Optimisation of the Structure of a Wind Farm—Kinetic Energy Storage for Improving the Reliability of Electricity Supplies

Abstract

An important issue in the correct operation of the power system is the reliability of the electricity supply from generation systems. This particular problem especially concerns renewable sources, the output power of which is variable over time and additionally has a stochastic character. The solution used in the work to improve the reliability indicators of wind farm sources is the partial stabilization of their output power achieved through cooperation with the kinetic energy storage. Excessive increase in storage capacity is associated with a large increase in investment and operating costs. It is therefore important to determine the minimum storage capacity required to maintain the accepted criteria for the reliability of energy supply. In this paper, a population meta-heuristics algorithm was used for this purpose. The obtained results confirm the possibility of limiting the energy capacity of the flywheels, they also indicate its non-linear character as a function of selected parameters of the reliability of energy supplies from wind farms.

1. Introduction

The prospect of exhausting fuel resources, air pollution, growing environmental awareness and legal regulations contribute to the increasing popularity of renewable energy sources for the production of electricity (RES-E). This trend can be observed for both low-power sources, which typically supply individual facilities (including, in particular, photovoltaic systems) and high-power facilities, such as water power plantsas well as wind and photovoltaic farms [1,2]. In some countries, the share of renewable energy sources (RES) in the power grid is greater than the share of conventional sources (Norway is the leader in this field; 98% of electricity in Norway is generated by renewable power plants—mainly water power plants) [3,4,5]. Few EU countries can boast such a high share of renewable sources in power generation. According to the “Renewable energy in Europe—2017 Update” report, on average, about 30% of electricity in member-states was generated from renewable sources [3].

One should pay attention to the fact that a dynamic increase in the number of renewable energy sources has some disadvantages related to a lack of power generation stability, which depends on the current weather conditions and low operating inertia. The problem applies especially to solar and wind sources. Heterogeneity can stabilise the operation of a power grid. It involves a high share of prosumer solutions and integration by means of intelligent systems (e.g., power generation predictions). Another solution is to use energy storage units, which can buffer the generated power (e.g., power generated by photovoltaic systems during the day will be released at night). Unfortunately, in the area of electrical power engineering, the existing energy storage systems are insufficient on a large scale, except for pumped-storage power stations (and thermal storages in southern countries). Moreover, it is expensive to build new energy storage systems (ESS) intended to work with RES-E. The cost can reach thousands of Euro/kWh in the case of electrical-chemical and kinetic storage units [6,7,8]. This is why scientific centres worldwide carry out research on ESS that can function in conjunction with RES-E, especially photovoltaic cells and wind turbines. References [9,10,11,12,13,14,15] focused on the issue of predicting the generating capacity of RES-E to adapt in advance with the power generated by other systems. There are the examples of papers [16,17,18,19,20,21,22,23,24] concern with the selection of the structure of energy sources and storages according to location needs and conditions. The assessment of the ESS effectiveness and profitability of their use is of paramount importance. The issue was mentioned in papers [2,12,22,23].

However, the literature, does not provide exhaustive information on the reliability of electrical energy generation in the context of the contribution of unstable renewable sources to the power grid. This is why the authors decided to concentrate on the topic of cooperation between wind farms and kinetic energy storage. Minimising the volume of ESS fulfils the set criterion including the reduction of costs related to the RES-E and ESS system installation. Analyses were carried out using historical wind speed data and, hence, the identified indicators are not probabilities or expected values but specific values identified for the specific analysis period.

2. Characteristics of a Wind Farm—Kinetic Energy Storage System

2.1. Description of the System, Power Flow Algorithm

A wind farm is an unstable source with random values of the instantaneous power P₁(t) in the range [0, P_WFn], where P_WFn stands for the total of the rated power values of all turbines. The annual and multi-annual deterministic components add to the stochastic nature of the wind speed changes, which means that a period of one year should be taken as the minimum period of analysis for such systems.

In relation to the above, even high rated power wind farms, where turbines are located at a certain distance from one another, do not guarantee stable power released to the power grid in either short (minutes) or long (hours, days) time intervals [16]. The stability and predictability of power generation by the sources within the power grid are essential in order to guarantee an adequate level of power supply reliability and determine the energy safety of a country. In the case of wind sources, the best results of the output power stability (power released to the power grid) are achieved by using energy storage devices with a capacity suited to the power plant efficiency, its geographical location and assumed period of continuous operation. The capacity of the storage not only depends on the farm’s efficiency and the mean annual wind speed v_wAvg, but also on the dynamics of wind speed changes specific for the local conditions, and on the statistical parameters of breaks in power generation: their mean duration, the number of subsequent intervals between the periods when the energy storage can be charged, etc. The author has presented detailed results of such studies in Reference [24].

