Application of the Linear Programming Method in the Construction of a Mathematical Model of Optimization Distributed Energy

Abstract

The Polish economy is facing a huge challenge regarding the future of energy in Poland. The current energy system is very inefficient, it consumes huge resources that, like countries with high energy efficiency, could be allocated to the development of low-carbon and renewable energy networks. At the moment, the Polish energy sector, related to the coal monoculture, lacks electricity and the entire transmission system is obsolete. The solution may be distributed energy, which can ultimately satisfy energy supplies in less urbanized areas and in rural areas, while guaranteeing the sustainable development of these areas. In order to take up the challenge of better understanding and explaining such a complex reality, it was decided that the research framework of this article will be distributed energy in the region. The aim of the article is to ensure energy security in the conditions of innovative, ecological and open to competition regional energy using local energy resources. Currently, it is believed that distributed energy can be an effective solution to the problem of conventional energy operation.

1. Introduction

The Polish economy is facing a huge challenge regarding the future of energy in Poland. The current energy system is very inefficient, absorbing huge resources that, like countries with high energy efficiency, could be allocated to the development of low-carbon and renewable energy networks. A carbon-based system from several large centrally controlled power plants that distribute electricity throughout the country is very expensive and highly unreliable.

The deposits of conventional energy sources have a colossal impact on the development of the energy system, which in recent decades has undergone increasing centralization, resulting in the creation of large centers of production. Powerful power plants provide electricity to mass consumers, in a large area and exceptionally in Poland they are based primarily on fossil fuels. From coal-fired power plants in Poland, just over half of the electricity generated reaches the final recipients, causing that an unexpected heat wave or other adverse atmospheric phenomena may contribute to the occurrence of a power failure in a significant area. Currently, the power of power plants in Poland, according to ENTSOE Installed Capacity per Production Type, is approximately 45.029 GW. Electricity in Poland is produced by thermal, hydro, wind and solar power plants. This power is generally sufficient to meet national needs. Thermal power plants using hard coal and lignite generate the most energy. The thermal power plant with the highest installed capacity is the Bełchatów lignite-fired power plant, the second largest coal-fired power plant in the world. At the moment, the Polish energy sector, related to the coal monoculture, lacks electricity and the entire transmission system is obsolete. The solution may be distributed energy, which can ultimately satisfy energy supplies in less urbanized areas and in rural areas, while guaranteeing the sustainable development of these areas. Distributed energy extended to the entire economy is the main guarantor of market mechanisms for shaping energy security [1].

An interesting thread is the issue of changing the role of the end user of energy: from a passive consumer to an active prosumer who not only shapes his behavior in such a way as to save energy, but strives to increase the efficiency of its use and to produce surplus energy that can be transferred to the local network. This is the way to make a civilizational leap in Poland, to build a knowledge society in our country. An advantage motive for the development of distributed energy is technological progress, conducive to the reduction of the cost of generating energy in renewable sources, as well as the development of local energy resources. The current development path of the Polish energy sector has been rather based on continuous improvement of existing technologies and services [2,3,4].

In order to take up the challenge of better understanding and explaining such a complex reality, it was decided that the research framework of this article will be distributed energy in the region. The aim of the article is to ensure energy security in the conditions of innovative, ecological and open to competition regional energy using local energy resources. The necessity to disaggregate the existing theoretical constructs became a premise for the identification of an auxiliary objective, which was formulated as follows: so far in Poland there is no interest in distributed energy, both from the central energy sector and from regional and local authorities. The use of various research approaches and methods enabled the use of methodological triangulation (data sources, research methods and research environment) increasing the reliability and validity of inference. The article uses literature on the subject, scientific articles, press, papers, conference materials, especially from countries where there is distributed energy (Germany, Great Britain, the United States), legal and official acts, reports, international agreements (European Union), publications.

2. Literature Review

There are many definitions of distributed energy in the literature. The International Renewable Energy Agency (IRENA) defines distributed energy as facilities located in distribution networks or at consumers, permanently using renewable or conventional resources. Distributed generation networks may include prosumers, energy associations and municipal power plants [5].

In contrast, distributed power generation is defined as the production of electricity by installations that are smaller than central power plants to ensure interconnection almost everywhere in the power system by the Institute of Electrical and Electronics Engineers (IEEE) defines [6,7,8].

