**4. Conclusions and future work**

This chapter has presented a detailed review of the interval-based simulation techniques and their application to the analysis and design of DSP systems. First, the main extensions of the traditional IA have been explained, and AA has been selected as the most suitable arithmetic for the simulation of linear systems. MAA has also been introduced for the analysis of nonlinear systems, but in this case it is particularly important to keep the number of noise terms of the affine forms under a reasonable limit.

Second, three groups of experiments have been performed. In the first group, a simple IIR filter has been simulated using IA and AA to detail the causes of the oversizing of the IAbased simulations, and to determine why AA is particularly well suited to solve this problem. In the second group, different deterministic traces have been simulated using intervals of different widths in some or all the samples. This experiment has revealed the most sensitive frequencies to the small variations of the signals. In the third group, the effect of including intervals in the computation of the statistical parameters using the Monte-Carlo method has been studied. Thanks to these experiments, it has been shown that intervalbased simulations can reduce the number of samples of the simulations, but the edges of the distributions are softened by this type of processing.

Finally, it is important to remark that interval-based simulations can significantly reduce the computation times in the analysis of DSP systems. Due to their features, they are particularly well suited to perform rapid system modeling, verification of the system stability, and fast and accurate determination of finite wordlength effects.


*Article* **Slow Mode-Based Control Method for Multi-Point Temperature Control System**

**Song Xu 1,2,†,‡, Seiji Hashimoto 1,2,\*,‡, Wei Jiang 2,‡, Yuqi Jiang 1,‡, Katsutoshi Izaki 3, Takeshi Kihara 3 and Ryota Ikeda 3**

Received: 10 July 2019; Accepted: 12 August 2019; Published: 14 August 2019

**Abstract:** In recent years, thermal processing systems with integrated temperature control have been increasingly needed to achieve high quality and high performance. In this paper, responding to the growing demands for proper transient response and to provide more accurate temperature controls, a novel slow-mode-based control (SMBC) method is proposed for multi-point temperature control systems. In the proposed method, the temperature differences and the transient response of all points can be controlled and improved by making the output of the fast modes follow that of the slow mode. Both simulations and experiments were carried out, and the results were compared to conventional control methods in order to verify the effectiveness of the proposed method.

**Keywords:** slow-mode-based control; multi-input multi-output (MIMO) temperature system; transient response; temperature differences


3.3.2. Wind Turbine Model

A generic dynamic wind turbine model was used to calculate the expected power output in the selected location using the following [49]:

$$P(t) = \frac{1}{2} \mathbb{C}\_p \rho(t) A V^3(t) (\eta\_m \cdot \eta\_\varepsilon) \tag{7}$$

where *Cp* is the power coefficient, *ρ*(*t*) is the air density at the hub height, *A* is the selected swept area in *m*2, *V*(*t*) is the wind speed in m/s at the time step *t*, and *ηm* and *ηe* are the mechanical and electrical efficiencies. The wind speed is usually measured at a different height compared to the hub height, *Zhub*. Therefore, the model uses the logarithmic law to derive the hourly wind speed at the hub height as follows [49]:

$$V\_{hub} = V\_{amcm}(\frac{\text{In}(Z\_{hub}/Z\_0)}{\text{In}(Z\_{amcm}/Z\_0)})\tag{8}$$

where *Z*0 is the surface roughness length (m), *Zanem* is the anemometer height (m), *Vhub* is the wind speed at the required hub height (m/s), and the *Vanem* is the measured wind speed at the anemometer height (10 m for the dataset used). For simplicity, *Cp* is evaluated by way of a 2D-look up table based on the four classes of wind turbines described in IEC 61400 standard [50]. The average wind speed and distribution was evaluated and the most appropriate characteristic was chosen from the four available classes ranked from low to high wind speeds [51]. The normalised power range for each class of "Offshore", "IEC-1", "IEC-2", and "IEC-3" are shown in Figure 5a. The energy output over the course of one year can also be determined analytically by assessing the wind speed distribution. The Rayleigh distribution, shown in Figure 5b, has been overlaid to show that the wind speed distribution data follow this statistical law, which indicates that the normalised power curves will operate effectively for the model.

**Figure 5.** Wind model input assumptions are primarily a combination of standardised wind turbine power coefficients and the load wind speed data measured at 10 m above sea level.

