Modeling of Autonomous Power Systems A Mathematical Model of a Hybrid Power System

Transcription

1 Modeling of Autonomous Power Systems A Mathematical Model of a Hybrid Power System Suvisanna Mustonen 1,* and Khamphone Nanthavong 2 1 Turku School of Economics, Finland Futures Research Centre, Finland 2 National University of Laos, Faculty of Engineering, Lao PDR Abstract: This paper discusses an optimization model for a hybrid power system. The most common criteria for optimal technical systems are based on minimal investment and operation costs. As a part of a pilot project developing a hybrid power system to a remote rural community, a linear mathematical program has been derived for an autonomous small scale village power system. The model considers a 24 hour operation period of a village power system. The aggregated load profile for the village power system was derived from the power consumption data of a typical rural community in Laos. Applying linear programming in this study involves the linearization of cost functions for power load, generation, energy storage, and power distribution. In reality, many components system of a power system have nonlinear characteristics. Further development of the model involves elaboration of the computational grid, objective function, load model and constraints. After developing the computation routine for the 24 hour operation cycle of the village power system, the optimization will be expanded to consider the availability of renewable energy for constant power generation on yearly level. Development of the LP model will be followed by a case study to test the model. The hybrid power system that will be the case study consists of a hydro power unit, solar PV system with energy storage units, and a diesel generator. After installation of the system, data will be gathered and used applied later in the project. Keywords: Autonomous Power System, Modeling, Linear Optimization, Hybrid Power System, Renewable Energy, Combustion Engine, Energy Storage, Load Model 1. INTRODUCTION Off-grid power systems, renewable energy sources, capacity building at the local level, and participation from the private and local sectors are often suggested as desirable, if not successful in developing sustainable energy systems. This is especially true for rural electrification projects and providing energy services to rural areas. One such case is in Lao PDR. According to Ministry of Energy and Mines, approximately 47% of the population in Lao PDR has access to electricity [1]. By 2020, the Government of Lao aims to increase this access to 90% [2]. To meet this target, electricity generated by large scale hydro power plants is foreseen to provide most of electricity for electrification in the country. However, Laos is a mountainous country where grid extension many cases is a financially unviable option. Remote rural communities have characteristically low power demand as a result of sparsely populated distribution areas. Poor road conditions further hamper grid construction. Therefore, the benefits of stand-alone autonomous power systems and renewable local energy sources are recognized in the Government of Lao s strategy for rural electrification. Best potential in renewable energy sources for Laos is in small-scale hydro power, solar energy, biomass, and wind power. Most solar PV systems implemented so far cater for the electricity needs of individual household only, and for a variety of equipment that could be used for income-generating purposes, larger power output is often required. Without a converter, PV system s output is usually DC, while electrical machines often operate on AC. Pico hydro applications on the other hand fail to provide reliable power output to operate machinery due to water level variation. Therefore, larger and more reliable power output is needed for electrical equipment such as a rice mill, husking, food processing, carpenter s tools, water pumps or irrigation. Hybrid power systems offer an alternative to meet the previously mentioned demands by combining two or more generating technologies. This paper will describe the development of a methodology to be used for a village power system optimization model. The case study where the model will be applied first is a hybrid power system developed under a joint private-public rural electrification pilot project to overcome the limitations of solar PV and pico hydro power applications. The pilot project involves the development of a village power system, consisting of the power generation and distribution infrastructure in the target village Nam Ka. The village power system will be a stand-alone hybrid power system, consisting of diesel generators, hydro power and solar PV panels with energy storages. Optimization and simulation models for rural electrification purposes already exist. Often these models have fixed assumptions, which limit their applicability. In the case of modeling rural electrification and power generation, assumptions involve issues such as how power sector is organized in the country, how power suppliers function, what kind of technical standards for equipment may be used in power production, or what is the standard of service in terms of electricity quality and reliability. In Laos, policies and standards for rural off-grid power systems and their operation do not exist yet. This implies that a model with inbuilt operation and performance standards may produce results negligent of the actual operating environment. Therefore, we pursue to develop a simple, transparent optimization model for village power systems, consistent with the context of Lao PDR. Corresponding author: suvisanna.mustonen@tse.fi 1

