Transactive Control of Electric Railways Using Dynamic Market Mechanisms

A wide range of traction system architectures and technologies have been developed for electrified railway systems, across low frequency (DC, 16.6Hz, 20 Hz and 25Hz) as well as industrial frequency (50Hz and 60Hz) networks which are powered by overhead and third rail traction distribution systems. Additionally, electric trains have been both powered by rail-side generators as well as grid-tied systems. In some scenarios, electrical connections to a large-scale electricity distribution or transmission system are complemented with distribution lines that travel along the rail, and can be designed such that they improve the reliability of the traction system by providing redundant power supplies to the traction substations feeding the traction system. These various architectures have a direct impact on the flexibility of the demand of the train and therefore a possible energy cost reduction.

A typical architecture of energy procurement and dispatch that occurs along the electric railway is provided in Fig. 1 and will be adopted for the discussions in this paper. This architecture is typically adopted in deregulated electricity markets such as Amtrak in the Northeastern United States, and will be used in the case study presented in Section 6. That is, we assume that Railway Operators face delivery charges as distribution-level customers of electric utilities which own and operate the distribution system. However, due to their energy requirements and the regulatory setting in which they procure energy, they could access wholesale markets for electrical supply. Additionally, we assume that third-parties could invest in rail-side DERs, seeking to provide energy services to the Railway Operator. Competitive retail sales of electricity from power marketers [18] and on-site Power Purchase Agreements (PPAs) offered by Energy Services COmpanies (ESCOs) [19] are both established and growing energy procurement mechanisms for commercial companies including electric railways.

With the architecture chosen as in Fig. 1, we now introduce a problem formulation for energy procurement and dispatch for optimal grid-railway interaction. A first component of this problem formulation is the Area Control Centers(ACCs) (see Fig. 1) managed by a Railway Operator. An ACC is charged with serving the electric railway traction system, limited to a portion of contiguous electric railway with the property that marginal injections or demands for power have the same cost to the operator for all time $t$. Along this portion of the track, all rail-side DERs and trains that interface with the traction system are enabled to provide price and quantity information regarding their dispatch, and are compensated and charged based on their actions by the ACC. The Railway Operator manages the ACC with the objective of reducing overall operational costs while maximizing the value of the DERs along the track and meeting all thermal and electric loads in the system. That is, the overall objective of the railway dispatch is to schedule energy resources, which includes trains, so as to minimize the cost of energy resources and trains along the railway system. Such an optimization has to be carried out subject to the various constraints of the grid and railway networks.

More formally, the underlying problem is the dispatch of electric railway power systems across $N$ ACCs for a time horizon $\tau=\left[t_{0},t_{f}\right]$ in an optimal manner. This includes the electric traction loads of $L_{n}$ trains, with power profiles denoted as $P_{n}^{l}$ for the electric profile of train $l$ in $ACC_{n}$ over the time interval $\left[t_{n,0}^{l},t_{n,f}^{l}\right]$. The Railway Operator is charged for its electric loads at a rate $\lambda_{n}\left(t\right)$.

On the generation side, we consider $D_{n}$ dispatchable generators (including DERs and electric utility imports) at each $ACC_{n}$ with electric and thermal power profiles $P_{n}^{d_{e}}$ and $P_{n}^{d_{th}}$. For compactness, we will use $P_{n}^{d}$, a tuple of the electric and thermal generation for each generator $d$. These agents incur a cost $C_{n}^{d}\left(P_{n}^{d}\right)$, which is private to their owner and operator.

All other electric and thermal power demand in the electric railway system, such as the thermal conditioning and lighting loads at passenger stations, are considered price-inelastic and are included within the power balance constraints of the problem but are not included in the objective function due to their fixed nature.

Given that the utility of the trains and generators are fixed, the social welfare maximization problem, which is commonly used for economic dispatch problems in power grids, reduces to a cost minimization problem with two terms in the objective function, the generator cost of generating the electricity and the energy cost of operating the trains.

