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Fig. 8. Custom-built experimental setup, where (1) Li-ion battery, (2) cooling fan, (3) current and voltage sensor, (4) acquisition board, (5) battery surface temperature senso (6) environment temperature sensor, (7) power supply and (8) security relay.

Figure 8 Custom-built experimental setup, where (1) Li-ion battery, (2) cooling fan, (3) current and voltage sensor, (4) acquisition board, (5) battery surface temperature senso (6) environment temperature sensor, (7) power supply and (8) security relay.

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Besides the main reaction, Li-ion batteries are prone to side reactions that may lead to capacity/power fade and can even com- promise the safe operation of the battery [1,22]. Degradations might have different origins, which could be correlated and are dependent on the specific chemistry of the battery [22]. Neverthe- less, once identified, such mechanisms can be modeled as operating constraints. The most relevant sources of degradation that have been identified for a variety of electrode chemistries include side reactions at the solid-electrolyte electrodes interfaces, and elec- trode phase changes [23]. Interestingly enough, the occurrence of these phenomena strongly depends on the state of operation of the battery and can be avoided by ensuring the following constraints: where C; and c;; are the average and surface lithium concentra- tion, respectively, whose normalization by the maximum allowed lithium concentration Csmax are lower and upper bounded by Te and r,,, respectively, and 7s, and Us; are the overpotential and the equilibrium potential of the side reactions, respectively. Eq. (7a) and (7b) hinders the situation where Li is extracted/deposited beyond the maximum concentration allowed by the electrode, which can cause phase transformations, active material dissolu-
Besides the main reaction, Li-ion batteries are prone to side reactions that may lead to capacity/power fade and can even com- promise the safe operation of the battery [1,22]. Degradations might have different origins, which could be correlated and are dependent on the specific chemistry of the battery [22]. Neverthe- less, once identified, such mechanisms can be modeled as operating constraints. The most relevant sources of degradation that have been identified for a variety of electrode chemistries include side reactions at the solid-electrolyte electrodes interfaces, and elec- trode phase changes [23]. Interestingly enough, the occurrence of these phenomena strongly depends on the state of operation of the battery and can be avoided by ensuring the following constraints: where C; and c;; are the average and surface lithium concentra- tion, respectively, whose normalization by the maximum allowed lithium concentration Csmax are lower and upper bounded by Te and r,,, respectively, and 7s, and Us; are the overpotential and the equilibrium potential of the side reactions, respectively. Eq. (7a) and (7b) hinders the situation where Li is extracted/deposited beyond the maximum concentration allowed by the electrode, which can cause phase transformations, active material dissolu-
Main DFN model equations* 4 Spatial and temporal dependency of state variables not explicitly indicated here for compactness. > Linearized equation [21]. © Assuming ao =0.5.
Main DFN model equations* 4 Spatial and temporal dependency of state variables not explicitly indicated here for compactness. > Linearized equation [21]. © Assuming ao =0.5.
A common simplification of the DFN model is to abstract each electrode as a single spherical particle (the so-called single-particle model framework) [1], as the one shown in Fig. 2(a). On the basis of this abstraction several models have been derived in the litera- ture. One of the most interesting ones is the equivalent-hydraulic model (EHM) proposed in [18]. This model, depicted in Fig. 2(b), allows one to interpret the transportation phenomena within one electrode as the mass flow between two tanks with levels q; and q2,
A common simplification of the DFN model is to abstract each electrode as a single spherical particle (the so-called single-particle model framework) [1], as the one shown in Fig. 2(a). On the basis of this abstraction several models have been derived in the litera- ture. One of the most interesting ones is the equivalent-hydraulic model (EHM) proposed in [18]. This model, depicted in Fig. 2(b), allows one to interpret the transportation phenomena within one electrode as the mass flow between two tanks with levels q; and q2,
and I(t) is the input current. The resulting EHM is a simple linear time-invariant second-order model. Yet, this model is still able to reproduce the main dynamic behavior of an ECheM with a typically small model mismatch when compared with the DFN model [25]. The state equation (9) is complemented by the output equation representing the voltage difference between the battery electrodes given by At this point the dynamics of the battery can be summarized by the state of the negative electrode The model can be further reduced by exploiting the fact that the dynamics of the positive electrode are much faster than the ones of the negative one. As such, it is reasonable to assume that the diffusion within the positive electrode is instantaneous (CSC*(t)=SOC*(t)) and it is linked to the state of the negative elec- trode through the algebraic equation SOC*(t)=— pSOC(t)+o, with pando as positive constants [25]. More details on the EHM can be found in [25].
and I(t) is the input current. The resulting EHM is a simple linear time-invariant second-order model. Yet, this model is still able to reproduce the main dynamic behavior of an ECheM with a typically small model mismatch when compared with the DFN model [25]. The state equation (9) is complemented by the output equation representing the voltage difference between the battery electrodes given by At this point the dynamics of the battery can be summarized by the state of the negative electrode The model can be further reduced by exploiting the fact that the dynamics of the positive electrode are much faster than the ones of the negative one. As such, it is reasonable to assume that the diffusion within the positive electrode is instantaneous (CSC*(t)=SOC*(t)) and it is linked to the state of the negative elec- trode through the algebraic equation SOC*(t)=— pSOC(t)+o, with pando as positive constants [25]. More details on the EHM can be found in [25].
