2.1.2.Constraints

Equality constraints (2) represent typical power balance and power flow equations. The power balance equation in distribution network with DG units with renewable and non-renewable energy sources can be expressed as follows:

, (11)

where NR is the number of DG units with renewable energy sources.

The backward/forward sweep power flow equations are given in [12,13].

Constraints (3) define feasibility region of the problem control variables such as: DG unit active power output limits, root node voltage limits, PV bus voltage limits, transformer tap setting limits and shunt VAR compensation limits:

(12)

(13)

(14)

(15)

(16)

Inequality constraints (4) are functional operating constraints consisted of: load bus voltage magnitude limits, DG units reactive power capabilities and branch flow limits:

(17)

(18)

, (19)

Inequality constraints of dependent variables contain load bus voltage magnitude, DG units reactive power output and branch loading is added to the objective function as quadratic penalty terms [6]. The new expanded objective function to be minimized becomes

, (20)

where λ_V, λ_QDG and λ_S are defined as penalty factors. x^lim is limit value of the dependent variable x and given as [11]:

(21)

1. Dg units modeling for optimal power flow

3.1 Diesel generator

Usually, diesel fuel consumption data in (L/h) at 25%, 50%, 70% and 100% of the diesel generator power (kW) rating are given by the manufacturer. Based on these data, fuel consumption characteristic of the diesel generator can be estimated as a quadratic function of its active power output:

, (22)

where Fuel_dg is diesel generator fuel consumption in (L/h), P_dg is diesel generator output power in (kW), a, b and c are coefficients of the fuel consumption characteristic.

Accordingly, the total ($/h) diesel generator fuel cost f_dg can be calculated as:

, (23)

where c_dg is the diesel fuel price to supply the diesel generator in ($/L).

We assume that MDJW 410 T6, 369 kW diesel generator is used in this paper. Diesel generator fuel consumption data is available in [14]. Based on these data and adopted fuel price ($/L), we obtain the fuel costs characteristic shown in Fig. 1.

Fig. 1. Diesel generator MDJW 410 T6 fuel cost curve for c_dg=0.6 ($/L).

3.2 Fuel cell

Fuel cost of the fuel cell depends on the active power output and the fuel cell efficiency [5,15]:

(24)

where is fuel cost for the fuel cell ($/h), is fuel cell output (electrical) power (kW), is fuel cell efficiency, is fuel (natural gas) price to supply the fuel cell ($/kWh).

The fuel cell efficiency can be expressed as the ratio of the actual operating voltage V and 1.482 V [16]:

(25)

In this paper the Proton Exchange membrane (PEM) fuel cell has been considered. The actual operating voltage of the fuel cell is the function of power level, and can be calculated using the following formula [16]:

(26)

where: - theoretical or thermodynamic fuel cell potential (0.9 V); - selected cell potential at nominal power (0.45 – 0.75 V); - power level; - nominal power of the fuel cell.

We assume the PEM fuel cell efficiency characteristic for and . Fig. 2 shows the fuel cost curve estimated for and ($/kWh).

Fig. 2. PEM fuel cost curve for V₀=0.9 V, V_Pnom=0.6 V, P_nom=300 kW and c_fc=0.05 ($/kWh).

3.3 Microturbine

Microturbine fuel costs f_mt in ($/h) can be calculated as the following [5]:

(27)

where is microturbine output electrical power (kW), is microturbine electric efficiency, is price of the fuel to supply the microturbine ($/kWh).

The efficiency of the microturbine increases with the increase of the output power [17]. The electric efficiency characteristic of the microturbine can be estimated as a quadratic function of its output power:

(28)

From a typical electrical efficiency curve of a 300 kW CGT301-302 microturbine [17] we obtain the microturbine fuel costs curve given in Fig. 3. It is assumed that the price of the fuel to supply the microturbine is ($/kWh)

Fig. 3. Microturbine 300 kW CGT301-302 fuel cost curve for c_mt=0.05 ($/kWh).

3.4 Wind turbine

To calculate an output power of the wind turbine, two main factors have been known:

• wind speed on certain location and

• power curve of the wind turbine

According to [18, 19], the power curve of the wind turbine can be modeled by means of the function split into four different parts:

(29)

where , , and are nominal power, nominal wind speed, cut-in wind speed, and cut-out wind speed of the wind turbine, respectively. and denoted output power of the wind turbine and the wind speed.

In this paper, we took into consideration Vestas V44/600 kW wind turbine model [20]. Parameters used to model the power curve obtained from the owner’s manual are as follows: P_nom=600 kW, v_ci=4 m/s, v_nom=16 m/s and v_co=20 m/s.

Fig. 4 shows wind speed data used to calculate the output power generated by the wind turbine generator in the OPF algorithm.

Fig. 4. Wind speed data.

3.5 Photovoltaic

Output power of the photovoltaic module is dependent on the solar irradiance and ambient temperature of the site as well as on the characteristics of the module itself [19]. The following equation can be used to calculate the output power of the photovoltaic module [21]:

(30)

where is photovoltaic module maximum power at Standard Test Condition (STC) (W), is solar irradiance on the photovoltaic module surface (W/m²), is photovoltaic module temperature coefficient for power (˚C^-1), is photovoltaic cell (module) temperature (˚C).

The photovoltaic module temperature can be calculated as a function of the solar irradiance and ambient temperature based on the module’s nominal operating cell temperature (NOCT). The NOCT model equation is given below [22, 23]:

(31)

where T_a is ambient temperature (˚C), T_NOCT is nominal operating cell temperature (°C) of the module.

In this paper, we have considered Sunmodule, SW 250 mono modules. Their performance characteristics are: , , .

Figs. 5 and 6 show the ambient temperature and solar irradiance data [5] used to calculate the output power generated by the photovoltaic modules in the OPF algorithm.

Fig. 5. Ambient temperature data.

Fig. 6. Solar irradiance data.

3.6 Electric grid

Market energy costs in ($/h) can be represented by means of square function as:

(32)

where is electrical power in (kW) from the electric grid, and a, b, c are cost coefficients.

In this paper we adopted: , and .