With the above definitions, the overall grid-railway optimization can be posed thus:

It is clear that the objective function in Eq. (1) is composed of the sum of the cost of the dispatchable generators (which we will denote as agents) and the energy expenses of the electric trains. These two costs are the fundamental building blocks of the two part railway dynamic market mechanism ($rDMM$) developed in Sections 3-5. When broken down into the individual ACCs, the first term captures the railway dispatch problem with fixed loads, which in turn is subject to power balance and agent capacity limit constraints (2)-(4). The second term corresponds to the train dispatch problem, and is in turn subject to train dynamics and scheduling constraints (5)-(10).

The optimization problem as in (1)-(10) is difficult to solve, highly intractable, and poses several challenges. First, its solution requires that all of the traction system agents share their private cost information $C\left(P_{n}^{d}\right)$ to appropriately capture and minimize the overall system cost. This feature would likely inhibit private investment in rail-side DERs. The timescale of electric train dispatch may not individually align with that of the $ACC_{n}$ dispatch. Trajectory optimization of electric railway power profiles occurs in the seconds timescale whereas energy asset dispatch is unlikely to be used at a timescale faster than 5 minutes. Additionally, the trains are entering and exiting each of the $ACC_{n}$ at different times which are dependent on the pricing signal. All of these cause an intricate coupling between spatial and temporal constraints that are nonlinear and convex. In an effort to simplify the process and make the problem more tractable, we propose a two step approach that iterates between the electric railway and the train dispatch problems in Sections 3 - 5.

Within each of the railway segments $n$ the Railway Operator must meet the electrical and thermal demand or load during the next $M$ future dispatch intervals which are indexed as $K=\{1,...,M\}$. Moreover, in managing the electrical system of the railway, the Railway Operator is tasked with dispatching and compensating the electrical agents or assets within $ACC_{n}$ in a way that minimizes the total cost of the operation. The electric railway dispatch problem described in this section determines optimal dispatch for each future time interval; however, the pricing and dispatch is only binding and executed for $K=1$ after the Railway Dispatch problem has been solved.

We define five types of dispatchable agents within the railway power system at each node $n$ that are classified into the following sets: heating assets (e.g. boilers), $\mathcal{H}_{n}$; electric generation assets (e.g. fuel cell, microturbines), $\mathcal{E}_{n}$; cogeneration assets (e.g. combined heat and power units), $\mathcal{C}_{n}$; storage assets (e.g. batteries), $\mathcal{S}_{n}$; low voltage side network connections (i.e. points of common coupling), $\mathcal{N}_{n}$. The set of all dispatchable agents at node $n$ is denoted as $\mathcal{A}_{n}\triangleq\mathcal{H}_{n}\cup\mathcal{E}_{n}\cup\mathcal{C}_{n}\cup\mathcal{N}_{n}$. It is assumed that within each dispatch interval $K$ there are two faster timescales, where the first corresponds to instances $j\in\{1,...,j^{**}\}$, at each of which forecast updates of all non-dispatchable loads and generation are received. Between each $[j,j+1]$, we introduce a faster timescale $k\in\{1,...,k^{*}\}$ where at each instance $k$, all dispatchable agents negotiate electric and thermal generation schedules and prices (see Fig. 2). It is assumed that these timescales with $j^{**}$ and $k^{*}$ are such that they are nested, and that the time-intervals permit a useful forecast data and sufficient negotiations, respectively.

In addition to the above agents, we also consider three types of non-dispatchable agents at each node $n$, classified into the following sets: renewable generators, ${re}_{n}$; electrical loads (e.g. lighting at the passenger station), ${e}_{n}$; and thermal loads (e.g. heating of the passenger stations), ${th}_{n}$. The set of these non-dispatchable agents at node $n$ is denoted as $\mathcal{F}_{n}\triangleq{re}_{n}\cup{e}_{n}\cup{th}_{n}$, all of whom inject and demand electric and thermal energy from the same network as the dispatchable agents but are not dispatched by the Railway Operator. Instead, the operators in charge of each of these passive resources (i.e. the renewable asset operator for ${re}_{n}$ and the passenger station operator for $e_{n}$ and ${th}_{n}$) communicate the best load estimate over $ACC_{n}$’s future time intervals $K$. This forecast update occurs every $j$ when new updates arrive for the renewable assets, or any of the loads. The periods associated with the time scales $k$ and $j$ need to be such that it should accommodate new information related to the forecast and at the same time, the convergence rates of the negotiations.

