Vivi

Vivi is a framework for energy integration optimization. It contains structures and functions to facilitate the selection/sizing of technologies (process and utilities), under constraints of mass, energy and heat cascade (pinch analysis) balances.

Optimization problem

Vivi will write a linear optimization problem as:

$$ \max_{\gamma^\omega,in_i,out_i,R_k} \left [ \sum_o^{O} out_o \cdot v_o - \sum_i^{I} in_i \cdot v_i \right ] $$

$s.t.$

$$ in_i + \sum_{\omega}^{\Omega} \gamma^\omega\left (T_{out,i}^{\omega} - T_{in,i}^{\omega} \right) = out_i \text{ } \forall \text{ } i \in \text{Resources} $$

$$ R_{k-1} + \sum_{\omega}^{\Omega} \left (\gamma^{\omega} \sum_n^N \dot Q_{n,k}^{\omega} \right) = R_{k} \text{ } \forall \text{ } k \in \text{ Heat cascade}$$

In which,

$$ \gamma^\omega \geq 0 $$

$$ in_i \begin{cases} \geq 0 &\text{if } in_i = Inf \\ = in_i &\text{otherwise} \end{cases} $$

$$ out_i \begin{cases} \geq 0 &\text{if } out_i = Inf \\ = out_i &\text{otherwise} \end{cases} $$

$$ R_k \begin{cases} = 0 &\text{if } k = 0 \text{ or } k = K \\ \geq 0 &\text{otherwise} \end{cases} $$

Name	Description
$in_i$ and $out_i$	amount of resource input and output "i", respectively
$v_i$	specific value of resource "i"
$\gamma^\omega$	size factor for technology $\omega$
$T_{in,i}^{\omega}$ and $T_{out,i}^{\omega}$	input and output amount of resource "i" for technology $\omega$, respectively
$R_k$	Net heat of temperature interval "k" in Heat cascade
$Q_{n,k}^\omega$	Heat transfer of stream "n" in temperature interval "k" for technology $\omega$
$i$ and $I$	input number and total number of inputs, respectively
$o$ and $O$	output number and total number of outputs, respectively
$n$ and $N$	stream number and total number of streams, respectively
$k$ and $K$	temperature interval number and total number of temperature intervals, respectively

In simple terms, vivi will determine the set of size factors $(\gamma^\omega)$, inputs $(in_i)$ and outputs $(out_o)$ that optimize the value of outputs, discounting the inputs value, and respecting the balance of resources and heat cascade (first and second constraint, respectively). If the amount of a certain input or output is fixed (e.g., $in_i\neq Inf$), this information it is also accounted as an additional constraint.

Note: "value" can be a monetary figure, energy, exergy, carbon, amoung others.

How to use?

See the jupyter notebook "tutorial.ipynb" in the folder "examples"

Citation

Nakashima, R. N. (2022). Modelling, simulation and optimization of biogas conversion routes integrated with fuel cell technology. Doctoral Thesis, Escola Politécnica, University of São Paulo, São Paulo. https://doi.org/10.11606/T.3.2022.tde-26082022-081436

Bibtex

@phdthesis{nakashima_modelling_2022,
    address = {São Paulo},
    title = {Modelling, simulation and optimization of biogas 
    conversion routes integrated with fuel cell technology.},
    url = {https://www.teses.usp.br/teses/disponiveis/3/3150/tde-26082022-081436/},
    language = {en},
    urldate = {2022-09-05},
    school = {Universidade de São Paulo},
    author = {Nakashima, Rafael Nogueira},
    month = feb,
    year = {2022},
    doi = {10.11606/T.3.2022.tde-26082022-081436},
}

Contact

For doubts, comments and requests:

Rafael Nogueira Nakashima (rafnn@dtu.dk)

References

Papoulias, S. A., & Grossmann, I. E. (1983). A structural optimization approach in process synthesis—II. Computers & Chemical Engineering, 7(6), 707–721. https://doi.org/10.1016/0098-1354(83)85023-6

Marechal, F., & Kalitventzeff, B. (1996). Targeting the minimum cost of energy requirements: A new graphical technique for evaluating the integration of utility systems. Computers & Chemical Engineering, 20, S225–S230. https://doi.org/10.1016/0098-1354(96)00048-8

Maréchal, F., & Kalitventzeff, B. (1997). Effect modelling and optimization, a new methodology for combined energy and environment synthesis of industrial processes. Applied Thermal Engineering, 17(8–10), 981–992. https://doi.org/10.1016/S1359-4311(96)00079-8

Kemp, I. C. (2006). Pinch Analysis and Process Integration. Elsevier. https://doi.org/10.1016/B978-0-7506-8260-2.X5001-9