2 2. METHODOLOGY 2.1 Approach to creation of methodology for optimum design of APS There are three fundamental concerns in forming operations research models: the choices available for decision makers, the constraints limiting decisions, and the objectives that express how some decisions are preferred to others [3]. The operations research problem here is to minimize the cost of electricity. A simple linear programming model will be used. This allows applying simplest LP solving algorithms. 2 A 2 A 4 ES 4 4 CT CT 6 RE A 1 RE 3 ES 3 A 3 Fig. 1 Sample scheme of autonomous electricity power system In this paper, CT has been used to denote combustion engine turbine fueled by renewable or non-renewable fuel like diesel, biodiesel and biogas. Therefore, RE denotes non-fuel renewable energy technologies, in this paper primarily solar PV and small hydro turbines. In Fig. 1, A n denotes appliance or load in node n, CT n and RE n generation units using combustion technologies and renewable energy in node n, respectively, and ES n and DL n denote energy storage units and distribution lines. The graph in Figure 1 represents a simple example of a stand-alone village power system as a network consisting of nodes, which are interconnected by power distribution lines. In the graph there are three types of nodes: generating nodes (nodes 5 & 6), load nodes (node 2) and combined nodes which may be connected to loads, RE and/or energy storages (nodes 1, 3, 4). Model of each node incorporates LP models of components included in the node. In matrix representation, the objective function and constraints of the optimization problem of an autonomous power system can be formulated as follows: min SAPS( x) = c x (1), A x = b (2). In (1) and (2), x is the vector of decision variables; c the matrix describing the coefficients of decision variables; A the matrix describing the coefficients of system s constraint functions that may be equalities or inequalities; and b the vector describing the values of system s constraints on the right hand side of constraint equations and inequalities The most commonly applied criteria to optimize technical systems are minimal investment and operation costs. The criteria can be applied in case of APS as follows: S APS = SCT + SRE + SBA + SDL (3) In (3), S CT, S RE, S BA, S DL, are the cost of investment and operation of combustion engines, renewable energy sources, energy storages, and distribution lines, respectively. Matrix A in (2) comprises of the following constraints: power balance at nodes APS at any given time, energy balance in the energy storages over the village power system s 24 hour duty cycle, rated power and capacity of energy storages, power flows in distribution lines, rated power of combustion engines, renewable energy sources and energy storage as well as rated capacity of energy storage units and rated nominal capacity of distribution lines. 2.2 Load model The load model is a central component in optimization problems that is needed in developing the objective function and constraints. Figure 2 represents the load model for a village power system. The power load has been approximated as a step function, which makes a distinction between low average load and peak load duration during a 24 hour operation cycle for the village power system. The linear programming (LP) model will look at this 24 hour duty cycle and optimize the use of the system over that time. The load and the availability of non-fuel renewable energy determine the timing and duration of the use of combustion engines and energy storages. 2

3 P τ P ES1 P ES2 P ES3 b) βτ P max a) t RE P RE c) P α.p max t P CT1 P CT2 P CT3 Fig. 2 Power load and generation during operation cycle in village power system d) t Fig. 2 depicts village power system s load profile in relation to power generation and storage during the daily 24 hour operation cycle. The diagram blocks are not in proportion with each other regarding quantity of power consumed or generated. The aggregated daily load profile for the village power system is derived from the power consumption data of a typical rural community in Laos, and approximated as a discrete function. The approximation is assumed to be sufficiently accurate provided that 1) the area between the curve and the ordinate axis is numerically equal daily power consumption, 2) the highest step equals to maximum power consumption (P max ), and 3) the lowest step equals to minimum power consumption (P min ). Simplifying the calculation, the profile of power generation is assumed to express a discrete step function just as the load profile. Renewable non-fuel energy sources are assumed to generate power during part of the minimum power demand (Fig. 2c). While combustion engines generate power continuously (Fig.2d), energy storage units charge energy during minimum load hours and eke out the power supply by discharging during the peak load. (Fig.2b). 