#### 3.3.3. Lithium Ion Battery Model

The battery model utilises a simplified version of the Shepard battery model [52], replacing internal and other resistive losses with a total charge *ηcharge* discharge *ηdischarge* efficiency for the hourly discharge case. The simplification allows for less information to be known about the chemistry and dynamics of the specific battery to perform calculations for the current capacity and state of charge (SOC). The battery system contains two parts: a charge model and a discharge model. The models take the power requirement from the battery and output the resulting SOC for the end of the timestep. These parts are defined as follows:

$$\begin{cases} \text{SOC}(t+1)\_{batt} = \frac{Q(t)\_{batt} + \int P(t)\_{batt} \eta\_{charg\epsilon'} dt}{Q(t\_0)\_{batt}} \cdot 100 & \text{charging} \\ \text{SOC}(t+1)\_{batt} = \frac{Q(t)\_{batt} - \int P(t)\_{batt} \eta\_{charg\epsilon'} dt}{Q(t\_0)\_{batt}} \cdot 100 & \text{discharging} \end{cases} (9)$$

where SOC*t*+1,*batt* is the next timestep battery SOC, *Qt*,*batt* is the battery state of charge at timestep *t*, *Qt*0,*batt* is the initial SOC, *Pt*,*charge* is the average charge power draw, and *Pt*,*discharge* is the discharge power draw. These outputs are subject to the minimum and maximum SOC limits *SOCmin* and *SOCmax*. The model includes degradation in the battery capacity linearly as a function of charge cycles, as shown below:

$$Q(l,t)\_{batt} = Q(t\_0)\_{batt} - \varkappa l \tag{10}$$

where *Q*(*l*, *t*)*batt* is the dynamic capacity in kWh as a function of cycles the cycles *l*, and *α* is the ageing factor (kWh/cycle).

#### 3.3.4. Regenerative Hydrogen Fuel Cell

The RHFC model provides an alternative energy storage facility to the electrochemical battery. The model consists of a PEM fuel cell and PEM electrolyser capable of consuming and producing hydrogen, respectively. The system also considers a hydrogen storage module with its own rated capacity and efficiency. The overall equations are like that of the simplified battery model in that the electrolyser and fuel cell analogously represent the charge and discharge elements. The system can therefore be shown as the following:

$$Q(t+1)\_{H2} = Q(t)\_{H2} - \int P(t)\_{fc} \eta\_{fc'} dt \qquad \text{Fuel cell} $$

$$Q(t+1)\_{H2} = Q(t)\_{H2} + \int P(t)\_{el} \eta\_{el'} dt \qquad \text{Electrolysis}$$

where *Q*(*t* + 1)*H*2 is the next timestep hydrogen energy stored (kWh), *Q*(*t*)*H*2 is the current timestep hydrogen energy stored (kWh) *P*(*t*)*fc* is the average fuel cell power production [kW] in the current one-hour timestep *t*, and *P*(*t*)*el* is the average electrolyser power consumption [kW]. *ηfc* and *ηel* are the average lifetime fuel cell and electrolyser efficiencies [%], respectively. Like the battery, these energy values are also subject to *QH*2,*min* and *QH*2,*max* limits.

#### 3.3.5. Model Input Assumptions

Table 2 contains the necessary input assumptions for the energy models, including efficiencies and other system dynamics that determine the output power generated or stored. The PV panel characteristics are based on the Sunpower Maxeon panel series, while the wind turbine is an approximation of common small-scale turbine systems on the market. The roughness length assumption of 0.05 is defined as rural, farmland area with low crops and without many trees [53]. The hydrogen system efficiency values are based on industry knowledge gathered from leading European fuel cell and electrolyser manufacturers.

**Table 2.** Hybrid renewable energy system design input assumptions across the different included technologies.

#### *3.4. Energy Management Strategy*

The energy management strategy for the hybrid storage system is shown in Figure 6. When generation supply is available in excess of demand, the battery charges first, followed by the larger capacity hydrogen storage via the electrolyser. When the demand outgrows the supply, the battery discharges first, followed by the activation of the fuel cell. In practical terms, the battery is actually being charged by the fuel cell while active, as the fuel cell cannot module its output without incurring performance losses. The charge and discharge states are is shown in Figure 6.