Figure 1 presents a schematic diagram of a wind farm cooperating with a global (common for all turbines) storage of kinetic type. The power generated in each turbine P_1(i)(t) (for i = 1, 2, …, N_WT, where N_WT is the number of turbines in the farm) is supplied to a node and then—depending on the instantaneous power of farm P₁ and the state of charge (SOC) of the ESS—to storage P₂ and/or power grid P₃. The storage can also operate with power P₂ when discharged, and then the stored energy is supplied to the power grid. The (+) sign of power P₂ means that the storage is discharged, while (−) stands for its charged conditions. Power P₃ is supplied to the power grid. The power is the sum of the power of the turbines and the storage, reduced by the ΔP power loss in the transformer and the line supplying power to the grid:

$$P_3\left(t\right)=P_1\left(t\right)+P_2\left(t,\mathrm{SOC}\right)-\Delta P$$

(1)

The CFEM (control flow of energy module) is responsible for controlling the flow of energy between the wind farm, storage, and power grid.

It is possible to improve the reliability of electrical energy supplies to the power grid from a wind farm-kinetic energy storage system when an appropriate control algorithm is implemented that stores the flow of energy between the turbines, storage and power grid.

This assumes that the minimum set power P_3min on the system outlet is maintained at all times with the maximum duration T_max, even when the wind speed drops below the turbine cut-in speed v_cut-in (then P₁ = 0). At an appropriately selected energy storage capacity A_FESSn [24], it is possible to recover power in the periods when the power generated by the farm is lower than the assumed minimum value P_3min. In this way, the system guarantees the maintenance of power supplied to the power grid regardless of weather conditions in a period not longer than T_max and with power not lower than P_3min. The implementation of energy storage results in an increase in the predictability of the source generation related to the partial stabilisation of its output power P₃.

From the point of view of the acquired functionalities of the system presented in Figure 1, the control system is among its most important elements. The control system responds to the instantaneous values of wind speed v_w(t) and the storage charging level SOC(t). Considering the turbine cut-in speed (v_cut-in), reaching the rated power (v_n) and the output power of the farm P_3min (v_3min), turbine cut-out time (v_cut-out) and the generated power values corresponding to them, one can identify four operating conditions of a wind farm-kinetic energy storage system: independent operation of the farm, storage charging, independent operation of the storage and simultaneous operation of the storage and farm. The conditions and system diagrams corresponding to them are presented in Figure 2. The arrows mark the energy flow directions that depend on the current wind speed, the acquired power level P_3min, the capacity of A_FESSn storage and its charging level, as well as the mutual relations between power P₁, P_2max, P₃, and P_3min, where P_2max is the maximum power of storage.

2.2. Wind Speed Measurement Data

A simulation of the operation of the system described in Section 2.1, establishing the course of instantaneous power, generated power, and the impact of a kinetic energy storage system on the reliability of electrical energy supplies to the power grid, requires the use of data which characterise the wind conditions. Measurements of the wind speed were used for this paper. Each sample represented the mean wind speed in the period Δt_M ≈ 47 s, while the number of samples used for one year was to 670,000. The duration of the kth sample is expressed based on the formula t_k = k∙Δt_M, for k = 1, 2, …, N, where N stands for the number of samples.

All measurements used in the paper were carried out with an NP-3 (Far Data) anemometer in the radiation transfer station in Strzyżów near Rzeszów—south-eastern Poland at a height of h_M = 10 m AGL. The reference measurement data were re-calculated for the actual altitude of the turbine nacelle h_WT, for the needs of the study, according to the relationship describing the exponential form of the vertical profile of wind speed changes [25]:

$$v_{\mathrm{WT}}(t_k)=v_{\mathrm{wM}}(t_k){\left(\frac{h_{\mathrm{WT}}}{h_{\mathrm{M}}}\right)}^{\alpha }$$

(2)

where: k is wind speed sample number, v_WT is wind speed at the height of the wind turbine nacelle h_WT, v_wM is wind speed at the measurement height h_M, and α is coefficient depending on the land roughness.

3. Mathematical Model of the System

3.1. Turbine and Wind Farm

The identification of a turbine output power by means of an analysis, based on the wind speed measurement values requires the use of a mathematical and numerical model of the turbine [25]. In order to examine the reliability of power supplies from the wind farm to the power grid, a simplified model was used for the wind speed sampling period Δt_M. In the simplified model, the wind turbine plays the role of a functional unit modelling the power characteristics P₁ = f(v_w(t)). Based on linear interpolation between two points of the turbine power’s discrete characteristics, the power generated for the kth sample of the wind speed v_w (t_k) can be identified based on the following formula:

$$P_1\left(v_{\mathrm{w}}\left(t_k\right)\right)=\{\begin{array}{c}{P}_1+\frac{P_2-P_1}{v_2-v_1}\left(v_{\mathrm{w}}\left(t_k\right)-v_1\right)\mathrm{for}{v}_{\mathrm{cut}-\mathrm{in}}\le v_{\mathrm{w}}\left(t_k\right)\text{<}{v}_{\mathrm{wn}}\\ \\ P_{\mathrm{WTn}}\mathrm{for}{v}_{\mathrm{wn}}\le v_{\mathrm{w}}\left(t_k\right)\le v_{\mathrm{cut}-\mathrm{out}}\\ \\ 0\mathrm{for}\mathrm{other}{v}_{\mathrm{w}}\left(t_k\right)\end{array}$$

(3)

where: v₁, v₂ are wind speed values on the discrete characteristics of the turbine power between which the speed v₁ ≤ v_w(t_k) ≤ v₂ is positioned, P₁ and P₂ correspond to the wind speed, v₁ and v₂ are turbine power values read from discrete power characteristics.

