Augmented Lagrangian approach for multi-objective topology optimization of energy storage flywheels with local stress constraints

Abstract

Flywheel energy storage systems (FESS) used in short-duration grid energy storage applications can help improve power quality, grid reliability, and robustness. Flywheels are mechanical devices that can store energy as the inertia of a rotating disk. The energy capacity of FESS rotors can be improved by choosing the optimal rotor geometry, operation conditions, rotor materials, and by tailoring the material properties. A multi-objective formulation is presented in this article to simultaneously improve the energy capacity and reduce the weight of energy storage flywheels using stress-constrained topology optimization to determine the material placement and optimize the stress distribution in the rotor. A Pareto-front of optimal solutions to the multi-objective problem demonstrates that the rotors with the best specific energy content have volume fractions of 55-65%. Local stress constraints with an Augmented Lagrangian formulation are introduced to improve the stress distribution and optimal design and are compared to designs obtained with global stress constraints based on P-norm aggregation approach. We show that designs with local stress constraints have a more uniform stress distribution and fewer stress concentrations compared to stress using P-norm.

Appendices

Appendix 1: Sensitivity analysis for the Augmented Lagrangian function

The adjoint form is used to analytically determine the gradient of AL function defined in Eq. (5). The sensitivity of ${\mathcal {L}}_{\mu ^{(k)}}(\hat{\varvec{\rho }}, \textbf{u})$ w.r.t design variable $\hat{\rho }_e$ (using the adjoint approach) is as follows:

$$\begin{aligned} \frac{\textrm{d} {\mathcal {L}}_{\mu ^{(k)}}}{\textrm{d} \hat{\rho }_e} = \frac{\partial {\mathcal {L}}_{\mu ^{(k)}}}{\partial \hat{\rho }_e} + \xi ^T\left( \frac{\partial K}{\partial \hat{\rho }_e} \textbf{u} - \frac{\partial F}{\partial \hat{\rho }_e} \right) \end{aligned},$$

(A.1)

where the adjoint vector $\xi$ is computed using the definition:

$$\begin{aligned} \xi = - (K^T)^{-1} \frac{\partial {\mathcal {L}}_{\mu ^{(k)}}}{\partial \textbf{u}} \end{aligned}.$$

(A.2)

Here, the partial derivative, $\frac{\partial {\mathcal {L}}_{\mu ^{(k)}}}{\partial \hat{\rho }_e}$ used in Eq. (A.1) is calculated using the definition of ${\mathcal {L}}_{\mu ^{(k)}}(\varvec{\hat{\rho }}, \varvec{u})$ in Eq. (5),

$$\begin{aligned} \frac{\partial {\mathcal {L}}_{\mu ^{(k)}}}{\partial \hat{\rho }_e}&= \frac{\partial f^{(k)}(\varvec{\hat{\rho }})}{\partial \hat{\rho }_e} + \frac{1}{N}\sum _{j=1}^N\nonumber \\&\quad \left[ \left( \lambda _j^{(k)} + \mu ^{(k)} h_j^{(k)}(\varvec{\hat{\rho }},\varvec{u}) \right) \frac{\partial h_j^{(k)}(\varvec{\hat{\rho }},\varvec{u})}{\partial \hat{\rho }_e} \right] \end{aligned}.$$

(A.3)

The first partial derivative term in Eq. (A.3), $\frac{\partial f^{(k)}(\varvec{\hat{\rho }})}{\partial \hat{\rho }_e}$, is defined as follows:

$$\begin{aligned} \frac{\partial f^{(k)}(\varvec{\hat{\rho }})}{\partial \hat{\rho }_e} = -\left( \frac{w_{ke}}{E_{\textrm{kin}}^0}\right) \frac{\partial E_{\textrm{kin}}(\varvec{\hat{\rho }})}{\partial \hat{\rho }_e} + \left( \frac{w_m}{m^0}\right) \frac{\partial m(\varvec{\hat{\rho }})}{\partial \hat{\rho }_e} \end{aligned},$$

(A.4)

where the partial derivatives of the kinetic energy $E_{\textrm{kin}}$ and mass m are calculated using their definitions in Eq. (4).