2.3 Formulation of the objective function Model of power source using combustion technology (CT) Combustion technology such as a diesel engine may use either renewable or non-renewable combustible energy carriers. The cost function of a CT unit can be presented as a sum of two components: investment deduction (appreciation) and fuel costs. The first depends on installed power; and, the second on the type of fuel consumed by the generator set, as well as amount of power produced. Generally the function of such cost may be expressed as following: S CT = ( CCT, inct, i + CF, iect, i) (4) i Ω CT In (4), C CT,i denotes investment into unit produced power at node - N CT,i is number of CT units at node E CT,i energy produced by CT energy source in node summed over duty cycle s all periods, C F,i fuel cost of in node I, and Ω CT number of nodes with available CT units Model of non-fuel Renewable energy sources (RE) RE in this model represents the use of renewable non-fuel energy carriers, primarily solar and hydro power. The cost function on RE is defined based on investment into installed power: S RE = ( CRE, ipre, i) (5) i Ω RE In (5), C RE,,i denotes investment into RE generation per unit in node -i; P RE,,i installed capacity of RE at node -i; and Ω RE number of nodes with available RE units Model of Energy storage (ES) It is assumed that the charging-discharging cycle of the energy storage (ES) is fully completed during the duration time τ of load graphic (Fig. 2d), and that charging of energy storages occurs simultaneously with the minimum load. The cost function of energy storage consists of the investment on rated power of the energy storage units and their capacity. S ES = ( C ES, i EES, i + C ES, i PES, i) (6) i Ω ES In (6), E ES,i denotes maximum storage capacity of energy storage units in node-i during τ, C B,i, C B,i unit investment coefficients into unit storage capacity and rated power of the ES at node- and Ω RE number of nodes with available RE units. The rated power is defined by following expression: P ES, i = max( PES, 1; PES, 2; PES, 3) (7) In (7), P ES i1, P ES,, 2 and P ES,3 are the rated power of ES at node i during the first, second and third sub-period respectively. Energy capacity (E ES,i ) of ES, stored during the 1 st and 2 nd sub-periods is defined as: 3

4 E [ PES, 1 tre + PES, 2 ( t )] = η βτ. (8) ES, i ES, i RE Model of power distribution line The cost function of a power distribution line (DL) consists of two components. The first one is an investment in DL construction, which is proportional to the maximum design power flow. The second component is the cost of compensating for active power loss. Assuming that the active power loss in the DL is proportionate to the power flow, a linear model can be used to express the following: n 3 S DL = Li( CDL, i PDLL, j + ( CL, j PL, j)), (9) j= 1 k= 1 In (9), L ij length of DL between nodes i and j, C DL,j denotes unit cost of DL, P DL,j nominal capacity of DL, k system s operation sub period, C L,j,j loss coefficient of DL and P L,j power flow between nodes i and j, and n total number of nodes. 2.4 Formulation of system s constraint functions The solution above should comply with the technical requirements in the system to describe a techno-economic model. The constraints in the hybrid power system include the following: observed power balance in the nodes; observed energy balance in ES; and maximal parameters of equipment. The technical constraints with the cost function of all APS s components will serve as the basis for optimal modeling of APS. To express the power balance requirements for each node in the system, a load model needs to be developed to function as the extraneous factor to which the power balance in each node has to be fitted. Given that these conditions are fulfilled, the load model of a given node can be expressed by the following expression: E τ = a P max β τ + P max(1 β ) τ (10) In (10), E τ denotes consumed energy during τ; α = P min /P max minimum load coefficient; τ operation cycle of power system, P max aggregated maximum load; P min aggregated minimum load; β = t min /τ relative duration of minimum load and t min duration of daily minimum load Power balance at the nodes Kirchhoff s Law states that in a power system the sum of power flows entering any node in must equal to the total power leaving the node. To simulate this physical law this power balance is required in each node of the power system model. The full duration of the load profile is divided into several sub-periods. For the initial stage of this study, 3 sub-periods were created (Fig. 2). Division of load profile into three sub-periods is arbitrary and could be replaced with some other suitable division. Power balance requirements