Technology and capacity planning for the recycling of lithium-ion electric vehicle batteries in Germany

Abstract

Currently, the German government strives for establishing Germany as a lead market for electric mobility. A successful growth of the electric vehicle market would be followed by a corresponding volume of spent lithium-ion batteries containing valuable and scarce metals. With their recycling, the high environmental load and energy use of primary production could be reduced, import dependencies could be diminished, and economic value could be created. Pioneering the mandatory deployment of an appropriate recycling network in Germany, the aim of this paper is to determine investment plans for the installation of recycling plants of different technologies and capacities and to derive recommendations for potential investors. For this purpose, a mathematical optimisation model is developed, allowing the economic selection of recycling technologies and capacities to be deployed in the network over time. In a comprehensive study, the impact of uncertain developments and mandatory minimum recycling rates on the timing of plant installations, on technology and capacity selection, on material flows, and on the profitability of recycling are analysed. The main finding is that despite the prevailing uncertainties a robust plan exists at least for initial decisions. It is shown that the recycling network can be operated highly economically in four out of five scenarios while meeting mandatory minimum recycling rates easily. By contrast, achieving both profitability and the minimum recycling rates will be a challenge if less valuable metals are used in the production of batteries, requiring the development of additional recycling steps to both reduce waste and increase profitability.

Appendix: Mathematical formulation of the optimisation model for technology and capacity planning

The model will be explained in two steps: Firstly, all symbols used are defined in Tables 9, 10 and 11. Decision variables are emphasised for convenience. Secondly, the objective function and the constraints of the model are formulated and explained.

Table 9 Sets and indices of the model

Table 10 Decision variables of the model

Table 11 Parameters of the model

The objective is to maximise the net present value of the cash flows in the planning horizon. For reasons of clarity and comprehensibility, the objective function (2) is split: The first summand of the objective function $DCF_{m}^{Invest}$ (3) represents the aggregate discounted cash flows related directly to the installation and operation of all units of modules of type $m$. For that, the corresponding decision variables $\varvec{x}_{{\varvec{m},\varvec{t}_{\varvec{B}} ,\varvec{t}_{\varvec{E}} }}$ are multiplied with module-specific discounted cash flows, which include initial expenses for installation and start-up, potential renewal expenses, fixed expenses for operation, and liquidation revenues. The latter factor is expressed by the net present values $p_{{m,t_{B} ,t_{E} }}^{inv}$ for all modules and all possible intervals of operation in the planning horizon, which have to be calculated by means of a predefined logic. While reducing the complexity of the model, this approach enables high flexibility such as pre-setting multi-periodic investment profiles, excluding certain combinations by using prohibitively high (negative) values, or permitting renewal investments to operate modules longer than their (initial) physical life.

The second summand of the objective function consists of three cash-flow terms, each being related to a specific period and discounted by multiplication with the discount rate $d_{t}$. The three terms depict cash flows caused by tactical decisions with regard to the recycling programme. The first one, $CF_{t}^{Operating}$ (4), is the aggregate sum of variable operation expenses over all activities executed in all modules. The second one, $CF_{t}^{Transport}$ (5), is the aggregate expense for the collection of products from the sinks and for the delivery of intermediates between modules. The third one, $CF_{t}^{Factors}$ (6), is the aggregate cash flow for the purchase of products and supplies and for the sale and disposal of recyclables and residues.

$$Max\ NPV = \mathop \sum \limits_{m} DCF_{m}^{Invest} + \mathop \sum \limits_{t} (CF_{t}^{Operating} + CF_{t}^{Transport} + CF_{t}^{Factors} ) \cdot d_{t}$$

(2)

with

$$DCF_{m}^{Invest} = \mathop \sum \limits_{{t_{B} }} \mathop \sum \limits_{{t_{E} \ge t_{B} }} p_{{m,t_{B} ,t_{E} }}^{inv} \cdot \varvec{x}_{{\varvec{m},\varvec{t}_{\varvec{B}} ,\varvec{t}_{{\mathbf{E}}} }}\quad \forall \ m \in M,$$

(3)

$$CF_{t}^{Operating} = - \mathop \sum \limits_{m} \mathop \sum \limits_{a} p_{t,m,a}^{op} \cdot\varvec{\lambda}_{{\varvec{t},\varvec{m},\varvec{a}}} \quad \forall \ t \in T,$$

