1
Optimal power flow solutions incorporating stochastic wind and solar power Partha P Biswas1, P. N. Suganthan1, Gehan A J. Amaratunga2
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1School
of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 2Department
of Engineering, University of Cambridge, UK
[email protected],
[email protected],
[email protected] Abstract: Generations from several sources in an electrical network are to be optimally scheduled for
economical and efficient operation of the network. Optimal power flow problem is formulated with all relevant system parameters including generator outputs and solved subsequently to obtain the optimal settings. The network may consist of conventional fossil fuel generators as well as renewable sources like wind power generators and solar photovoltaic. Classical optimal power flow itself is a highly non-linear complex problem with non-linear constraints. Incorporating intermittent nature of solar and wind energy escalates the complexity of the problem. This paper proposes an approach to solve optimal power flow combining stochastic wind and solar power with conventional thermal power generators in the system. Weibull and lognormal probability distribution functions are used for forecasting wind and solar photovoltaic power output respectively. The objective function considers reserve cost for overestimation and penalty cost for underestimation of intermittent renewable sources. Besides, emission factor is also included in objectives of selected case studies. Success history based adaptation technique of differential evolution algorithm is adopted for the optimization problem. To handle various constraints in the problem, superiority of feasible solutions constraint handling technique is integrated with success history based adaptive differential evolution algorithm. The algorithm thus combined and constructed gives optimum results satisfying all network constraints. Keywords: Optimal power flow · Emission · Wind power · Solar photovoltaic · Success history based adaptive differential evolution · Constraint handling technique
Nomenclature Abbreviations OPF optimal power flow SHADE success history based adaptive differential evolution
𝑔𝑗 ℎ𝑘 𝐾𝑅𝑤,𝑗
SF
superiority of feasible solutions
𝐾𝑃𝑤,𝑗
TG
thermal power generator
𝐾𝑅𝑠,𝑘
WG
wind generator
𝐾𝑃𝑠,𝑘
PV ISO PDF
photovoltaic independent system operator probability density function
Symbol 𝑃𝑇𝐺𝑖 𝑃𝑤𝑠,𝑗
power output of 𝑖-th thermal generator scheduled power from 𝑗-th wind power plant
𝐶𝑡𝑎𝑥 𝐺 𝑓𝑣 (𝑣) 𝑓𝐺 (𝐺) 𝑝𝑤𝑟 𝑃𝑠𝑟 𝑐, 𝑘
𝑃𝑠𝑠,𝑘
scheduled power from 𝑘-th solar PV plant
𝜇, 𝜎
𝑃𝑤𝑎𝑣,𝑗 𝑃𝑠𝑎𝑣,𝑘
actual available power from 𝑗-th wind power plant actual available power from 𝑘-th solar PV plant
𝑃𝑙𝑜𝑠𝑠 𝑉𝐷
27 28 29 30 31 32 33
1.
direct cost coefficient for 𝑗-th wind power plant direct cost coefficient for 𝑘-th solar PV plant reserve cost coefficient for over-estimation of wind power from 𝑗-th plant penalty cost coefficient for under-estimation of wind power from 𝑗-th plant reserve cost coefficient for over-estimation of solar power from 𝑘-th plant penalty cost coefficient for under-estimation of solar power from 𝑘-th plant carbon tax in $/tonne solar irradiance in W/m2 probability of wind speed 𝑣 m/s probability of solar irradiance 𝐺 W/m2 rated output power of a wind turbine rated output power of the solar PV plant Weibull PDF scale and shape parameters respectively lognormal PDF mean and standard deviation respectively real power loss in the network cumulative voltage deviation in the network
Introduction
Optimal power flow (OPF) remains a widely-cultivated topic within power system research community since its inception about half-a-century ago. The prime objective of OPF is minimization of generation cost with optimal settings of control variables which are the generated real power and generator bus voltages of the network. While optimizing the generation cost, system constraints on generator capability, line capacity, bus voltage and power flow 1
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balance are to be satisfied. Scheduled generator power, complex power flow in the lines and voltage vector of buses determined during the process of optimization represent the optimal operating state of the system. Classical OPF problem considers thermal power generators run on fossil fuels. With increasing penetration of wind power and solar PV in the network, the study of OPF becomes necessary incorporating uncertainties of these renewable sources. OPF with only thermal power generators has extensively been studied by researchers across the globe. A recent literature described application of state-of-the-art evolutionary algorithm (EA) is differential search algorithm (DSA) [1] where a few standard objectives in OPF are optimized for IEEE bus systems with thermal generators. Standard group search optimization algorithm is improved with adaptive group search optimization (AGSO) [2] to perform similar study on OPF. Reference [3] performs OPF calculation with more complex objectives of multi-fuel options and considers valve-point effect in thermal generators in applying backtracking search optimization algorithm (BSA). Improved colliding bodies optimization (ICBO) algorithm is proposed in [4] where number of colliding bodies are increased in each iteration to enhance performance of the algorithm when applied to the problem of OPF. A most recent literature [5] applies moth swarm algorithm (MSA) on numerous objectives of OPF for various bus systems to show effectiveness of the algorithm in terms of fast execution time and quick convergence. While abovementioned references deal with conventional generators only, a system consisting of thermal and wind power generators has recently been studied in pursuit of minimum generation cost in a few literatures. Gbest guided artificial bee colony (GABC) is applied in [6] to improve OPF