Multi-objective scheduling model

The proposed model is distinguished from existing models on two main counts.

First is how the degradation is handled given the non-linearity of degradation behaviour discussed before. Secondly, degradation is treated as a second objective along with revenue from market trading.

The multi-objective function to be maximized in given by Equation 9.1 where ζ is the fictitious composite objective.

9.5. Multi-objective scheduling model 143

Figure 9.4. Mapped degradation surface for 1 C and 2 C rates

ζ = ω · R − (1 − ω) · D (9.1)

The values of revenue R (e) and degradation D (A h) are scaled to be of the same order of magnitude. The weight ω is varied between 0 and 1 in a parametric sweep to generate Pareto solutions of scheduling strategies. The two objective functions are detailed in Equation 9.2 and Equation 9.3.

R =X

t

[λ_t· (P_t^dis,m− P_t^ch,m)] · ∆T ∀t (9.2) where

λ is the market clearing price, assumed to be a known parameter for the optimization (eW⁻¹h)

∆T is duration of a trading interval (h)

P_t^dis,m is the power supplied to the market in trading interval t (W)

P_t^ch,m is the power demanded from the market in one trading interval t (W) In this study, a single energy market is considered. However, multiple markets can be considered in this equation such as in in [262]. The proposed model can be readily embedded in other applications by replacing the primary objective function with, for e.g., the maximization of privacy or self-consumption.

The overall battery degradation D (A h) during the entire duration of the

Figure 9.5. Contour map of degradation for 1 C rate

decision making horizon is given by the sum of individual degradation components dt(A h) caused during each time interval as defined by Equation 9.3.

D =X

t

d_t ∀t (9.3)

The SOC of the storage system is updated in Equation 9.4 depending on whether charging or discharging takes place during a market interval.

SOCt= SOCt−1+(P_t^ch,b− P_t^dis,b) · ∆T

Vnom· Q ∀t > 1 (9.4) where

P_t^dis,b is the power supplied by the battery in trading interval t (W) P_t^ch,b is the power demanded by the battery in trading interval t (W) V_nomis the nominal voltage of the battery equal to 3.7 V

Q is the capacity of the battery equal to 2.15 A h

The lower bound of zero and the upper bound on the power capability of the battery are set through Equation 9.5 and Equation 9.6. Using the binary variable u_t, it also ensures that during any trading interval, the storage system either charges (u_t= 1) or discharges (u_t= 0).

9.5. Multi-objective scheduling model 145

0 ≤ P_t^ch,b≤ P^ch,max· ut ∀t (9.5)

0 ≤ P_t^dis,b≤ P^dis,max· (1 − ut) ∀t (9.6) where

P^ch,maxis the maximum input power to the battery (W) P^dis,maxis the maximum output power of the battery (W)

The process of energy conversion in the storage system is not 100 % efficient, and it is taken into account in Equation 9.7 and Equation 9.8. These equations distinguish the battery input/output power (subscript b) from that exchanged in the market (subscript m). A constant efficiency (η = 0.95) is assigned to the conversion and transmission processes. In reality, η is not a constant but it is also a factor of operating and environmental conditions. If this is factored in, it will introduce additional non-linearity and complexity in the work. However, given the overall high efficiencies of lithium-ion batteries (Table 1.1), it has been assumed a constant in this work.

P_t^ch,b= P_t^ch,m· η ∀t (9.7)

P_t^dis,b· η = P_t^dis,m ∀t (9.8)

The SOC of the battery (given in percentage) is constrained through Equation 9.9 because of its definition.

0 ≤ SOCt≤ 100 ∀t (9.9)

Since higher currents lead to greater degradation in batteries, the degradation for every market time interval can be expressed in a bilinear form in Equation 9.10 with the degradation at 1 C as a basis. The equation relates the actual degradation caused in the battery in a time interval, d_t(A h), to the degradation that will be caused in the battery for the same change of state but at 1 C current rate, d^1C_t (A h). It introduces through the scaling factor Ψ the non-linearity caused by the effect of current on degradation. Ψ is a function of the current rate, it (h⁻¹), which is calculated in Equation 9.11. In this equation, the current rate has been determined using a constant value of voltage (nominal battery voltage, Vnom). In reality, however, the voltage of the battery is not constant but a function of the state of charge. Including this dependency is outside the scope of this work as it will lead to additional non-linearity and computational burden.

dt= d^1C_t · Ψt ∀t (9.10)

it=(P_t^ch,b+ P_t^dis,b)

V_nom· I_1C ∀t (9.11)

where I_1C is the 1 C current equal to 2.15 A

The experimental tests on the cells under consideration have been conducted at two current values, at 1 C and 2 C. The values of Ψ are thus known for these two states. At no load conditions, no cycle aging takes place. Thus Ψ is equal to zero when the storage is idle. In this work, Ψ is assumed to scale linearly between these three currents in absence of more intermediate degradation values.