for a combined node type have been expressed in table 1. In combined node, power generating units, energy storage and load components may be presents. Therefore generating nodes and load nodes are simplified cases of combined nodes. Table 1 Power balance in a node during daily operation cycle Sub Duration Load Generation Units Storage Units Expression Equation period t RE Minimum RE and/or CT producing Charging 2 t RE - β τ Minimum CT producing Charging 3 β τ - τ Maximum All CT producing Discharging P + α RE, i PCT, i PES, i Pij ipi = i j=ω P α P CT, i PES, i + Pij ipi = i j=ω CT, i PES, i Pij = + In equations (11)-(13) of table 1, P ij denotes power flow between nodes i and j; P i load in node and P CT,i, P RE,i, P ES,i,P ES,I and P DL,I rated power of combustion engines, renewable energy generation units, energy storage units and rated nominal capacity of distribution lines, respectively Energy balance in energy storages During the period τ, charging and discharging energy in the energy storage have to be in balance. Since there are always energy losses in an energy storage unit, losses have to be accounted for by multiplying charged energy by energy storage efficiency η ES. The Balance equation for an energy storage unit: [ ES, 1 Fτ + PES, 2 ( βτ Fτ )] PES, p, i(1 β ) = 0 j= Ωi ηes P (14) In (14), η ES is the energy storage efficiency, and F and β relative durations of RE generation and minimum load, respectively (11) (12) (13) Rated power of the energy sources and distribution lines Design rating of CT and RE generation as well as energy storage units limit power allowed in each node of APS. In addition, 4

5 parameters of energy storage should not exceed power generation of combustion engines. These constraints for model can be expressed: PCT, i PCT, max, i ΩCT (15) PRE, i PRE, max, i ΩRE (16) PES, i PES, max, i ΩES (17) Ei P ES, i Pi, i ΩES (18) τ In (18), E i denotes energy consumed in the system during operation cycle τ. The availability of RE, expressed as operation time Fτ, is a statistical value. The energy supply data of RE technologies is usually defined as a yearly average, including days when no energy is generated from these sources. Therefore, a simple capacity reserve principle is applied in the calculation process. Combustion engines are designed on rated power, which is sufficient to supply power for the total demand of consumers during period τ. The requested reserve can be expressed: n PCT Pj i CT j= 1 In (19), i denote number of CT units and j number of customers. Power flow P DL,j in distribution line between nodes j, should not exceed highest rated power flow P DL,j,,max. This restriction is regulated by wire cross section for different types of wire. PDL, j PDL,, j max (20) (19) 3. RESULTS AND DISCUSSION A linear mathematical program has been derived for a small scale village power system. The model presented in the previous chapter considers a 24 hour operation period of a village power system. The following work for this model involves two integral parts. First, the linear program presented in this paper had to be developed into a computable LP problem. Secondly, the number of assumptions made to arrive to a linear program need to be justified or modified. However, seasonal variation in solar radiation and precipitation has an impact on the need of fuels and thus the price of electricity. Optimization of system s daily operation is not sufficient to address this problem. Therefore, after developing the computation routine for 24 hour operation cycle of the village power system, the optimization has to be expanded to consider the availability of renewable energy for constant power generation on yearly level. Finally, the model will be tested in a case study. To develop a computable LP problem, developing the computational grid in node representation, objective function, load model, and constraints need elaboration. Node representation requires the total number and type of nodes in the system, classifying nodes as generating, load or combined nodes. The generating units in each node are to be classified to CT technologies and non-fuel RE technologies. Based on technical and financial data, coefficients of decision variables in objective function need to be defined for each cost component i.e. CT, RE and ES units and distribution lines. The load model should express power demand allocated for each node. Lastly, power system s constraints containing the data of an actual power system are needed in a computable LP problem. Applying linear programming in this study involves the linearization of cost functions for power load, generation, energy storage, and power distribution. In reality, many components system of a power system have nonlinear characteristics. Therefore, both cost function and constraints may be reiterated to apply nonlinear programming. In addition, maintenance, replacement, and operation costs of the system, except for fuel cost, have been neglected in this presentation and will be included later. Usually they are handled as a percentage of overall cost of a unit, e.g. generator set. The aggregated load profile