results recorded in past literatures with similar experimental set up. Reference [7] proposes modified bacteria foraging algorithm (MBFA) and introduced doubly fed induction generator (DFIG) model in OPF framework to define limits on reactive power generation capability. Additional reactive power supporting devices, static synchronous compensator (STATCOM) is incorporated in [8] for system with wind and thermal power generators and the OPF problem is solved using ant colony optimization (ACO) and also MBFA. Authors in [9] proposes a paradigm for modelling the cost of wind-generated electricity. The problem on scheduling of generators for economic dispatch is more commonplace for system with thermal power and wind generators. In an OPF dispatching program, reference [10] presents a stochastic model of wind generation. In attempting the same problem, authors in [11] includes DFIG model of wind turbine. Literature [12] proposed dynamic economic dispatch (DED) model with penetration of large scale wind power considering risk reserve constraints. Reference [13] incorporates emission, valve-point loading effect of generator in DED problem. OPF management for an isolated hybrid system with solar PV, diesel generator and battery is presented in [14]. Pumped hydro storage is introduced in [15] as an alternate form of storage for a similar standalone hybrid system consisting of a solar PV, a wind turbine and a diesel generator. Integration of wind and solar PV power into the grid is studied in a few literatures. However, these literatures focus primarily on real-time scheduling of generators for economical operation considering various pricing strategies between the utility operator and the independent system operator (ISO). Economic dispatch being main objective, reference [16] considers minute-to-minute variation of renewable energy sources. Hybrid system in [17] includes diesel generator with optimization platform being basic MATLAB functions. In economic dispatch (ED) problem, system constraints especially limitations on network parameters may have often been ignored; however, complying with network constraints is a must in OPF. Literature [18] mentions of system constraints, but details on satisfying those constraints have not explicitly been addressed. Furthermore, voltage profile throughout the network, emission aspects are generally not addressed in ED problem, but in OPF problem. In summary, optimal power flow in a network consisting of thermal, wind power generators and solar PV needs further attention. The present study is dedicated to optimal power flow problem with detailed uncertainty modelling of wind and solar power. The biggest challenge in incorporating wind and solar PV power in grid integration is their intermittent nature. Normally wind or solar PV farms are owned by private operators. Grid / independent system operator (ISO) signs an agreement of purchasing scheduled power from these private operators. But as generations from these renewable sources are uncertain, sometimes the power output may be more than the scheduled power leading to underestimation of the available amount. ISO is to bear the penalty cost as surplus power goes wasted if not utilized. On contrary, overestimation is the scenario when the generated power is less than the scheduled power. To mitigate power demand, ISO needs to keep spinning reserve which adds up to the operating cost of the system. The objective function formulated in this paper considers direct cost, penalty and reserve cost of renewable sources in addition to generation cost of thermal power units. Wind distribution is modelled using Weibull probability density function (PDF), solar irradiance is modelled with lognormal PDF. IEEE-30 bus system is modified to accommodate wind generators and solar PV with reactive power capabilities. Generation cost is optimized and effect of change in reserve and penalty costs on optimal scheduling is studied. On the aspect of emission, fossil fuel driven thermal generators emit harmful gases into the environment, while renewable sources do not. Carbon tax [19] is imposed in some countries in proportion to the emitted greenhouse gases. In selected case studies, carbon tax amount is entwined with the objective function to study the effect on generator scheduling. Success history based parameter adaptation technique of differential evolution (SHADE) is employed for the optimization problem. SHADE, an advanced variant of differential evolution (DE), uses a historical memory of 2
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successful control parameter settings to guide the selection of future control parameter values [20]. This ensures accurate and fast convergence to the global optima of a constrained, multimodal, non-linear optimization problem. SHADE is combined with an effective constraint handling technique called superiority of feasible solutions (SF) [21]. In almost all the literatures of OPF, penalty function approach is adopted to check violation of constraints. This approach is sensitive to selection of penalty coefficient. Small penalty coefficient over-explores the infeasible region, delaying the process of finding feasible solutions, and may prematurely converge to an infeasible solution. On the other hand, large penalty coefficient may not explore the infeasible region properly, thereby resulting in untimely convergence [22]. The proposed SF method of constraint handling carefully compares between a pair of solutions where both can either be feasible, infeasible or a mix. Based on comparison, the search is directed towards the feasible region. With penalty function approach, there is a slight possibility of constraint violation if penalty factor selection is inappropriate, sometimes without cognizance of the programmer. However, using a proper constraint handling technique discards any such probability.