Lastly, we consider electric trains $l=\{1,...,L\}$ injecting and demanding electric power from $ACC_{n}$. We denote the set of all trains at node $n$ as $\mathcal{T}_{n}$. For the purpose of the Railway Dispatch problem, the trains are no different from the set of passive agents $\mathcal{F}_{n}$ in that they provide an update to their electric power profile for the dispatch intervals $K=\{1,...,M\}$ at each forecast instance $j$. However, as shown below, once we define the Train Dispatch problem faced by the train operators for each train in Section 4 and the composition of the transactive controller in Section 5, the train update will transform from a simple forecast update to a price-dependent minimum cost dispatch update.

The overall goal of Railway Dispatch is to arrive at pricing information that can yield optimal setpoints for generators and utility power import. Over the time horizon of $M$ dispatch intervals, the electric and thermal power profiles for each of the agents considered (dispatchable, non-dispatchable, and trains) is assumed to take positive (generation) and negative (load) values.

and over the multi-periods as

For the agents representing low voltage side network connections $\mathcal{N}$, the cost function can be updated as a function of the equilibrium price in an external market, such as a wholesale energy market. Labeling the external market price for the low voltage side network connections as $\pi^{\mathcal{N}}_{n,j}$, the cost function parameters $a_{i,K}$, $b_{i,K}$, and $c_{i,K}$ in (14) are determined for $i\in\mathcal{N}$ at each forecast instance $j$. Given that these market prices commonly represent marginal prices, the cost function parameters may be simply chosen as: $a_{i,K}=0$, $b_{i,K}=\pi^{i}_{n,j}$, and $c_{i,K}=0$.

Each of the passive agents in $\mathcal{F}_{n}$ determine their power output for the dispatch intervals $K=\{1,...,M\}$ at each forecast instance $j$ (see Fig. 2), which are denoted by $\hat{P}^{re}_{j,K}$ for renewable generators, $\hat{P}^{e}_{j,K}$ for electric loads and $\hat{P}^{th}_{j,K}$ for thermal loads. We drop the subscript $K$ to denote the power output vector over the dispatch intervals in $\mathbb{R}^{M}$ as $\hat{P}^{re}_{j}$, $\hat{P}^{e}_{j}$, and $\hat{P}^{th}_{j}$. In order to determine the train demand, suppose that train $l$ traverses $ACC_{n}$ over the period $[t_{l,n-1},t_{l,n}]$ over the dispatch interval $K$ and $P_{l}(t,n)$ is the corresponding forecasted demand at instance $j$, we denote this demand as $P_{l,j,K}^{*}$. This in turn can be summed over all $L$ trains to yield

For ease of exposition, we drop the subscript $K$ in (16) and simply denote the total train demand at forecast instance $j$ as $\hat{P}_{j}^{T}$.

With the overall costs and constraints related to all agents specified as above, we state the Railway Dispatch problem at $ACC_{n}$ at a fixed forecast instance $j=\{1,...,j^{**}\}$ over the dispatch intervals $K=\{1,...,M\}$ in $\mathbb{R}^{M}$ as:

The output values for each dispatchable agent $i\in\mathcal{A}_{n}$ over the dispatch interval $K=\{1,...,M\}$ at dispatch instance $j$ are denoted as $y_{i,j}\in\mathbb{R}^{M}$ and constitute the decision variables of the problem. These decision variables are bound at each dispatch interval $K$ by the sum of the electric loads per (18), the thermal loads per (19), the maximum capacity constraint per (20) where $\overline{y_{i,j}}=min\{\overline{P_{i}^{e}}/d_{i}^{e},\overline{P_{i}^{th}}/d_{i}^{th}\}$ and the minimum capacity constraint per (21) where $\underline{y_{i,j}}=max\{\underline{P_{i}^{e}}/d_{i}^{e},\underline{P_{i}^{th}}/d_{i}^{th}\}$.

At each forecast instance $j=\{1,...,j^{**}\}$, the power profile for each renewable generator, $\hat{P}^{re}_{j}\in\mathbb{R}^{M}$, electric load, $\hat{P}^{e}_{j}\in\mathbb{R}^{M}$, thermal load, $\hat{P}^{th}_{j}\in\mathbb{R}^{M}$, and the total traction load, $\hat{P}_{j}^{T}\in\mathbb{R}^{M}$, are updated for the dispatch interval $K=\{1,...,M\}$. The new forecasts are used to update constraints (18) and (19), with the resulting optimization problem in (17)-(21) solved again.

