The nomenclature used in the formulation of the problem is detailed at the end of the paper. All electrical variables are represented in phasor form. Consider an electrical network whose topology is represented by a graph G = ( B , L ) , where B = { 1 , … , n } is the set of buses (vertices) and L ⊂ B × B is the set of lines (edges). The lines are unordered pairs of buses ( i , k ) , where i and k are the pair of buses connected by the line.

A bus k is adjacent to another bus i if there is a line connecting them, i.e., if ( i , k ) ∈ L . The set of buses adjacent to bus i is denoted by A i and is defined as follows: A i = k ∈ B ∣ i , k ∈ L . (1)

The state of the network variables is physically related (Alexander and Sadiku, 2013). First, the apparent power at each bus S i is expressed using Equation 2, where I i * is the complex conjugate of I i . S i = V i ⋅ I i * . (2)

I i is an aggregation of currents, as stated in Equation 3. I i = ∑ k ∈ A i I i k + I i k 0 . (3)

Both I i k and I i k 0 can be calculated using Ohm’s law, as shown in Equations 4, 5, respectively. I i k = y i k ⋅ V i − V k , (4) I i k 0 = b i k ⋅ V i 2 . (5)

Power flow equations are obtained combining all the previous expressions. In the case of AC buses, the resulting expressions are Equations 6–7 (Montes and Castro, 1995; Samperio, 2023). P i = ∑ k ∈ A i c i k ⋅ e i 2 + f i 2 − e i ⋅ e k − f i ⋅ f k + s i k ⋅ e i ⋅ f k − e k ⋅ f i + P conv,ik , (6) Q i = ∑ k ∈ A i c i k ⋅ e i ⋅ f k − e k ⋅ f i + s i k ⋅ − e i 2 − f i 2 + e i ⋅ e k + f i ⋅ f k + b i k ⋅ − e i 2 − f i 2 2 . (7)

P conv,ik , which is the power injected or transferred from converters, is added to the active power load flow equation. In the case of DC buses, the expressions are Equations 8, 9. P i = ∑ k ∈ A i c i k ⋅ e i ⋅ e i − e k + P conv,ik , (8) Q i = 0 . (9)

In any case, P i and Q i are the sum of generation and demand (Equations 10, 11). P i = P gen,i + P load,i , (10) Q i = Q gen,i + Q load,i . (11)

At all times, P gen,i and Q gen,i are limited to the nominal power and the minimum power of the generators (Equations 12, 13, respectively). P gen,nom,i ≥ P gen,i , ≥ P gen,min,i , (12) Q gen,nom,i ≥ Q gen,i . ≥ Q gen,min,i . (13)

If two buses are connected through a converter, the expression that describes the power exchange between them is given by Equation 14. P conv,ik = P conv,ki η i k . (14)

The maximum and minimum voltage limits of buses are maintained, as shown in Equation 15. V max,i 2 ≥ e i 2 + f i 2 ≥ V min,i 2 . (15)

When grid-forming mode is activated in a converter, the voltage at the output bus of the converter is set to its nominal value, as described in Equation 16. e i 2 + f i 2 = 1 . (16)

When the converter is in grid-following mode, this restriction is not considered.

The total current that lines can transport is limited by Equation 17. I max,ik 2 ≥ c i k ⋅ e i − e k + s i k ⋅ f k − f i 2 + c i k ⋅ f i − f k + s i k ⋅ e i − e k 2 . (17)

Transformers cannot exceed their nominal power when operating, as shown in Equations 18, 19. S n , i k 2 ≥ c i k ⋅ e i − e k + s i k ⋅ f k − f i 2 + c i k ⋅ f i − f k + s i k ⋅ e i − e k 2 ⋅ e i 2 + f i 2 , (18) S n , i k 2 ≥ c i k ⋅ e i − e k + s i k ⋅ f k − f i 2 + c i k ⋅ f i − f k + s i k ⋅ e i − e k 2 ⋅ e k 2 + f k 2 . (19)

In the case of converters, this limitation is expressed using Equation 20. S n , i k ≥ P conv,ik . (20)

Four different objective functions h can be chosen for minimization in the AC/DC optimal power flow, depending on the objective to be achieved.

1. H1: Total active power generated. This function focuses on reducing energy losses (Equation 21).

The variables that are optimized when performing the power flow are e i , f i , P gen,i , Q gen,i , and P conv,ik .

• KPI1: Electrical energy generated. This indicator represents the amount of electrical energy generated per year.

• KPI5: Total life cycle income. This indicator represents the total income earned by the microgrid over its useful life. This index includes income from energy production and flexibility management.

The AC/DC optimal power flow simulation has been implemented in Python due to its wide range of available open-source optimization libraries. In this paper, the optimization has been performed using CasADi (Andersson et al., 2019), which is a software library equipped with specific tools focused on the modeling, optimization, and control of nonlinear dynamic systems. CasADi is widely used to define both mathematical models and constraints involved and allows us to utilize different solvers in order to optimize the problem. In this study, the solver IPOPT (Wächter and Biegler, 2006) has been used. IPOPT applies sequential quadratic programming (SQP) to solve constrained nonlinear optimization problems, which is the case in the AC/DC optimal power flow. The calculation time is not significant: it is only a few seconds.