According to the Act of 20 February 2015 on renewable energy sources (Journal of Laws of 2015, item 478 of the following dates) by distributed energy we define as solar, wind, hydro or geothermal energy, as well as biomass and biogas (coming mainly from the agri-food industry and directly from agriculture), but also municipal waste that can be used for energy purposes [9,10].

The International Council for Large Electricity Systems CIGRE identifies distributed power as all producing units of 50 MW with a most extreme limit to 100 MW that are generally connected to the distribution network and that are neither centrally arranged nor dispatched. The second piece of their description implies that distributed production units are past the control of the transmission network administrator. In this way, creating units built by the transmission network administrator as a substitute for network development and which have dispatching measures in place are not viewed as distributed generation, as indicated by this philosophy [11,12]. Popczyk, on the other hand, defines distributed energy as a model in which the recipient of energy, producing heat or electricity for his own needs, is at the same time its producer and consumer (prosumer). Groups of prosumers are households, farms and small and medium-sized enterprises, which are located mainly in the area of villages and suburban areas. [13,14,15].

Distributed generation, defined as “electricity generation in distribution networks or on the customer side” [16], seems to be a promising way to give admittance to energy in ruristic regions that are not joined to the matrix [17,18,19]. Actually, the low population thickness and low energy use of provincial consumers might harmonize with the adaptability and scalability of divided power plants. The mix of divided generation with environmentally friendly power sources (such as sun, wind, water, biomass and geothermal energy) can be depicted as Distributed Renewable Energy (DRE) [20,21].

The distributed energy model prevails over the centralized system in that it has superiority in the economic, technical, social and, of course, ecological fields. The more energy is generated in this way, the less energy will have to be generated in power plants. By causing that on the side of consumers with their own installations, there will be less demand for energy. The use of distributed energy makes everyone a winner, because with an adequately constructed system of public subsidies and a tax system, everyone can benefit from distributed energy. This also applies to the energy sellers themselves, where they can resell the excess energy produced by themselves to the grid, as well as to small producers, because the prosperity of distributed energy will contribute to the increase in the number of enterprises performing, assembling and servicing installations, as well as the creation of new jobs on local markets, and even more so in rural areas. Very often, distributed energy is also called “citizen energy”. Kassenberg defines citizen energy as “a system in which individuals, organizations, institutions and companies outside the energy sector take an active part in the generation, transmission and management of energy. It is a local and small-scale production of electricity and heat from renewable sources and the use of solutions that increase energy efficiency. Civic energy is also the participation of local communities in larger RES (Renewable Energy Sources) projects. It is also about building local alternatives to a centralized and company-dominated energy system” [22,23,24]. The foundation of civic energy is the direct involvement of citizens. As a democratic system, it provides all participants with the opportunity to profitably produce and use energy. Everyone benefits from its development: residents of cities and villages, local governments, entrepreneurs, housing communities and cooperatives as well as public benefit organizations. Among others, hospitals, clinics, schools, kindergartens and animal shelters will be able to become energy producers. Citizen energy can develop throughout the country, in every region, in every municipality and in every locality [25,26]. A low-carbon future is strongly dependent on small-scale, decentralized and distributed generation in rural regions, solving local problems while meeting energy demand with environmentally friendly power sources, belonging to local communities. Locally developed and embedded energy projects can benefit local residents and businesses, while increasing security of energy supply. Several case studies suggest that collaboration between local stakeholders is the basis for a community-scale RES turnaround [27,28].

Furthermore, there is a clear shift from examining the role of the state or capital in the transformation of the rural periphery towards a “new regionalism” with a clear bottom-up approach. In this approach, “the key issue in rural development is no longer the region’s ability to attract businesses from outside the region, but the use of its local resources to generate from a balanced transition.” In this context, local businesses and social actors have been identified as key drivers for the adoption and institutionalization of microgeneration of electricity with renewable energy [29,30]. National counseling firms, financial institutions, national and local governments assume a less significant part than the networkers who start the process of distributed energy. The basic idea of building sustainable communities is that these communities add to national energy security, advance social development and social business, enlarge occupation and regional development opportunities. For instance, wind energy projects where 100% ownership is local produce two times as many long-term jobs and 1–3 times the economic effect of wind projects held by absentees. At long last, it is important assuming nearby occupants are involved in the improvement of a RES project locally, they will quite often become mindful of the gracious patterns of energy use [31,32,33].