With each forecast update $j=j+1$, the decision variables of the problem, denoted as $y_{i,j}^{*}$ optimize the cost in (17). For the dispatch intervals $K=\{1,...,M\}$ these decision variables can in turn be mapped to the electric and thermal output of the agents as $[g_{i}^{e}(y_{i,j}^{*}),g_{i}^{th}y_{i,j}^{*})]\in\mathbb{R}^{M}$.

The underlying optimization problem that the Railway Operator has to solve is therefore given by the solution of (17)-(21) for a given set of forecasted profiles. We propose that each $ACC_{n}$ solves this through an iterative negotiation process amongst the dispatchable agents within this $ACC$. This is proposed to be carried out by the faster timescales, $k=1,2,...k^{*}$ for each $j$. That is, at each forecast instance $j$ the negotiation process starts at $k=0$, where $\hat{P}^{re}_{j}$, $\hat{P}^{e}_{j}$, $\hat{P}^{th}_{j}$, and $\hat{P}_{j}^{T}$ are fixed, allowing the Railway Operator at $ACC_{n}$ to establish the electric and thermal loads for the dispatch horizon. And at $k=0$, it is assumed that the Railway Operator for $ACC_{n}$ broadcasts an initial price duple $\lambda_{n,j,k=0}=\left[\lambda^{e}_{n,j,0},\lambda^{th}_{n,j,0}\right]\in\mathbb{R}^{M}$ consisting of electric and thermal prices.

With these initial conditions, using a Lagrangian and a gradient-based update, the decision variables $y_{i,j}^{k}\in\mathbb{R}^{M}$ and the prices $\lambda^{e}_{n,j,k}\in\mathbb{R}^{M}$ and $\lambda^{th}_{n,j,k}\in\mathbb{R}^{M}$ are updated at each negotiation instance $k$ as:

where $\beta_{y_{i}},\beta_{\lambda^{e}},\beta_{\lambda^{th}}\in\mathbb{R}\;\forall\;i\in\mathcal{A}_{n}$ are positive step-size parameters and $\mu_{i,j}^{\pm}\in\mathbb{R}\;\forall\;i\in\mathcal{A}_{n}$ is the penalty for violating capacity constraints (20) and (21). The penalty function updates are given by:

It is assumed that these iterations occur at each $k$ and converge as $k\rightarrow k^{*}$ for some $k^{*}$. As outlined in [9], under suitable convexity conditions, it can be shown that convergence to unique optimal values takes place.

Defining $\hat{P}^{F}_{j}=[\hat{P}^{re}_{j}+\hat{P}^{e}_{j},\hat{P}^{th}_{j}]$ as the estimate of the fixed assets at $j$, these estimates can be updated with $j$ as the new forecasts arrive. The forecast update enables an improved dispatch of the agents across the dispatch intervals $K=\{1,...,M\}$.

Note that in practice, exit conditions based on the agent output and price profile updates within the negotiation process can be established such that the forecast update is promptly initiated. These exist conditions follow the form: $\left|y_{i,j}^{k+1}-y_{i,j}^{k}\right|\leq\gamma_{k}^{y}\;\&\;\left|\lambda_{n,k+1,j}-\lambda_{n,k,j}\right|\leq\gamma_{k}^{\lambda}$. If these conditions are met, the equilibrium agent output and price profiles can be set as $y_{i,j}^{*}=y_{i,j}^{k+1}\forall i\in\mathcal{A}_{n},$ and $\lambda_{n,j}^{*}=\lambda_{n,k+1,j}$ respectively.

Similarly, exit conditions can be established for the forecast update process, such that the agent output and price profile updates can be communicated for dispatch and settlement if so desired. These exit conditions follow the form: $\left|y_{i,j+1}^{*}-y_{i,j}^{*}\right|\leq\gamma_{j}^{y}\;\&\;\left|\lambda_{n,j+1}^{*}-\lambda_{n,j}^{*}\right|\leq\gamma_{j}^{\lambda}$. If these conditions are met, dispatch for the first interval $K=1$ can be established using agent output profiles $y_{i}^{**}=y_{i,j+1}^{*}\forall i\in\mathcal{A}_{n}$ and price profiles $\lambda_{n}^{**}=\lambda_{n,j+1}^{*}$.