Four scenarios have been tested, one per objective function. In all scenarios, TIGON CEDER storage has been considered to act as a load that consumes electricity at half its nominal power ( P stor /2), and the external grid has been limited to power consumption (if required). The results obtained in each scenario are given in Table 5.

In all scenarios, all the power demanded by loads and storage is delivered. In none of them exists the demand for reactive power, so all the reactive power in the microgrid is generated in the AC section at bus 2 because of the transformer and AC line reactance.

In scenarios H1 and H3, the exact active power is generated to supply the demand and compensate for losses. The difference between them is that the wind turbine reduces its active power generation in scenario H3 because this technology has higher operational costs than the PV, whereas more power is saved in scenario H1.

In scenario H2, V i at all buses is the nominal value or very close to it. To achieve it, the generation of active power and the consumption of the external grid are precisely adjusted.

In scenario H4, the PV and wind turbine generate the maximum active power, which is then supplied to the external grid.

All the scenarios have been recreated in the real environment of the TIGON CEDER microgrid. The two quantities accessed for measurement are active power and voltage. The values obtained in the measurements taken for each scenario are shown in Table 6.

The measures obtained are very similar to the calculated values in the simulations, as shown in Figures 3, 4. In Table 7, the mean relative error (MRE) has been calculated for both active power and voltage using Equation Equation 33. M R E % = 100 N ⋅ ∑ i = 0 N − 1 | a ̂ i − a i | | a i | . (33)

The largest differences are observed between the expected and actually measured PV voltages, with a maximum relative error of 3.11%. This occurs because the PV voltage (bus 3) varies considerably as the delivered active power changes. Small differences in accuracy arise from the monitoring devices and the difficulty of precisely obtaining at the same time the proposed values of active power generation from both the wind turbine and PV due to their inherent variability depending on the weather conditions and technical restrictions.

The results confirm that the AC/DC optimal power flow works correctly and accurately enough to assess the operation of the microgrid.

The techno-economic assessment calculations have been implemented in Python. Again, the calculation time is very short, only a few seconds. Four scenarios in which the TIGON CEDER microgrid could operate have been studied: baseline, no battery, battery flexibility, and virtual battery (VB) flexibility.

In the baseline scenario, the microgrid elements have been considered with the same characteristics presented in Section 4. Nevertheless, in this situation, which is real, the battery is being used only for performing tests and research experiments. For this reason, we have considered three more scenarios in which the microgrid could be more profitable. In the no-battery scenario, the battery and its converter have been removed from the microgrid, along with their investment costs and residual values. In the battery flexibility scenario, the battery has not been eliminated, but it has been deemed to be used in the Spanish upward tertiary regulation market. Batteries are loads with inherent electrical flexibility as they can be charged and discharged at the most convenient time, keeping the daily balance of the generated and consumed electricity unchanged. The upward tertiary regulation market has been chosen because it is the balance market with the highest average price in Spain in 2022: 224.17 €/MWh ³ . The power of the battery is very small to participate in the Spanish tertiary regulation market on its own as it is necessary to make bids of at least 1 MW (Red Eléctrica de España, 2021). Therefore, it needs to be part of an aggregation that participates in the market as a unitary market agent, following the methodology explained by Martín-Crespo et al. (2023). Lastly, in the VB flexibility scenario, the battery has been replaced by a VB consisting of an aggregation of thermostatically controlled loads (TCLs) already in place, with the same nominal power P stor,i . These TCLs have the capacity to store thermal energy equivalent to a given electricity consumption. They could be heat pumps or water heaters, for instance. The aggregation of TCLs in the VB has been achieved using the method proposed by Martín-Crespo et al. (2021). This method is a control algorithm executed in real time and consists of three steps: Check of TCLs, Aggregation, and Priority Control. As a result of these stages, the controller decides which TCLs should be switched on or off to meet the stated power requirement. For both flexibility scenarios, storage participation in the market for 1 h per day is assumed to operate at its nominal power P stor,i and the aggregation shares its benefits proportionally between its loads. This results in an additional flexibility income F I of 2,045.55 € per year.

The KPI values obtained for each scenario are given in Table 8.

KPI2 (CO₂) proves that carbon dioxide emissions are very low. This is caused by the usage of renewable generation technologies, specifically PV and wind turbines. KPI3 (self-consumption percentage) shows that the microgrid produces more energy than required by the loads per year, thus compensating for energy losses.

KPI7 (payback) shows that in the baseline scenario, the microgrid is not profitable. It makes sense as the TIGON CEDER microgrid is a small-scale network where the elements are high-cost prototypes. The battery, which is the storage element of the grid, is key for increasing or decreasing costs and incomes. The battery flexibility scenario demonstrates that using the battery for making bids in the Spanish electricity balance markets enhances the incomes and makes the microgrid profitable. Nevertheless, the flexibility remuneration is not high enough to make the microgrid payback lower than that when the battery is removed, i.e., the no-battery scenario. In the no-battery scenario, costs are highly reduced, as well as in KPI8 (LCOE). This is at the expense of reducing KPI4 (storage flexibility) to zero. The most profitable scenario is VB flexibility as it combines the additional income obtained by participation in the tertiary regulation market with the cost savings caused by the disappearance of the physical battery. KPI7 (payback) is reduced to 17 years and KPI8 (LCOE) to 113.92 €/kWh, whereas KPI4 (storage flexibility) remains at 25 kW.