3. Energy Programming Methods in the Region

In the development of an energy model in a given region, consideratrion ought to be paid to the issue of picking choices and arranging energy production. Every choice in some cases has extremely far-reaching consequences, and its outcomes are generally expected very complex. If the optimal option is chosen in the planning of energy production in a given area, the decision should be multifaceted, considering different issues. When assessing the possibilities of obtaining energy, one cannot rely solely on the financial analysis of the investment, but also take into account important issues such as, for example: environmental aspects (environmental outlays, loss of soil fertility), agro-energy aspects, innovative aspects, organizational aspects or social.

The choice to pick the option of project execution in the planning of energy creation in a given region additionally requires examining the options as far as their good and bad influence. The positive sides incorporate advantages and open doors, while the negative ones incorporate components connected with expenses and risks. The issue with evaluating these aspects is regularly the trouble of expressing them in mathematical terms. For instance, several of the advantages are qualitative, like ecological benefits or risk components [34,35].

The choice is often an arbitrary or absolutely intuitive decision, not upheld by any strategy planning or examination. To keep away from blunders and randomness of decision, it is important to set up criteria and objectives for activity prior to proceeding to project the energy model of the studied region. Along these lines, the objectives can be of a different nature:

-

competitive goal—when increasing the value of one of the goals results in a decrease in the implementation of the other, e.g., maximizing profit and increasing the size of its risk,

-

conjugated objectives, between which there is a link in that progress towards one objective is accompanied by an increase in the other,

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complementary objectives that support each other,

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supplementary objectives—independent of each other, reducing or increasing the achievement of one does not affect the size of the other objective [36,37].

The links depicted between the different objectives are not constant. Any of them can move to another one, standing on the size of absolute generation of renewable energy. Objectives can likewise be complementing, i.e., supplement each other in the apply of one element of production, and simultaneously contend with each other for another factor.

The essence of the relationships among the various rules is hard to identify. Their organization can be noticed just during optimizing the mathematical model of the energy generation plan in a given region. The errand in model planning is to create a production plan that would supremely achieve individual aims in accordance with its advantages [38,39,40,41,42,43].

Table 1. The scheme of the conducted study.

Literature analysis	Analyzing the literature in the field of energy in order to identify research and theoretical gaps and to select a research problem.
Research problem	As a result of literature studies, the ineffectiveness of the current energy system has been noticed. Indication of an alternative which may be the development of low-emission and renewable energy networks. Indication of distributed energy as an effective solution to the problem of conventional energy operation.
Re-analysis of the literature	Leading to the identification of variables in the model (conceptualization and operationalization of variables).
Selection of research methods	The use of triangulation on three levels (data sources, research methods and research environment) increasing the credibility and accuracy of inference. Choosing the method of searching for an optimal solution—linear programming with a multi-criteria objective function.
Research	Collecting quantitative data that meet the substantive criteria and legal conditions. Construction of a mathematical model.
Data analysis	Construction of the objective function—its coefficients and limiting conditions resulting from the legal conditions of the Polish energy sector. Searching for the optimal solution with the use of linear programming methods.
Conclusions from the research	Interpretation of the obtained optimal solution ensuring energy security in the conditions of innovative, ecological and open to competition regional energy using local energy resources.

Source: own study.

below shows a schematic of the survey conducted.

In this way, when arranging energy creation, it is important to appeal to proven scientific methods. These involve linear programming methods.

Application of the linear programming method in the construction of a mathematical model of energy production optimization.

The use of the linear programming method allows you to build a mathematical model of electricity production from renewable sources in the studied area, the solution of which will take into account different types of costs.

The following decision variables shall be entered in the model, which can be found in

Table 2. Decision variablesused in the model.