In order to approximate the farm model to its actual operating conditions, differences in the position and, hence, the actual wind speed on the nacelle level were taken into consideration for each N_WT turbine. To that end, random changes (±5%) in the wind speed (in the moment t_k) were considered for each turbine as compared to measurement samples. Thus, for the ith turbine and kth sample, the wind speed (in the moment t_k) is:

$$v_{\mathrm{w}\left(i\right)}\left(t_k\right)=\left(1\pm rnd(-0.05,0.05)\right)v_{\mathrm{w}}\left(t_k\right)$$

(4)

where: rnd (−0.05, 0.05) is a random value (normal deposition) from the [−0.05, 0.05] range. Then, the power released to the power grid from the reference farm is:

$$P_3\left(t_k\right)={\displaystyle \sum }_{i=1}^{N_{\mathrm{WT}}}{P}_{1\left(i\right)}\left((1\pm rnd(-0.05,0.05\left)\right)v_{\mathrm{w}}\left(t_k\right)\right)-\Delta P$$

(5)

3.2. Kinetic Energy Storage

Kinetic energy storage systems use an indirect method of electrical energy storing in the form of flywheel rotational kinetic energy. Its shape, material and maximum rotational speed determine the storage dimensions, capacity, and weight, and are simultaneously the basic criteria of their division [9,24,26,27,28]. Figure 3 presents a general construction diagram of a kinetic energy storage system using a three-phase AC machine.

Analyses carried out to identify the reliability of power supply to the power grid do not require the consideration of all aspects of kinetic storage modelling and, especially, of dynamic states. A change in the storage charging level as a function of time (subsequent wind speed measurement samples) is important for the issue in question. Therefore, storage modelling was limited to the following equation of the rotational movement dynamics:

$$J_{\mathrm{F}}\frac{{d\omega }_{\mathrm{F}}}{dt}=\frac{P_2}{{\omega }_F}+{\omega }_{\mathrm{F}}{K}_1+K_2$$

(6)

where: J_F is the moment of inertia of the system, P₂ is the storage charging/discharge power, ω_F is the rotational speed of the rotor, K₁ is the coefficient of friction, K₂ is the Coulomb’s friction.

Solving Equation (6), when the storage capacity, dimensions of the rotational mass and scope of changes in the working speed and efficiency of electrical systems, which are part of the storage, are known, one can identify the course of changes in the power stored in the storage system when it is connected to a wind turbine and power grid, and execute the algorithm given in Section 2.1.

3.4. Power Electronics Systems Controlling the Flow of Power

From the point of view of the executed algorithm (point 2.1), power electronics systems related to DC electrical energy conversion into AC and vice versa (in the kinetic storage structure) and control of the energy flow between the turbine, storage and power grid according to Equation (1), are all important elements of the system presented in Figure 1. The efficiency of the abovementioned systems is a parameter that greatly affects the effectiveness of power conversion. A typical relationship of a power electronics converter η_INV as a function of load η_INV = f (P/P_INVn) is used in the acquired model, where P is converted power, and P_INVn is the rated power of the converter assembly. Figure 4 presents the assumed efficiency characteristics of power electronics systems.

4. Indicators of Electrical Energy Supplies from a Wind Farm—Kinetic Energy Storage System to the Power Grid

An assessment of electrical energy generation reliability for the assumed period involves the identification of the extent to which power demand P_z(t) is covered by the capacity of subsystem P_w(t). Information about reliability established for future periods (e.g., following changes in the structure of the interesting part of the grid) includes load connection, additional unstable sources etc. A model of power generation reliability, as a difference in the value of stochastic processes P_z(t) and P_w(t), is also defined as a stochastic process D(t) which is a power deficit. This is described by the general relationship [29]:

$$\mathrm{D}\left(t\right)=\{\begin{array}{c}{P}_{\mathrm{z}}\left(t\right)-P_{\mathrm{w}}\left(t\right)\mathrm{for}{P}_{\mathrm{z}}\left(t\right)>P_{\mathrm{w}}\left(t\right)\\ 0\mathrm{for}{P}_{\mathrm{z}}\left(t\right)\le P_{\mathrm{w}}\left(t\right)\end{array}$$

(7)

Indicators related to the duration of power deficit, power not supplied to customers and frequency and duration of occurrence, loss of energy expectation (LOEE), frequency and duration indices (F&D), expected capacity deficiency (ECD), probability of capacity deficiency (PCD), expected loss of load (XLOL) etc.—are used as quantitative parameters describing a stochastic process of a power deficit D(t) [29,30,31,32,33].