for the village power system was derived from the power consumption data of a typical rural community in Laos. To handle the load in node representation, the aggregated load is allocated for each node. Allocating the load involves considering the number and type of customers connected to each node. The power load has been approximated as a step function, which makes a distinction between low average load and peak load duration during a 24 hour operation cycle for the village expressed power system. This approximation is used initially for convenience and needs to be looked at in more detail before we can justify or develop it further. Power generation and energy storage have to equal to the power demand in the system. Without this balance power systems perform weakly. The balance requirement between power demand and supply makes the load model a central element in the overall village power system model. Energy storage units charge energy during minimum load hours and eke out the power supply by discharging power during the peak load. The load and the availability of non-fuel renewable energy determines the usage of combustion engines and energy storages units in timing and duration as depicted in Fig. 2. The availability of RE, as daily RE operation time Fτ, is a statistical value. The energy supply data of RE technologies is usually defined as yearly average, which includes also the days when no energy is generated from these sources. Therefore, a simple capacity reserve principle is applied in the calculation process. Combustion engines are designed on rated power, which is sufficient to supply power for the total demand of consumers throughout the daily operation cycle. This condition refers to RE availability only, and does not consider peaking or contingency reserve. It is characteristic for distributed and dispersed power systems to trade-off between electricity price and power quality [4]. The village power system in this study is very small in scale, thus it is difficult to maintain a reserve capacity that would guarantee high standard stability in the system without compromising affordable electricity for villagers. For this reason, there is a need for elementary demand side management, which involves regulating what kind of electrical equipment and machinery can be connected to the village grid. The hybrid power system that will be the case study for testing the LP model consists of a 25 kw hydro power plant unit, 3 kw solar PV system with energy storage units, and a 15 kw diesel generator operated with diesel and augmented with bio fuel with the aim to gradually replace all the diesel. This hybrid system is planned to provide electricity to 200 households and one commercial 5

6 electricity user. The average power demand per household is projected to be 2.4 kwh per day. At present, construction of the hybrid power system is underway. Measured data from the operation of the system will be gathered during following months and used applied later in the project. 4. CONCLUSION A linear mathematical program has been derived for an autonomous small scale village power system. The model considers a 24 hour operation period of a village power system. Applying linear programming in this study involves the linearization of cost functions for power load, generation, energy storage, and power distribution. The aggregated load profile for the village power system was derived from the power consumption data of a typical rural community in Laos. The power load has been approximated as a step function, which makes a distinction between low average load and peak load duration during a 24 hour operation cycle for the village expressed power system. To develop a computable LP problem, the computational grid in node representation, objective function, load model, and constraints need elaboration. In reality, many components system of a power system have nonlinear characteristics. After developing the computation routine for 24 hour operation cycle of the village power system, the optimization has to be expanded to consider the availability of renewable energy for constant power generation on yearly level. Development of the LP model will be followed by a case study to test the model. The hybrid power system that will be the case study consists of a hydro power unit, solar PV system with energy storage units, and a diesel generator. Finally, the model will be tested in a case study. Measured data from the operation of the system will be gathered during following months and used applied later in the project. 5. ACKNOWLEDGMENTS The authors gratefully acknowledge the contribution of the Academy of Finland in financing this project. 6. REFERENCES [1] Private Communication (2006), Ministry of Energy and Mines, Government of Lao PDR. [2] Power Sector Development Plan for Lao PDR (2004), Final Report. Maunsell Ltd and Lahmeyer GmbH, Auckland, New Zealand. [3] Rardin, R.L. (2000) Optimization in operations research, Prentice-Hall, New Jersey, USA [4] Willis, H.L. and Scott, W.G. (2000) Distributed power generation: planning and evaluation, Marcel Dekker, New York, USA 6