In summary, the Railway Dispatch algorithm for $K=1$ starts at $k=1,j=1$ with a forecast of the power profile for each renewable generator, $\hat{P}^{re}_{j}\in\mathbb{R}^{M}$, electric load, $\hat{P}^{e}_{j}\in\mathbb{R}^{M}$, thermal load, $\hat{P}^{th}_{j}\in\mathbb{R}^{M}$, and the total traction load, $\hat{P}_{j}^{T}\in\mathbb{R}^{M}$, and returns the optimal agent output profiles $y_{i}^{**}$, the total traction demand profile $P^{T**}$ and the price profiles $\lambda_{n}^{**}$ that can be used for dispatch and settlement of dispatch interval $K=1$. The dispatch interval window then shifts over, with $K=2$ corresponding to the active dispatch interval and the process repeats.

The power consumption of the electric train depends on the traction force, which in turn depends on the overall train dynamics. In this section, we derive the underlying dynamic model of the train and the corresponding power consumption profile. The position, velocity and acceleration of train $l$ in the direction of motion $\boldsymbol{\hat{\i}}$ are denoted by $x_{l}$, $\dot{x_{l}}$, and $\ddot{x_{l}}$ respectively. We proceed by defining the three forces with a component in the direction of motion of the train $\boldsymbol{\hat{\i}}$ as it travels at an angle $\alpha_{l}(x_{l})$ from the horizon. The gravitational force on the train can be decomposed into the direction of motion $\boldsymbol{\hat{\i}}$ as $-m_{l}gsin(\alpha_{l}(x_{l}))$ and the direction normal to the ground $\boldsymbol{\hat{\j}}$ as $-m_{l}gcos(\alpha_{l}(x_{l}))$, where $m_{l}$ is the total mass of the train.

The electric motors converting electrical into mechanical power is assumed result in the traction force $F_{T,l}$ in the $\boldsymbol{\hat{\i}}$ direction. An opposing force $F_{DF,l}$ is also included in the model, which includes a drag force and a friction force [20]. With these definitions, we can derive the equation of motion in the direction of motion $\boldsymbol{\hat{\i}}$ as

The traction force $F_{T}l$ is a function of both the electric power $P_{l}$ and $\dot{x}_{l}$, and is in general a mapping of the ratio $P_{l}/\dot{x}_{l}$. The drag-friction force can be represented as

In the analysis henceforth, we consider the cost minimization problem of train $l$ that departs location $x_{0}$ at time $t_{0}$ and arrives at a final destination $x_{f}$ at time $t_{f}$. The train stops at passenger stations denoted by $s\in\{0,...,D\}$ between $t\in[t_{l,a}(s),t_{l,d}(s)]$, where $t_{l,a}(s)$ is the arrival time and $t_{l,d}(s)$ is the departure time from station $s$ and $x_{l,s}$ denotes the position of station $s$. And as indicated by (26), $x_{l}$ is determined by the train dynamics.

The train trajectory is assumed to traverse $n$ subsections, each dispatched by an Area Control Center, denoted as $ACC_{n},\forall n\in\{1,...,N\}$. It should be noted that these sections may not coincide with the railway sections between passenger stations. Section $n$ of the track is bound by the position interval $[x_{n-1},x_{n}]$ and, as mentioned earlier, train $l$ is assumed to travel within this section during the time interval $[t_{l,n-1},t_{l,n}]$ for $ACC_{n}$. The operator of train $l$ must choose the electric power demand profile given by the set of functions $P_{l}(t,n)$, over this interval, $\;\forall\;n\in\{1,...,N\}$. The power demand profile of train $l$ is therefore composed of the profiles at each ACC:

We assume that the railway is level at the passenger stations (i.e. $\alpha(x_{l,s})=0,\;\forall\;s\in{0,...,D}$) implying that the power demand of the train during the stop is equal to zero, $P_{l}(t)=0\in[t_{l,a}(s),t_{l,d}(s)],\;\forall\;s\in{0,...,D}$.