A method based on an analysis of the changes in the level of the output power P₃ for a one-year period was used to compare the reliability of electrical energy supplies to the power grid from a wind farm—kinetic energy storage system with various structures. The application of an energy storage system helps to shape the changes in the power supplied to the system as compared to the power level resulting from changes in the wind energy. The applied indicators are similar to the parameters defined for a power deficit stochastic process D(t) but they still take into consideration the acquired power flow control algorithm (Section 2.1). Furthermore, they are identified based on historical data. The following indicators are used further in this paper:

• total annual time of power generation with a value lower than P_3min, covering only the periods lasting up to T_max (according to the system operation algorithm, the generated power deficit is replenished by the energy from the storage) at the storage capacity A_FESS:

  $$T_{sumTmax}\left(A_{FESS}\right)={\displaystyle \sum }_{i=1}^M{T}_{Tmax\left(i\right)}$$

  (8)

  where: T_Tmax(i) is the duration of the ith interval which is shorter than or equal to T_max in which the power supplied to the grid P₃ is lower than the assumed minimum power P_3min, and M is the number of intervals T_Tmax in which power deficit occurs,
• percentage coefficient of elimination of periods when the generated power value is lower than P_3min, considering only the periods which last up to T_max:

  $$\Delta T_{sumTmax\%}=\frac{T_{sumTmax}\left(0\right)-T_{sumTmax}\left(A_{FESS}\right)}{T_{sumTmax}\left(0\right)}\times 100\%$$

  (9)

  where: T_sumTmax(0) is time T_sumTmax identified for a system without storage, T_sumTmax(A_FESSn) is the time T_sumTmax identified for a system with energy storage with a rated capacity A_FESS (the diagram explaining the difference between T_max, T_sumTmax and T_sumTmax(0) was presented in Figure 5),
• power deficit accounting only for periods lasting up to T_max:

  $$\Delta A={\displaystyle \sum }_{i=1}^M{T}_{Tmax\left(i\right)}(P_{3min}-P_{3avg\left(i\right)})$$

  (10)

  where: P_3avg(i) is the mean power supplied to the grid from the wind farm—kinetic energy storage system in the ith interval shorter than or equal to T_max, in which the power released to the grid P₃ is lower than the assumed minimum power P_3min.

Equation (8) identifies the total time composed of periods no longer than T_max, in which the deficit of power generated by the system (P₃ < P_3min) was not completely covered by the energy in the storage. Efforts are made to minimise the indicator, while its value close to zero means complete elimination of power outages in the assumed maximum duration T_max. High values of Equation (9) mean elimination of the majority of outages with duration up to T_max. Efforts are then made to keep the value at 100%. Equation (10) characterises the system operation on the power side and helps to identify the power which was not supplied to the power grid in periods lasting up to T_max. Efforts are made to keep the value of Equation (10) as low as possible, and ideally at 0.

5. Optimisation of a Wind Farm—Kinetic Energy Storage System

5.1. Purpose of Optimisation, Function and Constraints

The purpose of the optimisation task solved in the paper is to identify the structure of a wind farm—kinetic energy storage system, which for the set rated power of the farm P_WTn and its geographical location minimises the capacity of A_FESS storage limiting the annual time of supplies of power values to the power grid below P_3min (considering only the periods which last up to T_max) to T_sumTmaxY. The vector of decisive variables x includes: wind turbine type N_WT (x₁), turbine height h_WT (x₂), number of turbines n_WT (x₃), type of kinetic storage N_FESS (x₄), and the number of connected energy storage modules n_FESS (x₅). The minimisation task defined in this way takes the following form:

$${\dot{A}}_{FESS}=\min \left\{A_{FESS}\left(x\right)\right\}=f(N_{\mathrm{WT}},n_{\mathrm{WT}},h_{\mathrm{WT}},N_{FESS},n_{FESS},w)$$

(11)

where: ${\dot{A}}_{FESS}$ is the minimum capacity value meeting the requirements for the set operating conditions of the system, as w is the collection of algorithm parameters of energy flow between the farm, storage, and power grid.

The type of turbine used (variable x₁) determines its power curve and at the same time the power generated in the moments specified by available wind speed samples. In the case of turbine, several variants are available differing in the installation height of the rotor (variable x₂), which, according to Equation (2) to the vertical profile of wind speed variations, also affects the level of power generated by the turbine. The number of turbines specified by the variable x₃ allows obtaining the nominal power of the P_WFn farm assumed in the algorithm. With the assumed minimum power P_3min (definition in Section 2.1), the number and duration of periods of power generation by the farm below the level P_3min depend on the variables x₁, x₂ and x₃ in Figure 5. Individual modules of kinetic magazines are characterized by different energy capacities, but the most important from the point of view of the analysed problem are the values of their maximum charging and discharging power (these parameters are determined by the variable x₄). The last of these parameters in various ways adapt the properties of the energy storage in the field of exchange the energy, to the real dynamics of wind speed changes at a specific farm location (the course of changes in wind speed). The last variable x₅ affects the total capacity and power of the used kinetic energy storage system.

