5.19a illustrates that the objective function cost (21a), calculated for each MPC run, decreased monotonically over time. This does not prove stability, but highlights the Lyapunov-like properties of the objective function [7] as MPC drives the system back to the Level 1 (economically optimal) equilibrium point.

The maximum line temperatures for the base case and MPC are illustrated in Figure 5.19b. Note that MPC is able to avoid excessively high temperatures, and in fact drives all line temperatures below their respective limits by around minute k ≈ 75. Later, a few lines hover slightly above their temperature limits. However, this is due to model inaccuracy arising from MPC’s use of an approximate linear temperature model and the DC power flow. In particular, over that latter phase, the largest temperature deviations above limits are associated with 138 kV lines that exhibit X/R = 3.83 < 4. This relatively low X/R ratio engenders errors in the DC approximation of the nonlinear AC network equations. The DC model incorrectly informs the controller that losses are sufficiently low, implying that negligible control action is required for the temperature to drop below its limit in the next time step. But the actual power system, described by the AC power flow, has higher than predicted losses, and the temperature stays slightly above the limit. The controller repeats these incorrect estimates of losses until control action is required for other reasons, or load patterns autonomously reduce line loadings below limits. For k > 50, all line loadings are less than 5% above their thermal ratings, which is within expected error levels [40,41]. These results suggest that despite the presence of approximate models, the MPC scheme is able to reject the disturbance through feedback and return the system to an acceptable state. A thorough discussion of the impact of model approximations is provided in [42].

Level 2 MPC performs a balancing act between ensuring safety criteria and restoring economically optimal set points. This balance is highlighted in Figure 5.19c, where the cost of conventional generation is shown for both MPC and the base case. To ensure acceptable line temperatures, MPC initially sacrifices economic optimality by deviating from the Level 1 set points. For k > 120, the system returns to economically optimal levels, with inaccuracy in the MPC model causing some minor discrepancies. Interestingly, over the first 15 minutes or so, the generation cost achieved by MPC is actually less than the optimal cost given by Level 1. Two factors contribute to this apparent anomaly. Firstly, Figure 5.19c shows the post-disturbance Level 1 schedule, whereas the generators were initially operating according to less-costly pre-disturbance set points. Secondly, the updated Level 1 reference schedule enforces hard line-flow constraints, while MPC allows line flows to temporarily exceed limits.

As discussed in Section 6.1, the control actions available to Level 2 MPC for reducing line temperatures include: load reduction, wind curtailment, and energy-storage injections. Figure 5.19d and 5.19e illustrate the main controls employed to alleviate excessive temperatures for this case study. Contrasting MPC response with the base case, it is clear that load and energy-storage controls were crucial immediately following the disturbance. By initially curtailing energy-storage discharge (Figure 5.19d) and reducing the aggregate load by less than 5% (Figure 5.19e), line temperatures were brought to within their limits. For k ∈ [75, 240], storage discharge exceeded reference levels in order to bring SOC back to economical reference levels as displayed in Figure 5.19f. Wind curtailment was employed as cheap control over the longer term to bring and keep line temperatures below their limits.

Finally, with a shorter horizon M and the terminal constraint requiring greater use of expensive load and storage control, the MPC enjoys smaller departures in generation from the Level 1 economic reference. Such an outcome is displayed in the generation costs of Figure.5.19c. However, with a short horizon (e.g., M = 5), the controller is unable to predict the line overloads far enough in advance, which causes inferior management of energy storage and load. For example, notice in Figure.5.19f how much further away from the Level 1 reference the MPC with horizon M = 5 is compared with M ≥ 10. Considering the degree of load and storage control and line temperature profiles, a prediction horizon of M = 20 provides the best sacrifice between computational complexity and controller performance.

6.5 Data management and communication

The MPC control scheme requires a model of the network, together with measurements of the conductor temperature of (potentially) overloaded lines, SOC of energy-storage devices, output power from both conventional and renewable generation, power demand, and the operating points of all FACTS devices would be required. These data establish the initial point for the MPC prediction trajectory, and therefore must be updated every time MPC reinitializes, at the time step T_s. These measurement requirements are consistent with existing energy management system (EMS) capabilities, with topology processing establishing the network model, and state estimation providing generation and load information. Technology for measuring conductor temperature is available, though telemetry of such measurements is not currently common. It is argued in [14] and references therein, in the context of dynamic line rating, that gathering line temperatures is quite feasible. Also, a trivial modification to the MPC formulation would allow some lines to be subject to standard (hard) power-flow limits, while modeling temperature dynamics for lines that were outfitted with temperature sensors. Participation of energy-storage devices in electricity markets will likely require telemetry of their SOC. This is already the case in NYISO [43] and PJM [44].

In addition to well-defined initial conditions, MPC prediction also requires forecasts of demand, the power available from renewable generation sources, and the ambient weather conditions governing line temperatures. Generation and load forecasts are already available and used in EMS contingency analysis. Short-term weather forecasts are also typically available. Given that the MPC prediction horizon will generally be on the order of 15-30 minutes, a persistence forecast (which assumes those external influences remain unchanged) will often be adequate.

MPC broadcasts control signals at an interval of T_s ≈ 1 minute, which is much slower than other controls, such as AGC [45]. Thus, the input/output communications and data management requirements of the MPC scheme are consistent with the capabilities of existing EMS installations.