Variable	VariableName	Units	Comments
x1	the volume of non-renewable energy produced	kWh
x2	the amount of energy produced from co-incineration	kWh
x3	the volume of hydropower produced	kWh	from existing installations (by the end of 2017)
x4	the volume of hydropower produced	kWh	in new installations (from January 2018) *
x5	the volume of solar energy produced	kWh
x6	the amount of energy produced from windmills in the household	kWh
x7	the amount of energy generated from wind farms	kWh	from existing installations (by the end of 2017)
x8	the volume of energy and wind generation	kWh	in new installations (from January 2018) *
x9	the amount of energy produced from biogas	kWh	from existing installations (by the end of 2017)
x10	the amount of energy produced from biogas in high-efficiency cogeneration with a total installed electrical capacity of less than 1 MW	kWh
x11	the amount of energy produced from biogas	kWh	in new installations (from January 2018) *
x12	the amount of energy produced from biofuels	kWh
x13	The amount of energy produced from the combustion of biomass from existing boilers	kWh	by the end of 2017
x14	the amount of energy generated from biomass combustion from new boiler installations	kWh	since January 2018 *
x15	the amount of energy generated from geothermal energy	kWh
x16	total annual electricity production from different energy sources	kWh
x17	the size of the cultivation of raw materials for biomass combustion—miscatus	kWh
x18	the size of the cultivation of raw material for burning biomass—poplar	kWh
x19	the size of the cultivation of the raw material for burning biomass—slazowiec	kWh
x20	the size of the cultivation of raw material for biomass combustion -Jerusalem artichoke	kWh
x21	the size of the cultivation of raw material for burning biomass—rapeseed	kWh
x22	the size of the cultivation of raw materials for the combustion of biofuels—cereals	kWh
x23	the size of the cultivation of raw material for the combustion of biogas—corn	kWh
x24	the size of the cultivation of raw material for the combustion of biogas—beets	kWh

* On 29 June 2018, the President of the Republic of Poland signed the Act of 7 June2018 amending the Act on renewable energy sources and certain other acts, on the basis of which the definition of a structure was changed in the Construction Law and the definition of a wind farm in the Act of 20 May 2016 on investments in wind farms. The amended regulations became applicable retroactively from 1 January 2018. Source: Own study.

:

The target function of the energy production optimization model (coefficients with decision variables from x₁ to x₁₅) will consist of:

-

costs related to production,

-

costs related to ecological certificates,

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costs of EUA entitlements,

-

energy storage costs for each type of energy,

and with:

-

the costs of loss of soil fertility caused by their exploitation in the production of raw materials for biomass, biogas, biofuels and commodity agricultural production (coefficients with decision variables from x₁₇ to x₂₄).

The coefficient as a function of the target for variables from x₁₇ to x₂₄ denote the unit cost of producing energy from energy crops. It is understood as the amount of individual effort that should be incurred in connection with the loss of soil fertility caused by their exploitation, in order to return to the initial state.

In the optimization model, only one function is minimized, $L\left(\mathit{x}\right)$ which is a component of the above components.

The objective function of the decision model takes the following form:

$$L\left(\mathit{x}\right)={\displaystyle \sum }_{i=1}^{24}{k}_i·x_i=k_1·x_1+\cdots k_{24}·x_{24}\to min$$

(1)

where:

$\mathit{x}=\left[x_1, x_2,\dots x_{24}\right]—$vector of decision variables, $k_i—$means the unit cost to produce the amount of energy (from $x_1 \mathrm{to} x_{15})$to produce energy from energy crops (from$x_{17}\mathrm{to}{x}_{24})$.

All right-hand sides in the following limiting Equations (2)–(14) are marked: $e_{j }or e_{j-k}$ are numerical values resulting from the assumptions made for the model.

Under restrictive Equations (2)–(14) resulting from the assumptions of the Draft Energy Policy, the Polish until 2040. Of 23 November 2018:

$$x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8+x_9+x_{10}+x_{11}+x_{12}+x_{13}+x_{14}+x_{15}=x_{16}$$

(2)

Equation (2) implies that the sum of production and energy from non-renewable and renewable sources represents the total volume of production and energy.

$$x_{16}\ge e_{16}\mathrm{kWh}$$

(3)

Equation (3) assumes that the energy production in the test area will be not less than the energy volume determined a $e_{16}$ kWh. The project assumes that the excessenergy producedin a given region canbe exported to other areas.

$$x_3+x_4+x_5+x_6+x_7+x_8+x_9+x_{10}+x_{11}+x_{12}+x_{13}+x_{14}+x_{15}=0.3·x_{16}$$

(4)

Equation (4) means that 30% of total production will come from renewable energy.

$$x_6+x_7+x_8\ge e_{6-8 }\mathrm{kWh}$$

(5)

Equation (5) assumes that the production of wind energy will be not less than the size of$e_{6-8}$ kWh.

$$x_8\le e_8\mathrm{kWh}$$

(6)