Equation (11) can be solved only when considering the collection of constraints, i.e., the relationships (equality and/or inequality) which identify the size of the acceptable solution area X. Structural constraints in the analysed task (related directly to the vector of decisive variables) expressed in a standardised form include:

• True power of the farm P_WFr(x):

  $$1-\frac{P_{WFr}\left(x\right)}{0.9P_{WFn}}\le 0\cap \frac{P_{WFr}\left(x\right)}{1.1P_{WFn}}-1\le 0$$

  (12)
• Size of turbine database (number of designs and turbine heights) and kinetic storage units (number of energy storage types).

Constraints in the true power of the farm P_WFr(x) are related to the inclusion in the calculation process of a finite collection of turbine designs whose multiple power values are not always equal to the assumed rated power of the farm P_WFn.

Functional constraints (not related to vector x) cover the parameters which are a result of a numerical analysis of the system operation (output power P₃) and are related to characteristic indicators of the control algorithm of the energy flow between the farm, storage, and power grid. In the analysed task, the functional constraints in a standardised form cover:

• time T_sumTmax(x) − Equation (6):

  $$\frac{T_{sumTmax}\left(x\right)}{T_{sumTmax}}-1\le 0$$

  (13)

  where: T_sumTmaxY is the total annual boundary time of power generation below P_3min assumed in the optimisation task, covering only the periods lasting up to T_max;
• maximum power of kinetic storage P_FESSMax(x):

  $$1-\frac{P_{FESSMax}\left(x\right)}{P_{3min}}\le 0$$

  (14)

Interrelations between the parameters of the selected turbine (power characteristic, height) and a single module of energy storage (rated energy capacity, maximum charging and discharging power) directly affect, for the given location (wind conditions), the T_sumTmax parameter in Figure 5. Equation (13) of limiting the indicated parameter to the value of T_sumTmaxY therefore requires the use of different storage capacities for various types of turbines and energy storage. Therefore, the parameters determined in Equation (11) determine the value of the energy storage capacity, and the application of the optimization algorithm described in the paper allows the determination of the optimal structure of the wind farm—kinetic energy storage, minimizing the capacity of the storage working in it, while fulfilling all technical requirements for the cooperation of the farm with power system.

5.2. Selection of Optimisation Method, Description of Application Developed

The selection of the optimisation method dedicated to the solution defined in Section 5.1 of the task requires a detailed analysis of several elements, including in particular: the form of the function of the objective, the number of decisive variables and the size of the acceptable solution area [25,34,35,36]. Additionally, one should consider the possibilities of including in the optimisation complex numerical calculations used for identifying the value of the function of the objective and constraint control. The experience of the authors in the effective use of the method to solve similar tasks is very important [37].

The multi-modal nature of the acquired function of Equation (11) requires the use of stochastic methods or heuristics in order to seek its global extreme. Even though only five variables were defined in the reference task, the task remains complex and time-consuming, as a result of the multiple identification of changes in the power value in the analysed system, the integer nature of decisive variables and the hidden occurrence of variables in the criterion function. In relation to the above, a genetic algorithm population method is used to optimise the structure of a wind farm-kinetic energy storage system, which leads to the minimisation of storage capacity A_FESS, according to the system operation algorithm described in Section 2.1. An important advantage of the method is related to its ability to modify the basic parameters in order to improve its effectiveness when performing detailed tasks. A disadvantage in the case of tasks with highly complex calculations is the reduced reproducibility of the results [35,36]. It is also necessary to perform initial tests whose results help to identify the appropriate AG parameters that would improve the reproducibility of the results and reduce calculation time.

The effectiveness of the proposed method in solving tasks of global optimisation in the area of electrical engineering and renewable energy was proven in a number of scientific publications, e.g., [38,39] and the authors used it successfully to optimise the structure of complex electrical light systems, the shape and parameters of high-power lines, and to minimise the cost of power generation in hybrid generation systems [37,40,41,42,43,44,45].

Based on the mathematical model described in Section 3 and a selected optimisation method in Matlab (Lincence no: 975466, Version 2014b, Poznan University of Technology, Poznan, Poland, 2014) and MS Visual Studio environment, an application was developed intended to optimise the structure of a wind farm—kinetic energy storage system with a view to minimising energy storage capacity. The application uses proprietary structures and classes related to different types of energy storage, wind turbines, PV modules, power electronics systems and indicators of the reliability of power supplies to the power grid mentioned in Section 4, as well as functions from the Global optimisation toolbox of the Matlab environment to implement the modified genetic algorithm method.

The assessment of the quality of solutions obtained through optimisation required an analysis of power: generated by farm P₁, energy storage P₂, and supplied to the power grid P₃ in the analysed system. To that end, measurements of the wind speed in a one-year period and data from wind turbines and PV modules collected in a database developed for the purpose of the application were used.