(4)

$$CF_{t}^{Transport} = - \left( {\mathop \sum \limits_{{f \in F_{P} }} a_{t,f} \cdot q_{t,f} \cdot \mathop \sum \limits_{i} \varvec{z}_{{\varvec{t},\varvec{i}}} \cdot p_{f,i,t}^{coll} + \mathop \sum \limits_{{m_{S} }} \mathop \sum \limits_{{m_{D} \in {\text{M}}{ \setminus }\{ m_{S} \} }} \mathop \sum \limits_{{f \in F_{P} \mathop \cup \nolimits F_{M} }} \varvec{y}_{{\varvec{t},\varvec{m}_{\varvec{S}} ,\varvec{f},\varvec{m}_{\varvec{D}} }}^{{\varvec{ship}}} \cdot p_{f,t}^{ship} } \right) \quad \forall \ t \in T,$$

(5)

$$CF_{t}^{Factors} = \mathop \sum \limits_{m} \mathop \sum \limits_{f} \left( {\varvec{y}_{{\varvec{t},\varvec{m},\varvec{f}}}^{{\varvec{in}}} \cdot p_{t,f}^{in} + \varvec{y}_{{\varvec{t},\varvec{m},\varvec{f}}}^{{\varvec{out}}} \cdot p_{t,f}^{out} } \right)\quad \forall \ t \in T.$$

(6)

The solution space of the model is constrained by the following side-conditions. Beyond these, non-negativity and binary constraints exist that correspond to the number systems and decision variables described above.

Product collection rate and availability constraint (7): the collection rate given for every product and period must be achieved exactly. The constraint assures that the aggregate input of products into the system (left-hand side of the equation) is equal to the rate of products that has to be collected (right-hand side).

$$\sum\limits_{m} {\varvec{y}_{{\varvec{t},\varvec{m},\varvec{f}}}^{{\varvec{in}}} } = a_{t,f} \cdot q_{t,f} \quad \forall \ t \in T ,\ f \in F_{P}$$

(7)

Acceptance constraint (8): product or intermediate $f$ may be an input into a module $m$ only if specified by $c_{m,f}^{acc}$. $BigM$ is a sufficiently large number, e.g. the upper limit of the underlying data type (signed doubleword: $2^{31} - 1$).

$$\varvec{y}_{{\varvec{t},\varvec{m},\varvec{f}}}^{{\varvec{in}}} + \mathop \sum \limits_{{m_{D} \in {\text{M}}{ \setminus }\{ m_{S} \} }} \varvec{y}_{{\varvec{t},\varvec{m}_{{\mathbf{S}}} ,\varvec{f},\varvec{m}_{\varvec{D}} }}^{{\varvec{ship}}} \le c_{m,f}^{acc} \cdot BigM\quad \forall \ t \in T,\ m_{S} \in M,\ f \in (F_{P} \mathop \cup \nolimits F_{M} )$$

(8)

Material-flow constraint (9): each factor that is input into a module has to be purchased externally (first term on the left-hand side of the equation) or delivered from another module (second term) and is consumed (supplies like energy) or is transformed into another factor analogously to the activities executed in the module (third term). The corresponding output is either delivered to other modules (first term on the right-hand side; only intermediates) or sold or disposed of externally (second term; intermediates, recyclables, and residues).

$$\varvec{y}_{{\varvec{t},\varvec{m},\varvec{f}}}^{{\varvec{in}}} + \mathop \sum \limits_{{m_{S} \in {\text{M}}{ \setminus }\{ m\} }} \varvec{y}_{{\varvec{t},\varvec{m}_{\varvec{S}} ,\varvec{f},\varvec{m}}}^{{\varvec{ship}}} + \mathop \sum \limits_{a}\varvec{\lambda}_{{\varvec{t},\varvec{m},\varvec{a}}} \cdot v_{m,a,f} = \mathop \sum \limits_{{m_{D} \in {\text{M}}{ \setminus }\{ m\} }} \varvec{y}_{{\varvec{t},\varvec{m},\varvec{f},\varvec{m}_{\varvec{D}} }}^{{\varvec{ship}}} + \varvec{y}_{{\varvec{t},\varvec{m},\varvec{f}}}^{{\varvec{out}}} \quad\forall\ t \in T,\ m \in M,\ f \in F$$

(9)

No short-circuits constraint (10): a factor may not be input and output at the same time in a module.

$$\varvec{y}_{{\varvec{t},\varvec{m},\varvec{f},\varvec{m}}}^{{\varvec{ship}}} = 0\quad \forall\ t \in T,\ m \in M,\ f \in F$$

(10)

Minimum recycling-rate constraint (11): the aggregate mass of factors that are rated as recyclables ($F_{R}$) and are brought to sinks outside the system boundary must be at least as high as the mass of products collected multiplied by the mandatory recycling-rate in that period, $r_{t}$.