Equation (6) assumes that in accordance with the amendment to the Act on investments in wind farms, commonly known as Anti-Windmill, the overall energy potential of the area from this type of energy will be no greater than the size of $e_8$ kWh.

$$x_9+x_{10}+x_{11}\ge e_{9-11}\mathrm{kWh}$$

(7)

Equation (7) assumes that the production of energy from biogas will be not less than the size of $e_{9-11}$ kWh.

$$x_{12}\ge e_{12} \mathrm{kWh}$$

(8)

Equation (8) assumes that energy production from biofuels will be at least at the level of $e_{12}$kWh.

$$x_{13}+x_{14}\ge e_{13-14}\mathrm{kWh}$$

(9)

Equation (9) assumes that the production of energy from biomass will be not less than the size of $e_{13-14}$ kWh.

$$x_{15}\ge e_{15}\mathrm{kWh}$$

(10)

Equation (10) assumes that the production of energy from geothermal energy will notbeless than the level of $e_{15 }$ kWh.

$$x_1\le e_1\mathrm{kWh}$$

(11)

The size of $e_1$ kWh in Equation (11) means the maximum volume of production and conventional energy.

$$x_2\le e_2\mathrm{kWh}$$

(12)

Equation (12) assumes that the energy production from co-incineration will be no more than the size of $e_2$ kWh.

$$x_3+x_4\ge e_{3-4} \mathrm{kWh}$$

(13)

Equation (13) assumes that the production of hydropower will be not less than the size of $e_{3-4}$ kWh.

$$x_5\ge e_5\mathrm{kWh}$$

(14)

Equation (14) assumes that the production of solar energy will be not less than the size of $e_5$ kWh because of the innovative improvement of photovoltaic cells and there will be an effect of scale resulting in a decreasein the expense of producing kWh of energy.

At the condition edges given in general by the pattern (14). They add that all decision variables must be non-negative.

$$\bigwedge i x_i\ge 0\hspace{1em}for i\in \left\{1, \dots ,24\right\}$$

(15)

Optimization of the target function given by the formula (2) with limiting Equations (3)–(14) and boundary Equation (15) gives the solutions included in the work part of the optimization $\mathit{x}=\left[x_1, x_2,\dots x_{24}\right]$results.

Technical information necessary to determine the values of the coefficients of the target function and the limit values for the limiting conditions.

For the construction of the model and optimization, the values of technical and economic parameters were first calculated and the minimum or maximum levels of balance sheet states (and not limiting conditions) were set.

Taking into account the characteristics of disturbances in the nN distribution network, as well as a list of technical characteristics, operational requirements in selected technologies. The requirements for energy conditioning systems equipped with an energy storage tank and installed in nN networks for the implementation of individual types of network services have been compiled.

The model has selected five types of system services to be provided using energy storage tanks:

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Service 1: stabilization of RES capacity;

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Service 2: voltage regulation with active and reactive power;

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Service 3: reactive power and deformation compensation;

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Service 4: stabilization of the power of restless receivers;

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Service 5: Power reduction on demand.

Predispositions to provide individual services using specific technology are summarized in

Table 3. The possibility of providing a system service by a given electricity storage technology.

Functionality *	Compensation of Reactive Power and Deformations	Stabilization of RES Capacities	Stabilization of the Power of Restless Receivers	Voltage Regulation by Active and Reactive Power	Power Reduction on Demand
System 1 (EDLC)	+	+	+	-	-
System 2 (LIC)	+	+	+	-	-
System 3 (LFP)	+	+	+	+	+
System 4 (LTO)	+	+	+	+	+
System 5 (VRLA)	+	-	-	+	+

* EDLC (Electric Double-Layer Capacitors) supercapacitor storage; container with LIC (Lithium-Ion Capacitor) supercapacitors; tray with LFP batteries (Lithium Ferro Phosphate LiFePO4); container with LTO batteries (Lithium Titanate Oxide); VRLA (Valve Regulated Lead Acid) battery tray. Source: own study.

.

In the model of distributed energy development in the West Pomeranian Voivodeship, energy storage technology was selected, meeting the criteria of five types of system services. This technology is characterized by the lowest cost of energy storage.

4. Result of Optimization

In the model of development of distributed energy in the West Pomeranian Voivodeship, the LTO energy technology was selected, meeting the criteria of five types of system services (Table 3) and with a lower energy storage cost (

Table 4. A list of the designated investment and operating costs of containers made in selected technologies.