The Augmented Lagrangian Genetic Algorithm (ALGA) interior penalty function method [46] was employed to consider the constraints of the optimisation tasks (Equations (12)–(14)). The ALGA method was implemented in the Matlab environment.

5.3. Optimisation Calculations

A search for the structure of a wind farm—kinetic energy storage system which minimises the capacity of energy storage A_FESS (Equation (11)) for the assumed output parameters of the system, was carried out for wind farms with two power values: P_WFn(1) = 5 MW and P_WFn(2) = 10 MW (indices expressed as integers stand for a parameter option). Six time values were taken into account concerning power generation periods with power values below P_3min: T_max(1) = 5 min, T_max(2) = 10 min, T_max(3) = 15 min, T_max(4) = 20 min, T_max(5) = 25 min and T_max(6) = 30 min; two minimum power values P_3min(1) = 10%P_WFn, and P_3min(2) = 20%P_WFn; and four values of the acceptable total time: T_sumTmax(1) = 75 h, T_sumTmax(2) = 100 h, T_sumTmax(3) = 125 h, and T_sumTmax(4) = 150 h.

The calculations were carried out using the developed optimisation algorithm and application described in Section 5.2 in order to conduct a preliminary study involving changes in the AG parameters to solve the test task. The obtained results helped to draw conclusions relating to the value of the algorithm parameters which contribute to the final solution before the maximum number of iterations, and reduce computing time to ca. 3 h. Finally, a genetic algorithm was applied using the remainder selection method, the Gaussian mutation and elite strategy with a transfer of three best individuals. The number of individuals N_p = 50 and the number of generations N_g = 50 were established experimentally.

Figure 6 presents the changes in the value of the best individual’s (solution) adaptation as a function of the generation number for five activations of the algorithm (Figure 6b) for a farm with rated power P_WFn(2) = 10 MW, power P_3min(2) = 20%P_WFn = 2 MW, T_max(5) = 25 min, and T_sumTmax(1) = 75 h.

Figure 7 presents the results of optimisation calculations as changes in the minimum capacity of a kinetic storagesystem which meets the required parameters of a system controlling the flow of energy between the farm, storage and power grid as a function of time T_max for two wind farms with power P_WFn(1) = 5 MW (Figure 7a) and power P_WFn(2) = 10 MW (Figure 7b). In each case, the calculations were made for two minimum power values P_3min(1) = 10%P_WFn and P_3min(2) = 20%P_WFn and T_sumTmax(1) = 100 h.

An essential issue for the analysed class of systems is to identify the relationship between the minimum capacity of the storage A_FESSmin and the time of eliminated outages T_max depending on the maximum total time T_sumTmax. Optimisation calculations representing the aforementioned task were performed for a farm with power P_WFn(2) = 10 MW, two power values P_3min(1) = 10%P_WFn = 1 MW and P_3min(2) = 20%P_WFn = 2 MW and four times T_sumTmax:T_sumTmax(1) = 75 h, T_sumTmax(2) = 100 h, T_sumTmax(3) = 125 h and T_sumTmax(4) = 150 h. The results (value of the minimum capacity of energy storage A_FESSmin meeting the assumed algorithm of the system operation as a function of time T_max) are presented in Figure 8a (P_3min(1) = 10%P_WFn = 1 MW) and 8b (P_3min(2) = 20%P_WFn = 2 MW).

Figure 9 presents the relationship between the minimum capacity of the storage unit A_FESSmin and the maximum total time T_sumTmax for two values of time T_max(3) = 15 min and T_max(5) = 25 min.

Details of the optimisation results of the wind farm-kinetic energy storage system for rated power P_WFn(2) = 10 MW and selected parameters P_3min, T_sumTmax, and time T_max are presented in

Table 1. Results of optimising calculations for selected configurations of a wind farm—kinetic storage system for a farm with rated power P_WFn(2) = 10 MW and different time values T_max.