Fig. 3. Proposed RG feedback control scheme. The closed-loop system consists of a state feedback controller (K and K.) with an integrator (Ac, Bc, Xc). I(t) is the current generated by the state feedback controller and the block Cy is used to select the “measured” SOC(t) in the vector x(t). The outer loop implements the RG scheme that uses the current state of the system (Aq, Bg, x) and the desired reference r(t) to generate the admissible reference v(t).
Fig. 3. Proposed RG feedback control scheme. The closed-loop system consists of a state feedback controller (K and K.) with an integrator (Ac, Bc, Xc). I(t) is the current generated by the state feedback controller and the block Cy is used to select the “measured” SOC(t) in the vector x(t). The outer loop implements the RG scheme that uses the current state of the system (Aq, Bg, x) and the desired reference r(t) to generate the admissible reference v(t).
Fig. 4. Battery cell operating region obtained from the cell static characteristic, where the green and red areas are the admissible and non-admissible regions, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Battery cell operating region obtained from the cell static characteristic, where the green and red areas are the admissible and non-admissible regions, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where X,)(€|Xqj, Vv) = A“Xel +(1-Ag) 1 = Ao )B,\v represents the ¢ step ahead prediction given the initial state x,.; and applying the constant reference v. The set of all the initial states x,; and applied references v for which (25) is satisfied is denoted as the maxi- mal output admissible set O,,. In general, the computation of 0, involves an infinite number of constraints. However, if the con- straints (24) define a compact set, for any arbitrarily small ¢>0 it is possible to compute a finite integer £° (see [30]) such that the finitely-computable set 0,, is an inner approximation of O,,, where All the matrices and the vectors defined above have the same num- ber of rows Nc “cc x (€ +2). Note that Ox can be made arbitrarily close to O. by reducing ¢.
where X,)(€|Xqj, Vv) = A“Xel +(1-Ag) 1 = Ao )B,\v represents the ¢ step ahead prediction given the initial state x,.; and applying the constant reference v. The set of all the initial states x,; and applied references v for which (25) is satisfied is denoted as the maxi- mal output admissible set O,,. In general, the computation of 0, involves an infinite number of constraints. However, if the con- straints (24) define a compact set, for any arbitrarily small ¢>0 it is possible to compute a finite integer £° (see [30]) such that the finitely-computable set 0,, is an inner approximation of O,,, where All the matrices and the vectors defined above have the same num- ber of rows Nc “cc x (€ +2). Note that Ox can be made arbitrarily close to O. by reducing ¢.
Values for the constraints (22). Table 3 considered to take into account the model mismatch between the EHM and the DFN model. Extensive numerical simulations have demonstrated that for the reported values, the satisfaction of con- straints (22) in (17) and (18) always ensures the satisfaction of constraints (7) on the DFN model, even in presence of a bounded uncertainty on the state reconstruction.
Values for the constraints (22). Table 3 considered to take into account the model mismatch between the EHM and the DFN model. Extensive numerical simulations have demonstrated that for the reported values, the satisfaction of con- straints (22) in (17) and (18) always ensures the satisfaction of constraints (7) on the DFN model, even in presence of a bounded uncertainty on the state reconstruction.
Fig. 5. DFN model critical conditions for 7%. at the positive electrode/separator interface (electrode dependent), 7; = 0.0 V at the negative electrode/separator interface and Ce=1molm-~? at the current collector/negative electrode interface (common for both batteries).
Fig. 5. DFN model critical conditions for 7%. at the positive electrode/separator interface (electrode dependent), 7; = 0.0 V at the negative electrode/separator interface and Ce=1molm-~? at the current collector/negative electrode interface (common for both batteries).
Fig. 6. Current and voltage profiles from the DFN simulator using CCCV and the proposed RG method for LCO and LMO.
Fig. 6. Current and voltage profiles from the DFN simulator using CCCV and the proposed RG method for LCO and LMO.
Fig. 7. Profiles of the constrained variables from the DFN simulator using CCCV and the proposed RG method for LCO and LMO.
Fig. 7. Profiles of the constrained variables from the DFN simulator using CCCV and the proposed RG method for LCO and LMO.
Fig. 9. In (a), DFN critical conditions for 7; =0.0 V at the negative electrode/separator interface. In (b), current profile using CCCV and the proposed RG method for Turnig oattery.
Fig. 9. In (a), DFN critical conditions for 7; =0.0 V at the negative electrode/separator interface. In (b), current profile using CCCV and the proposed RG method for Turnig oattery.
EHM parameters for the Turnigy battery. Table 4 Table 6 Table 5
EHM parameters for the Turnigy battery. Table 4 Table 6 Table 5
Values for the constraints (22) in the implementation. 5. Experimental results
Values for the constraints (22) in the implementation. 5. Experimental results
Experimental comparison of the proposed RG against commercial CCCV charging strategies.
Experimental comparison of the proposed RG against commercial CCCV charging strategies.
Fig. 10. Voltage and SOC profiles using CCCV and the proposed RG method for Turnigy battery.
Fig. 10. Voltage and SOC profiles using CCCV and the proposed RG method for Turnigy battery.
Related topics:
Chemical EngineeringComputer ScienceConstrained controlState of ChargeElectrochemical Model
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