Tray Type *	Cost of Energy Storage [PLN/kWh]
EDLC	0.17
LIC	0.15
LiFePO4	4.03
LTO	0.22
VRLA	2.78

* EDLC (Electric Double-Layer Capacitors) supercapacitor storage; container with LIC (Lithium-Ion Capacitor) supercapacitors; tray with LFP batteries (Lithium Ferro Phosphate LiFePO4); container with LTO batteries (Lithium Titanate Oxide); VRLA (Valve Regulated Lead Acid) battery tray. Source: own study.

) amounting to PLN 0.22. Electricity generation in the West Pomeranian Voivode ship, conforming to the forecast contained in the draft Energy Policy Polish until 2040, will increase by approx. 30% in 2030 compared to 2015. Thus, the values of energy production in 2015 and 2030 will amount to 10015.6 GWh and 11956.08 GWh, respectively. The model in 2030 assumes the use of the potential of energy production from biomass in the West Pomeranian Voivodeship in 27%, in accordance with the provisions (Politics the energy Polish by 2040. Project PEP2040 v. 2.1–08.11.2019, Annex 2, Conclusions from forecasting analyses for the fuel and energy sector, Ministry of Energy, Warsaw 2019, p. 12) (of the draft Energy Policy Polish until 2040. The capacity of energy created from biomass will add up to a minimum of 3228.14 GWh. Whereas, as per the Energy Development Program in the West Pomeranian Voivodeship by 2015 with a viewpoint until 2030, the settled capacity of wind energy in 2030 will increment to 1000 MW.

The coefficients of the target function, i.e., the unit cost for each type of energy produced are included in

Table 5. Cost factors for each type of energy [PLN/kWh].

Types of Energy	Unit Costs
	Production	Certificates	Ecological (EUA)	Storage (LTO)	$\mathbf{Amount}{\mathit{k}}_{\mathit{i}} **$
x1	0.72	0.00	0.032	0.00	0.75
x2	0.87	0.00	0.025	0.00	0.90
x3	0.04	0.03	0.001	0.22	0.29
x4	0.04	0.03	0.001	0.22	0.29
x5	0.17	0.11	0.001	0.22	0.49
x6	0.07	0.04	0.001	0.22	0.33
x7	0.07	0.04	0.001	0.22	0.33
x8	0.02	0.01	0.001	0.22	0.25
x9	0.34	0.21	0.012	0.22	0.78
x10	0.34	0.21	0.012	0.22	0.78
x11	0.34	0.21	0.012	0.22	0.78
x12	0.59	0.36	0.012	0.22	1.18
x13	0.09	0.06	0.000	0.22	0.37
x14	0.04	0.03	0.000	0.22	0.29
x15	0.09	0.06	0.001	0.22	0.37

Analysisof the current formal, legal and market situation as well as technical conditions in terms of the use and development of renewable energy sources in the West Pomeranian Voivodeship, Szczecin 2021. ** for $i\in \left\{1, \dots ,15\right\}.$Source: own study.

 $\left(\mathrm{from}{x}_1 \mathrm{to} x_{15}\right)$, while the unit costs of energy production from energy crops are included in

Table 6. Cost of energy production from energy crops [kWh/ha].

Energy Resources	x17	x18	x19	x20	x21	x22	x23	x24
$\mathrm{Unit}\cos \mathrm{t}\mathrm{of}\mathrm{production}{k}_j$	0.18	0.35	0.17	0.07	0.04	0.50	0.34	0.50

Analysisof the current formal, legal and market situation as well as technical conditions in terms of the use and development of renewable energy sources in the West Pomeranian Voivodeship, Szczecin 2021. for $i\in \left\{17, \dots ,24\right\}$. Source: own study.

$(\mathrm{from}{x}_{17} \mathrm{to}{x}_{24})$.