No.	P3min (MW)	Tmax (min)	TsumTmax (h)	TsumTmax (AFESSmin) (h)	TsumTmax(0) (h)	ΔTsumTmax (%)	AFESSmin (kWh)	Turbine’s Number/Power(−)/(MW)	PWFnr (MW)	ΔA (MWh)
Tmax(2) = 10 min
1	1	10	75	73.7	214.3	65.6	125	10/0.9	9	21.1
2	1	10	100	96.2	247.1	61.0	72	5/1.8	9	31.6
3	1	10	125	122.5	232.1	47.2	36	4/2.3	9.2	38.5
4	1	10	150	144.8	225.5	37.8	25	6/1.8	10.8	47.6
5	2	10	75	73.4	241.6	69.6	108	10 0.9	9	37.3
6	2	10	100	92.7	198.0	53.2	100	4/2.3	9.2	49.3
7	2	10	125	120.5	231.2	48.1	72	3/3	9	60.4
8	2	10	150	144.2	241.5	40.2	36	6/1.8	10.8	72.0
Tmax(3) = 15 min
9	1	15	75	74.9	329.8	77.3	630	12/0.9	10.8	25.7
10	1	15	100	93.3	312.2	70.1	240	10/0.9	9	30.0
11	1	15	125	118.1	327.6	63.9	125	4/2.3	9.2	36.9
12	1	15	150	142.7	327.6	56.4	72	4/2.3	9.2	42.8
13	2	15	75	74.9	283.9	73.6	1470	4/2.3	9.2	50.9
14	2	15	100	98.2	283.9	65.4	400	4/2.3	9.2	63.4
15	2	15	125	123.6	283.9	56.4	175	4/2.3	9.2	75.6
16	2	15	150	145.2	283.9	48.9	108	4/2.3	9.2	84.5
Tmax(4) = 20 min
17	1	20	75	74.7	384.9	80.6	1800	12/0.9	10.8	37.8
18	1	20	100	99.5	397.7	74.9	684	12/0.9	10.8	34.1
19	1	20	125	123.3	384.9	67.9	350	10/0.9	9	40.3
20	1	20	150	147.9	384.9	61.6	175	10/0.9	9	46.7
21	2	20	75	74.9	302.8	75.3	2670	11/0.9	9.9	52.0
22	2	20	100	98.8	285.6	65.4	840	10/0.9	9	62.9
23	2	20	125	124.9	314.2	60.2	474	12/0.9	10.8	76.5
24	2	20	150	149.5	353.0	57.7	288	4/2.3	9.2	96.7
Tmax(5) = 25 min
25	1	25	75	74.9	447.2	83.3	5400	10/0.9	9	29.3
26	1	25	100	99.3	482.8	79.4	1440	4/2.3	9.2	39.2
27	1	25	125	124.9	447.2	72.1	750	10/0.9	9	44.0
28	1	25	150	148.7	447.2	66.8	390	10/0.9	9	54.6
29	2	25	75	74.9	355.5	78.9	6200	11/0.9	9.9	54.4
30	2	25	100	99.8	355.5	71.9	2340	11/0.9	9.9	70.3
31	2	25	125	124.7	334.1	62.7	1584	10/0.9	9	87.3
32	2	25	150	148.9	334.1	55.5	500	10/0.9	9	95.8
Tmax(6) = 30 min
33	1	30	75	-	-	-	-	-	-	-
34	1	30	100	99.3	552.0	82.1	2100	6/1.8	10.8	54.0
35	1	30	125	124.3	552.0	77.5	1188	6/1.8	10.8	65.0
36	1	30	150	148.9	552.0	73.0	780	6/1.8	10.8	75.9
37	2	30	75	-	-	-	-	-	-	-
38	2	30	100	99.8	426.9	76.6	4050	12/0.9	10.8	70.9
39	2	30	125	124.9	398.8	68.7	2070	11/0.9	9.9	88.3
40	2	30	150	149.7	378.9	60.5	1075	11/0.9	9	103.0

.

Figure 10 presents the changes in the percentage value of the coefficient of elimination of power generation periods with power values below P_3min for solutions optimum as a function of time T_sumTmax for two power values P_3min(1) = 10%P_WFn and P_3min(2) = 20%P_WFn, and two time values T_max(3) = 15 min (Figure 9a) and T_max(4) = 20 min (Figure 9b). The data were developed based on information given in Table 1.

5.4. Discussion

Following a thorough analysis of the results it was concluded that the applied optimisation methods, the parameters of the control algorithm of the energy flow between a wind farm, energy storage system and the power grid, and the system model were correct for the characteristics of the analysed task. Its solution had high reproducibility in minimising the capacity of the kinetic storage A_FESSmin (Figure 6a) and fast computation times (Figure 6b). In the analysed examples, the tasks were solved between the 30th and 40th generations.

Based on the completed studies, it was established that the relationship between the minimum capacity of the energy storage system A_FESSmin meeting the required criteria of a wind farm—kinetic energy storage system and time T_max was non-linear (Figure 7 and Figure 8). It was demonstrated that the dependence points A_FESSmin = f(T_max) could be approximated by polynomial functions of order 3. The values of the determination coefficients identified for the functions amount to over 0.97, confirming the good fit of a polynomial model for the calculation points. A non-linear character was also observed for changes in the capacity A_FESSmin as a function of the optimisation time constraint T_sumTmax − A_FESSmin = f (T_sumTmax) (Figure 9).