For each ACC, the operator of train $l$ faces the price of energy given by the set of functions $\lambda_{l}(t,n)\;\forall\;t\in[t_{l,n-1},t_{l,n}],\;\forall\;n\in\{1,...,N\}$. The price of energy for train $l$ is therefore composed of the profiles at each ACC:

Note that these profiles are a function of time as well as the $ACC$ traversed by the train. The price faced by the operator of the train can be determined from the Railway Dispatch problem equilibrium prices $\lambda_{n}^{**}$ as follows. Suppose we consider the time horizon $[t_{l,n-1},t_{l,n}]$ over which train $l$ traverses $ACC_{n}$, and we chose an arbitrary time $t_{K}\in[t_{l,n-1},t_{l,n}]$, and suppose it corresponds to the $K$’th interval (for example, the time horizon corresponds to $K=1$ in the example shown in Fig. 2). Then $\lambda_{l}(t,n)$ corresponds to the $K$th element of $\lambda_{n}^{e**}\in\mathbb{R}^{M}$. A similar procedure can be utilized to determine $\lambda_{l}(t)$ for all intervals in (29). In practice, trains may need to preemptively terminate the portion of the schedule included in the Train Dispatch problem or use proxy pricing if they operate within a particular $ACC_{n}$ outside of the dispatch horizon included in the Asset Dispatch problem.

The energy cost minimization of the train can now be formulated as:

In the above, (30) represents the energy cost incurred by the train across all ACCs between $t_{0}$ and $t_{n}$. Constraint (31) enforces the train motion dynamics described in Section 4.1, rearranging (26) to solve for the traction force $F_{T,l}$ as a function of the position, velocity and acceleration of the train. The power demand of train $l$ from the traction system at each ACC, $P_{l}(t)$, is constrained by (32) which imposes a lower bound at $\underline{P_{l}}(x_{l})$ and an upper bound at $\overline{P_{l}}(x_{l})$. Note that the limits are functions of the ACC in question as the particular track segment might not be able to receive or provide more than a given power magnitude.

Next, we define the limits of the traction force $F_{T,l}$ in (33) reducing the feasible traction force window $[\underline{F_{T,l}}(\dot{x_{l}}),\overline{F_{T,l}}(\dot{x_{l}})]$ based on the traction force curve of the manufacturer. The acceleration rate of the train $\ddot{x_{l}}$ is limited per (34) due to safety considerations of the passengers, who may be standing during moments of deceleration $\underline{a_{l}}$ or acceleration $\overline{a_{l}}$. Similarly, the velocity of the train $\dot{x_{l}}$ is bound from below by $\underline{v_{l}}(x_{l})$ and from above by $\overline{v_{l}}(x_{l})$ through equation (35) where the limits are functions of the position $x_{l}$. This dependency traces back to civil speed limit restrictions which reduce the window of allowed speeds in sections of the track with rail crossings, densely populated areas and passenger stations. Finally, constraints (36) and (37) represent the schedule constraint of train $l$, achieved by limiting the time spent at the stations to the minimum arrival time $\underline{t_{l}}(s)\forall s\in\{0,...,f\}$ and the maximum departure time $\overline{t_{l}}(s)\forall s\in\{0,...,f\}$ respectively.

In summary, in this section we have posed the problem of optimizing the power consumption of a train for a time-schedule in the form of a non-convex constrained optimization problem in (30)-(37). The resulting solution is in the form of power demand profiles $P_{l}^{*}(t)$ for a train $l$ at time $t$ for a given set of price profiles $\lambda_{l}(t)$, constructed by determining the corresponding time interval $K$ and the corresponding equilibrium price $\lambda_{n}^{e**}$. The solution of the Train Dispatch problem may be determined using any one of several commercial software packages such as Matlab’s fmincon [22]. Note that the decision variables used by the optimization problem are position $x_{l}$ and velocity $\dot{x_{l}}$, the two state variables used to represent the dynamical system. Also, numerical solvers can be used to transform the continuous variables (including the decision variables)to discrete-time variables of length $(t_{f}-t_{0})\delta\tau$, where $\delta\tau$ is the time step.