The scaling factor, Ψ, is equal to zero at no load conditions (0 C) as no cycle aging takes place when the battery is idle. Being based on 1 C current rate, it has the value of 1 at 1 C. From the data in Section 7.4.4, its value at 2 C is determined to be about 1.3. Since intermediate experiments at other current values were not carried out, it has been assumed that the parameters scale linearly between 0 C -1 C and -1 C - 2 C. For example, if the current rate is -1.5 C, the scaling factor is the average of 1 and 1.3.

Assuming a linear scaling factor in the absence of more precise data for intermediate current values can be a reasonable assumption because increasing current causes increasing degradation in the battery. Thus the scaling factor will be a monotonically increasing function of current. Accuracy can be improved by adding the intermediate values of the scaling factor if experimental results for more current values are available. This can be achieved without any change in the structure of the developed model.

In order to determine d^1C_t for each change in battery state, the experimental data on degradation is first expressed in cumulative form as shown in Figure 9.6.

The cumulative degradation function, δ^1C_t (A h), is made up of n segments (where n=10). This cumulative degradation function (δ_t^1C) does not have a physical significance unlike d^1C_t . It is a mathematical tool, conceived in order to implement the non-linear degradation behaviour represented in Figure 9.4, in the framework of MILP. Using the cumulative function facilitates modelling as degradation during a time interval (d^1C_t ) can be simply determined as a difference of the cumulative degradation values before and after the trading interval. This is expressed in Equation 9.12.

d^1C_t = |δ^1C_t − δ_t−1^1C| ∀t (9.12) The value of δ_t^1C at any time interval is determined using the incremental cost formulation [263, 264]. This formulation allows accessing a piecewise linear function such as the one of Figure 9.6 in a MILP. To implement the formulation at any trading interval t, the piecewise linear cumulative degradation function of Figure 9.4 is specified by the points (st,i, δ^1C(st,i)) ∀i ∈0, ..., n where n is the total number of segments. Thus, st,i is the x-coordinate and δ^1C(st,i) is the y-coordinate of the points highlighted in Figure 9.6.

9.5. Multi-objective scheduling model 147

Figure 9.6. Cumulative degradation function. Annotations to clarify the implementation of the incremental cost formulation (Equations 9.13-9.17) are shown in a lighter shade of

blue.

Next let,

lt,i = st,i - st,i−1 and mt,i = δ^1C(st,i) - δ^1C(st,i−1) ∀ i.

Any value of the state of charge SOC_t such that s_t,0 ≤ SOC_t ≤ s_t,n can be written as in Equation 9.13.

SOCt= st,0+

Then the cumulative degradation function value for time interval t and SOCt

is given by Equation 9.14.

δ_t^1C= δ^1C_t (s0) +

if the following holds true:

v_t,i < l_t,i, v_t,i+1 = 0 ∀i ∈1, ..., n − 1 .

This can be forced in the MILP through the following constraints and introducing the binary variables zt,i, i ∈1, ..., n − 1 .

lt,1· zt,1≤ vt,1≤ lt,1 (9.15)

lt,i· zt,i ≤ vt,i≤ lt,i−1· zt,i−1 ∀i ∈2, ..., n − 1 (9.16)

0 ≤ v_t,i≤ lt,i· zt,n−1 (9.17) The product of d^1C_t and Ψtwhich gives the actual degradation dtis calculated using following identity:

d^1C_t .Ψ_t= d^1C_t + Ψ_t 2

2

− d^1C_t − Ψ_t 2

2

(9.18) These two squared terms are approximated by piecewise linear formulations.

To determine values of the squared terms as well as Ψtfrom their piecewise linear functions, incremental cost formulation is used, similar to the implementation in Equations 9.13 - 9.17. This method introduces n-1 binaries and 2n constraints where n is the number of segments of the piecewise linear function. The absolute value in Equation 9.12 is determined by adding the inequalities of Equation 9.19 and Equation 9.20 to the model:

|δ^1C_t − δ_t−1^1C| ≥ δ_t^1C− δ^1C_t−1 (9.19)

|δ^1C_t − δ_t−1^1C| ≥ δ_t−1^1C − δ^1C_t (9.20) Since maximizing the objective function penalizes degradation, this formulation using the two inequalities computes the absolute value of δ_t^1C-δ^1C_t−1 which is equal to d^1C_t .