The application of higher constraints (lower values of T_sumTmax) amounting to 150, 125, 100, and 75 h/year contributed to an increase in the required capacity of energy storage A_FESSmin. The increase was diversified for different values of time T_max and power P_3min, but between subsequent values of T_sumTmax it usually ranged from 1.5 to 3.0 times (Figure 8). In the case of optimising calculations made for time T_max = 30 minutes and the strongest constraint T_sumTmax = 75 h, the task was not solved (Figure 8) for neitherpower P_3min = 10%P_WFn nor for P_3min = 20%P_WFn. Tests were repeated many times and an additional detailed analysis of the system operation with optimised parameters (type and number of turbines) was performed with a simultaneous increase in the power storage capacity. It was demonstrated that T_sumTmax could not be limited at the level of 75 h/year by further increasing storage capacity. This was caused by the characteristics of the annual changes in wind speed for the reference geographical location (especially for the mean annual wind speed, which amounts to 4.1 m/s at an altitude of 1 m AGL, and to 6.2 m/s at 100 m AGL), and the nominal parameters of the analysed turbines and energy storages systems.

Depending on the geographical location of the farm and the parameters of the turbines and storage systems, it is possible that no solutions can occur for different values of time T_max. Therefore, it is important to identify the maximum value of time for the assumed wind farm power P_WFn, location and basic parameters of the system for which elimination of power generation periods with power values below P_3min is possible on the T_sumTmax level. Still, the developed algorithm and application require significant modifications, alongside additional studies. The authors plan to address this in future works on the cooperation between wind farms and kinetic energy storages.

The level of the guaranteed output power P_3min (Figure 7 and Figure 8) in periods lasting up to T_max greatly affects the minimum energy storage capacity A_FESSmin. Before establishing the parameter value, one should analyse the actual wind farm operating conditions, especially its mean working power. Increasing the power P_3min above the value results in a significant increment in the required energy storage capacity and may mean that the storage unit cannot be fully charged. Consequently, the system is not able to eliminate all assumed energy deficits lasting up to T_max and does not meet a certain group of constraints T_sumTmax (e.g., Table 1—results for T_max = 30 min and constraints T_sumTmax = 75 h). A lack of solutions for the cases analysed in Table 1 applies to a farm with low mean annual power of ca. 1 MW. In cases where solutions were found, the optimum mean annual power values of the wind farms ranged from about 1.5 MW to 3 MW.

The optimisation of the kinetic energy storage capacity helps to achieve high percentage values of the coefficient of elimination of power generating periods when the power value is below P_3min (ΔT_sumTmax% − Equation (9)) which amount to 40–80% (mean 53%) of the analysed cases depending on the value of time T_max and constraint value T_sumTmax (Figure 10). The value of the indicator ΔT_sumTmax% increased as the system operation constraints become more stringent (shorter times T_sumTmax), entailing a significant increase in the required storage capacity. When the value of the constraint time is reduced twice (from T_sumTmax = 150 h to T_sumTmax = 75 h), the optimised values of the energy storage capacity were 5 to 14 times higher, depending on time T_max.

Values of power not supplied to the power grid ΔA (Equation (10)) in the system operation periods with power below P_3min and time up to T_max, analysed in a one-year period (Table 1), were established for the optimised structures of a farm with a kinetic energy storage system. The power value increased as time T_max and constraint time T_sumTmax increase. In the case of the analysed farm with power P_WFn = 10 MW, the values were low (Table 1) compared to the total energy generated in a one-year period and they did not exceed 0.5%.

It should be emphasised that it was possible to achieve the presented results of a wind farm—kinetic energy storage system optimisation due to a short (ca. 47 s) period of wind speed averaging. For a typical value of 10 min, it was not possible to precisely simulate the system cooperation with the power grid due to an approximated analysis of the energy storage operation. In such cases, the results for the minimum energy storage capacity A_FESSmin suffered from a much higher error. Based on the authors’ experience in the modelling of energy storage using RES, they suggest that the averaging period of wind speed measurements and other parameters related to power generation in RES—irradiation in particular—should be shorter than the period used currently and should amount to one or a few seconds.

6. Conclusions

The paper presents a new algorithm for identifying the minimum capacity of a kinetic energy storage system which helps to improve (on the assumed level) the reliability indicators of electrical energy supplies from an unstable source (i.e., wind farm) to the power grid. The concept of the method can be transferred onto different types of energy storage and unstable sources. In such a case, the relevant mathematical and numerical models of the system elements need to be replaced and preliminary studies conducted to establish the optimisation algorithm parameters.

The analyses helped to draw an important conclusion concerning changes in the values of reliability of power supplies from a wind farm to the power grid. It is possible to improve the value and stabilise the operation of the wind farm at relatively low capacities of energy storage (in cases of moderate parameters of the analysed system and optimisation constraints). It is still important to take such measures towards the majority of farms located close to one another, as it will help to predict the level of power generated from such sources more precisely. Any increase in the requirements concerning the system operating parameters results in a significant (up to several times) increase in the required capacity of the storage, and hence means higher investment and operating costs.

The authors of the paper focused on an essential matter related to the relative increase in the installed power of unstable sources in the generating systems operating in many countries. The situation causes a stochastic variance of electric power generation. The application of the presented methods and results can help to stabilise the output power of unstable sources at limited costs of necessary investments. Local compensation of temporary power deficiencies by storage system included in the structure of large wind farms helps to limit power transmission losses and to improve the total energy efficiency of the system.