$$\begin{aligned} \frac{\partial E_{\textrm{kin}}}{\partial \hat{\rho }_e}&= \frac{1}{2} \frac{\partial I(\hat{\varvec{\rho }})}{\partial \hat{\rho }_e} \omega ^2 = \frac{1}{2} \nonumber \\&\quad \left[ (p \hat{\rho }_e^{p-1} \rho _0) \left\| x_{e,c}^{2} + y_{e,c}^{2} \right\| v_e \right] \omega ^2 \nonumber \\ \frac{\partial m(\hat{\varvec{\rho }})}{\partial \hat{\rho }_e}&= (p \hat{\rho }_e^{p-1} \rho _0) v_e \end{aligned}.$$

(A.5)

The second partial derivative term in Eq. (A.3), $\frac{\partial h_e^{(k)}}{\partial \hat{\rho }_e}$, using the definition of local stress constraints in Eq. (7), can be written as follows:

$$\begin{aligned} \frac{\partial h_e^{(k)}}{\partial \hat{\rho }_e} = {\left\{ \begin{array}{ll} 0, &{} \text { if } g_e^{(k)}(\varvec{\hat{\rho }},\varvec{u}) < -\frac{\lambda _e^{(k)}}{\mu ^{(k)}} \\ q \hat{\rho }_j^{q-1} \frac{\sigma _{\textrm{vm},e}}{\sigma _{y}}, &{} \text { otherwise } \end{array}\right. } \end{aligned}.$$

(A.6)

Next, in order to calculate the adjoint vector in Eq. (A.2), the term $\frac{\partial {\mathcal {L}}_{\mu ^{(k)}}}{\partial \textbf{u}}$, with the small deformation assumption ($\frac{\partial f^{(k)}(\varvec{\hat{\rho }})}{\partial \textbf{u}} = 0$), is defined as follows:

$$\begin{aligned} \frac{\partial {\mathcal {L}}_{\mu ^{(k)}}}{\partial \textbf{u}} =\frac{1}{N}\sum _{j=1}^N \left( \lambda _j^{(k)} + \mu ^{(k)} h_j^{(k)}(\varvec{\hat{\rho }},\varvec{u}) \right) \frac{\partial h_j^{(k)}}{\partial \textbf{u}} \end{aligned}.$$

(A.7)

The term $\frac{\partial h_j^{(k)}}{\partial \varvec{u}_j}$ in Eq. (A.7) is defined as follows:

$$\begin{aligned} \frac{\partial h_j^{(k)}}{\partial \varvec{u}_j} = {\left\{ \begin{array}{ll} 0, &{} \text { if } g_j^{(k)}(\varvec{\hat{\rho }},\varvec{u}) < -\frac{\lambda _j^{(k)}}{\mu ^{(k)}} \\ \frac{1}{\sigma _{y}} \hat{\rho }_j^{q} \frac{\partial \sigma _{vm,j}}{\partial \textbf{u}_j}, &{} \text { otherwise} \end{array}\right. } \end{aligned},$$

(A.8)

where $\frac{\partial \sigma _{vm,j}}{\partial \textbf{u}_j}$ is determined using the definition of the von Mises stress at the cell centroid, as seen in Eq. (9):

$$\begin{aligned} \frac{\partial \sigma _{vm,e}}{\partial \textbf{u}_e} = \frac{1}{\sigma _{vm,e}} \textbf{B}^T \textbf{C} \textbf{V} \sigma _{e}^{(c)} \end{aligned}.$$

(A.9)

The terms $\frac{\partial K}{\partial \hat{\rho }_e}$ and $\frac{\partial F}{\partial \hat{\rho }_e}$ in Eq. (A.1) are, respectively, based on the definitions of the local stiffness matrix and the local forcing terms using the SIMP power law:

$$\begin{aligned} \frac{\partial \textbf{k}_{ij}^e}{\partial \hat{\rho }_e}&= p \hat{\rho }_e^{p-1} \textbf{k}_{ij}^{0} , \nonumber \\ \frac{\partial \textbf{f}_i^{e}}{\partial \hat{\rho }_e}&= p \hat{\rho }_e^{p-1} \textbf{f}_i^{0} \end{aligned},$$

(A.10)

where $k_{ij}^{0}$ and $f_i^0$ are, respectively, the original local stiffness matrix and local forcing vector for the solid material.

Appendix 2: Non-dominated optimal designs for the multi-objective Augmented Lagrangian formulation

The non-dominated optimal solutions to the multi-objective flywheel design problem obtained with different objective function weights are recorded in Table 3, which indicates the kinetic energy, mass, specific energy, maximum von Mises stress, volume fraction, and measure of non-discreteness for each design.

Table 3 Comparison of non-dominated optimal rotor designs obtained with different objective function weights for the multi-objective AL formulation