**Figure 6.** Energy management strategy of the hybrid storage system.

Virtual trading was used to fairly satisfy the community members based on the shared energy available. The excess energy available is shared equally, satisfying each load in ascending order of magnitude. This means that it is more likely that a member's electricity demand will be fully satisfied if smaller. It should be noted that this algorithm can be modified to suit any location-specific REC policy.

#### *3.5. Economic and Environmental Indicators*

A selection of three different system configurations: the best economic outcome, best environmental outcome, and a midpoint configuration between the two would be assessed in the model. It is important from a financial perspective to understand the investment requirements and expected returns for prospective REC members. The net present value (NPV) is commonly employed to determine economic feasibility, as well as the internal rate of return (IRR), simple return [%], payback period [years], and levelized cost of electricity (LCOE) for energy specific cases. Generally, if the NPV is positive compared to the base scenario, the investment is worthwhile [54].

$$\text{NPV} = \sum\_{n=1}^{N} \frac{\mathbb{C}\_{\text{Od\&M,n}} + \mathbb{C}\_{f,n}}{\left(1 + R\right)^{n}} - \text{Co} \tag{12}$$

where *CO*&*M*,*t* is the operation and maintenance cashflow for year *t*, *Cf* ,*t* is the fuel input cashflow, *R* is the discount rate, and *C*0 is the initial capital investment. It was assumed that any grid consumption is included in *Cf* ,*t* in units of EUR/year. The capital requirement and operating cashflows are summed for each generation and storage asset to solve for the system NPV. The IRR evaluates the rate of return if the NPV is set to zero, at which point the project breaks even.

$$\text{NPV} = \sum\_{n=1}^{N} \frac{C\_n}{\left(1 + IRRR\right)^n} \tag{13}$$

Calculating the LCOE is beneficial when assessing the economic feasibility of different technologies. The LCOE was evaluated against the grid cost to assess the cost savings per unit of electricity which could be expected by the community members. LCOE is defined as the total cost or lifetime cost of the asset divided by the total electricity delivered to the consumer [55].

$$\text{LCOE}(/\text{kWh}) = \frac{\sum\_{n=1}^{N} \mathbb{C}\_{0,n} + \mathbb{C}\_{\text{O\&M,n}} + \mathbb{C}\_{f,n}}{\sum\_{n=1}^{N} \mathbb{E}\_n} \tag{14}$$

where *C*0,*n* is the capital cost of the asset, and *En* is the lifetime energy delivered. A range of different assessments exist for the economics of renewable assets, as it is highly dependent on the capital requirement, location, delivery and installation cost, and available labour among other factors. The resulting CAPEX, OPEX, and lifetime parameters are shown in Table 2. The costs include the balance of plant (BOP), such as DC-AC inverters and IoT control equipment. The project has an assumed discount rate R of 5% and an estimated inflation rate of 2% per year, as well as a year-one electricity grid unit cost of 0.30 EUR/kWh for each building. Where the asset lifetime is less than 20 years, the asset is retired and the cost of a new equivalent system was included in the NPV assessment in that given replacement year. This method assumes the BOP cost is relatively low.

The environmental impact was estimated through the global warming potential (GWP) of the assets, which when summed together and divided by the total energy delivered over the system lifetime derives the emissions intensity, measured in *extgCO*2*eext*/*kWh*. The values are then compared with the grid emissions intensity for the island, for which the total decarbonisation potential was evaluated. The grid emissions were found using generation data gathered from the national TSO (Red Electrica de Espana) for the year 2021 and found to have an average of 325 gCO2*e*/kWh.

$$EI\_{total} = \frac{\sum\_{j=1}^{m} \left( EI\_j \cdot E\_j \right)}{\sum\_{j=1}^{m} \left( E\_j \right)} \tag{15}$$

*EIj* is the emissions intensity and *Ej* is the energy output for *m* number of generators and energy storage systems. This calculation was performed for each timestep of the simulation to find the dynamic emissions value depending on the instantaneous energy mix of the REC. The emissions intensity found within the literature can vary due to the range of manufacturing techniques and factors considered when performing the life cycle assessment (LCA). For this reason, some values such as those used for the hydrogen system are taken as an educated estimation of the emissions impact based on a variety of sources. The GWP embedded during manufacturing and installation for the assets and technology costs are shown in Table 3.

**Table 3.** Hybrid renewable energy system economic and climate impact assumptions for the different modelled technologies.