The optimization model takes the following form:

Objective function

$$\begin{array}{cc}\hfill L_1\left(\mathit{x}\right)=0.75·x_1& +0.90·x_2+0.29·x_3+0.29·x_4+0.49·x_5+0.33·x_6+0.33·x_7+0.25·x_8+0.78·x_9\hfill \\ & +0.78·x_{10}+0.78·x_{11}+1.18·x_{12}+0.37·x_{13}+0.29·x_{14}+0.37·x_{15}+0.18·x_{17}\hfill \\ & +0.35·x_{18}+0.17·x_{19}+0.07·x_{20}+0.04·x_{21}+0.50·x_{22}+0.34·x_{23}+0.50·x_{24}\to min\hfill \end{array}$$

(16)

Under restrictive Equations (17)–(29) resulting from the assumptions of the Draft Energy Policy Polish until 2040. of 23 November 2018. The values of the legal side of the following limiting conditions result from the adopted assumptions and are consistent with the analysis of the current formal, legal and market situation and the technical conditions for the use and development of renewable energy sources in the West Pomeranian Province, Szczecin 2021.

$$x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8+x_9+x_{10}+x_{11}+x_{12}+x_{13}+x_{14}+x_{15}=x_{16 }$$

(17)

The Equation (17) means that the sum of the quantities of energy produced ($\mathrm{from} x_1 \mathrm{to} x_{15}$) is equal to the total energy production $x_{16}.$

$$x_{16}\ge 1195608000$$

(18)

e16 = 1195608000 is the amount of energy in KWh equal to the energy demand for the West Pomeranian region.

The Equation (18) assumes a limitation of the amount of energy produced for the needs of this region at the level of $195608000\mathrm{kWh}$.

$$x_3+x_4+x_5+x_6+x_7+x_8+x_9+x_{10}+x_{11}+x_{12}+x_{13}+x_{14}+x_{15}=0.3·x_{16}$$

(19)

The Equation (19) ensures that 30% of the total energy production for the region comes from renewable energy.

$$x_6+x_7+x_8\ge 3011640000$$

(20)

Equation (20) assumes that the minimum level of production and wind energy will be $3011640000\mathrm{kWh}.$

$$x_8\le 84000000$$

(21)

Equation (21) guarantees that energy production in new wind farms from 1 January 2018 will not be higher than the level of $84000000\mathrm{kWh}.$

$$x_9+x_{10}+x_{11}\ge 60850000$$

(22)

Equation (22) assumes that the production of energy from biogas will not be lower than the level of 60850000 kWh.

$$x_{12}\ge 25500$$

(23)

The Equation (23) assumes that the production of energy from biofuels will be equal to or greater than $25500$ kWh.

$$x_{13}+x_{14}\ge 238200000$$

(24)

The Equation (24) assumes that the production of energy from biomass combustion will be not less than$238200000 \mathrm{kWh}.$

$$x_{15}\ge 0\mathrm{kWh}$$

(25)

Equation (25) assumes that the minimum amount of energy production from geothermal energy can be 0 kWh, i.e., it is allowed that this type of energy will not be generated.

$$x_1=0\mathrm{kWh}$$

(26)

Equation (26) assumes that conventional energy will not be produced, i.e., the volume (in the general model, the limiting Equation (10) is in the form of inequalities $\le$. Due to the value of 0 adopted by the right side of the condition in the scenarios, the notation was modified to preserve its mathematical sense) of production and conventional energy will be $0$kWh.

$$x_2\le 6907531938$$

(27)

Equation (27) assumes that the maximum energy production from co-incineration will not be greater than 6907531938 kWh.

$$x_3+x_4\ge 313770000$$

(28)

The Equation (28) assumes that the production of hydropower will be at least $313770000$kWh.

$$x_5\ge 9590000$$

(29)

The Equation (29) assumes that the production of solar energy will be not less than 9590000 kWh, and as a result of the technological development of photovoltaic cells and there will be an effect of scale resulting in a reduction in the cost of generating kWh of energy.

Boundary conditions given generally by the formula (30) assume that all variables must be non-negative.

$$\bigwedge i x_i\ge 0\hspace{1em}\hspace{1em}\hspace{1em}dla i\in \left\{1, \dots ,24\right\}$$

(30)

The optimal model solution for the indicated target function under limiting Equations (18)–(28) and data boundary conditions with the general formula (30) is contained in

Table 7. Optimal solution.

Types of Energy	Energy Production	Energy Resources	Crop Size
x1	0.00	x17	0.00
x2	4200.70	x18	0.00
x3	320.50	x19	0.00
x4	4.50	x20	0.00
x5	187.99	x21	0.00
x6	0.00	x22	0.00
x7	3575.08	x23	0.00
x8	245.50	x24	0.00
x9	30.45
x10	12.96
x11	54.06
x12	30.00
x13	238.20
x14	3049.94
x15	6.83
x16	11,956.08

Source: own study.