$$\mathop \sum \limits_{{f \in F_{R} }} \mathop \sum \limits_{m} m_{f} \cdot \varvec{y}_{{\varvec{t},\varvec{m},\varvec{f}}}^{{\varvec{out}}} \ge r_{t} \cdot \mathop \sum \limits_{{f \in F_{P} }} \mathop \sum \limits_{m} m_{f} \cdot \varvec{y}_{{\varvec{t},\varvec{m},\varvec{f}}}^{{\varvec{in}}} \quad\forall\ t \in T$$

(11)

Capacity constraint (12): the capacity required for the execution of activities in a certain period and module type (left-hand side of the equation) may not exceed the capacity of the module that is installed at the same time (right-hand side). On the left-hand side, the intensity of every activity that is executed in the specified module $m$ and in the specified period $t$ is multiplied with its corresponding capacity coefficient to obtain the aggregate capacity utilisation. On the right-hand side, the capacity installed at the specified period is determined by the double sum. For that, the number of modules that have been activated previously or simultaneously ($t_{B} \le t$) and that have not been deactivated previously ($t_{E} \ge t$) is multiplied with the technical capacity of each module.

$$\mathop \sum \limits_{a} c_{m,a} \cdot\varvec{\lambda}_{{\varvec{t},\varvec{m},\varvec{a}}} \le \mathop \sum \limits_{{t_{B} \le t}} \mathop \sum \limits_{{t_{E} \ge t}} \varvec{x}_{{\varvec{m},\varvec{t}_{\varvec{B}} ,\varvec{t}_{\varvec{E}} }} \cdot c_{m}^{max}\quad \forall\ t \in T, \ m \in M$$

(12)

Minimum operating-time constraint (13): where appropriate, a pre-set minimum time of operation of a module $t_{m}^{min}$ has to be satisfied. Firstly, all intervals of operation are excluded that are shorter than the pre-set number of periods: let $t_{B}$ be the first period of operation and $t_{E}$ the last—so that $\varDelta t = t_{E} - t_{B} + 1$ is the operating time—then the decision variable $\varvec{x}_{{\varvec{m},\varvec{t}_{\varvec{B}} ,\varvec{t}_{\varvec{E}} }}$ may be greater than 0 only if $t_{E} - t_{B} + 1 \ge t_{m}^{min}$, or $t_{E} \ge t_{B} + t_{m}^{min} - 1$. Formulated as a constraint, all intervals shorter than that are excluded, so that $\varvec{x}_{{\varvec{m},\varvec{t}_{\varvec{B}} ,\varvec{t}_{\varvec{E}} }}$ is 0 for all $t_{E} \in \left\{ {t_{B} .. \left( {t_{B} + t_{m}^{min} - 2} \right)} \right\}$. Secondly, this rule is disabled for all intervals that end in the last period of the planning horizon ($\bar{t}$) by using a minimum function. This is to avoid the effect that investments are antedated only to achieve the minimum operating time, which would eventually occur due to the limited planning horizon.

$$\varvec{x}_{{\varvec{m},\varvec{t}_{\varvec{B}}, \varvec{t}_{\varvec{E}} }} = 0\quad\forall \ m \in M, \ t_B \in T, \ t_{E} \in \{ t_{B} ..{ \hbox{min} }(t_{B} + t_{m}^{min} - 2;\bar{t} - 1)\}$$

(13)

Collection cost level selection constraint (14): the collection cost level selected in a specified period must equal the number of installed collecting modules, which is determined analogously to (12) at the right-hand side of the equation. On the left-hand side, the index of the collection cost levels $i$ is multiplied with the binary collection cost level variable $\varvec{z}_{{\varvec{t},\varvec{i}}}$, which is 1 if and only if the collection level $i$ is selected.

$$i \cdot \varvec{z}_{{\varvec{t},\varvec{i}}} = \mathop \sum \limits_{{m \in M_{c} }} \mathop \sum \limits_{{t_{B} \le t}} \mathop \sum \limits_{{t_{E} \ge t}} \varvec{x}_{{\varvec{m}, \varvec{t}_{\varvec{B}} ,\varvec{t}_{\varvec{E}} }}\quad \forall \ t \in T, \ i \in I$$

(14)

Collection cost level uniqueness constraint (15): exactly one collection cost level must be selected in every period.

$$\mathop \sum \limits_{i} \varvec{z}_{{\varvec{t},\varvec{i}}} = 1\quad \forall \ t \in T$$

(15)