 $L_1\left(\mathit{x}\right)$below.

5. Interpretation of the Model Solution

The total energy production is 11,956.08 GWh (covers the demand of the studied area). The main sources of energy production obtained in the optimization model are:

-

4200.07 GWh is the general amount of energy, produced from co-combustion,

-

320.50 GWh is hydropower from hydroelectric power plants commissioned by 31 December 2017,

-

4.5 GWh is hydropower that can be produced in new installations from 1 January 2018,

-

187.99 GWh is the sum of solar energy from installations until 31 December 2018 (9.59 GWh) and new installations that were built from 1 January 2018 (295.61 GWh),

-

3575.08 GWh is wind energy generated in existing wind farms until 31 December 2017,

-

245.50 GWh is the energy that can be produced in new wind farms built from 1 January 2018,

-

30.45 GWh is energy produced from existing agricultural biogas plants until 31 December 2017,

-

12.96 GWh is the energy that can be produced in biogas plants with high-efficiency cogeneration with a total installed electrical capacity of less than 1 MW, from 1 January 2018,

-

54.06 GWh is the energy that can be produced in new biogas plants from sewage treatment plants and biogas from landfills, created from 1 January 2018,

-

30.00 GWh generation of energy from biogas in new installations from 1 January 2018,

-

238.20 GWh is the energy generated in existing biomass boilers until 31 December 2017,

-

3049.94GWh is the energy that can be produced in new biomass boilers from 1 January 2018,

-

6.83 GWh is the energy that can be generated in new geothermal installations from 1 January 2018,

In the case of energy crops (x17, x18, x19, x20, x21, x22, x23, x24) the economic potential has a big influence. In order to use this potential, farmers must obtain the biomass price they receive for their current food production and, additionally, a risk premium for new production. In the presented model, the economic potential is 0.

The obtained value of the minimum cost for the optimal solution allowed to calculate the average cost of energy. Obtaining the construction of 1 MW of energy will amount to PLN 9,315,120 or EUR 2,076,479.

In order to calculate the optimal solution of the presented equations, the Excel Solver tool was used.

6. Conclusions

At the moment, the Polish energy sector, related to the coal monoculture, lacks electricity and the entire transmission system is obsolete. The solution may be distributed energy, which can ultimately satisfy energy supplies in less urbanized areas and in rural areas, while ensuring the sustainable development of these areas. One of the issues that needs attention, especially in the context of ensuring the state’s energy security, is the remodeling of the approach to energy from central to local.

So far, few authors have comprehensively dealt with distributed energy. So far, no detailed studies on the issue of dispersed energy in the region have been published, and no detailed analysis of the possibility of obtaining alternative energy sources has been carried out. The undertaken research issues contribute to the development of knowledge about distributed energy. The development of local energy is beneficial not only because of the possibility of securing energy security and continuity of supply, but also because of the necessity to implement the assumptions of EU directives, which place particular emphasis on energy efficiency and renewable energy sources. In order to operate effectively in this area, it is necessary to involve local authorities and local governments, to correlate the government’s strategy with investment plans of municipalities, to adapt the necessary legislation, to improve monitoring systems, and to solve the financing issue. So far, there has been no interest in distributed energy in Poland, both from the central energy sector and from regional and local authorities, caused by the volatility of EU and national regulations, which are not conducive to the development of prosumer and renewable energy. The current energy policy of the state is not conducive to the creation of autonomous regional energy systems, where the main decision-maker on the size and structure of the produced energy would be the local government, and not energy companies.

In the context of the development of regional energy, the restructuring of infrastructure, expansion of transmission networks and their adaptation to the current needs of recipients in rural areas, as well as local monitoring of power demand are also of great importance.

The use of energy storage in distributed energy for the provision of energy supply services is becoming an increasingly important issue for the transition to low carbon energy. This has been made possible by the advances in battery and supercapacitor technology over the past decades. This, in turn, was caused by the appearance of new types of loads on the electricity market, which are related to the development of electromobility and renewable energy sources.

The developed distributed energy optimization model is also very useful in regional energy planning and supports the development of an energy and climate plan for the municipality to achieve energy efficiency, renewable energy and greenhouse gas emission reduction targets.