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Table 1 summarizes basic parameters of IEEE-30 bus network under study. The adapted network consists of thermal, wind generators and solar PV. Both wind and solar PV power outputs are variables. The fluctuation in power output must be balanced by the combination of all generator outputs and reserve. Total generation cost thus consists of operation costs of all the generators, penalty cost and reserve cost which are explained subsequently.
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The organization of rest of the paper is done in following way. Section 2 includes a review of mathematical model including applicable constraints pertaining to OPF problem. In section 3, modelling of uncertainties in wind and solar power output is presented. Description and application of SHADE-SF algorithm are elaborated in section 4. Section 5 discusses case studies and simulation results followed by concluding remarks in section 6.
2.
Mathematical models
Table 1: Summary of IEEE-30 bus system under study Items Quantity Details Buses Branches Thermal generators (𝑇𝐺1, 𝑇𝐺2, 𝑇𝐺3) Wind generators (𝑊𝐺1, 𝑊𝐺2) Solar PV unit (𝑆𝑃𝑉) Shunt compensation (fixed) Control variables
30 41 3 2 1 2 11
Connected load Load bus voltage range allowed
24
[23] [23] Buses: 1 (swing), 2 and 8 Buses: 5 and 11 Bus: 13 Bus 5 (0.19 MVAr); Bus 24 (0.04 MVAr) Scheduled real power for 5 nos. generators: 𝑇𝐺2, 𝑇𝐺3,𝑊𝐺1, 𝑊𝐺2 and 𝑆𝑃𝑉; bus voltages of all generator buses (6 nos.) 283.4 MW, 126.2 MVAr [0.95 – 1.05] p.u.
2.1 Cost model of thermal power generators Thermal generating units need fossil fuel for operation. The association between fuel cost ($/h) and generated power (MW) is approximately given by the quadratic relationship: 𝑁𝑇𝐺 2 𝐶𝑇0 (𝑃𝑇𝐺 ) = ∑ 𝑎𝑖 + 𝑏𝑖 𝑃𝑇𝐺𝑖 + 𝑐𝑖 𝑃𝑇𝐺𝑖
(1)
𝑖=1
30 31 32 33 34 35 36
where 𝑎𝑖 , 𝑏𝑖 , 𝑐𝑖 are the cost coefficients of the 𝑖-th thermal generator producing power output 𝑃𝑇𝐺𝑖 . Total number of thermal generators is 𝑁𝑇𝐺 . Valve-point effect needs to be considered for more realistic and precise modeling of cost function. The thermal generating units with multi-valve steam turbines exhibit a greater variation in the fuel-cost functions [4]. The valve loading effect of multi-valve steam turbines is modelled as sinusoidal function, the absolute value of which is added to the basic cost function in equation (1). Total generation cost ($/h) of thermal units becomes: 𝑁𝑇𝐺 2 𝑚𝑖𝑛 𝐶𝑇 (𝑃𝑇𝐺 ) = ∑ 𝑎𝑖 + 𝑏𝑖 𝑃𝑇𝐺𝑖 + 𝑐𝑖 𝑃𝑇𝐺𝑖 + |𝑑𝑖 × sin (𝑒𝑖 × (𝑃𝑇𝐺𝑖 − 𝑃𝑇𝐺𝑖 ))|
(2)
𝑖=1
37 38 39
𝑚𝑖𝑛 where, 𝑑𝑖 and 𝑒𝑖 are the coefficients that represent the valve-point loading effect. 𝑃𝑇𝐺𝑖 is the minimum power the 𝑖-th thermal unit generates when in operation. All cost and emission coefficients for the thermal generating units used in the calculations are provided in Table 2.
3
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2.2 Direct cost of wind and solar photovoltaic power Unlike conventional thermal power generators, solar PV and wind power generators require no fuel. In the case where the wind/solar PV plants are owned by the independent system operator (ISO), the cost function may not exist as the wind/solar PV plants require no fuel, unless ISO wants to assign some payback cost to the initial outlay for the wind/solar PV plants or to assign this as a maintenance and renewal cost [24]. But when wind or solar PV plants are owned by private parties, ISO pays a price proportional to the scheduled power contractually agreed. Direct cost involved with wind power from 𝑗-th plant is modelled as a function of scheduled power: (3)
𝐶𝑤,𝑗 (𝑃𝑤𝑠,𝑗 ) = 𝑔𝑗 𝑃𝑤𝑠,𝑗
10 11 12 13
where, 𝑔𝑗 is the direct cost coefficient associated with 𝑗-th wind power plant, 𝑃𝑤𝑠,𝑗 is the scheduled power from the same plant. Similar to wind power plant, direct cost pertaining to 𝑘-th solar PV plant is: (4)
𝐶𝑠,𝑘 (𝑃𝑠𝑠,𝑘 ) = ℎ𝑘 𝑃𝑠𝑠,𝑘
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where, ℎ𝑘 is the direct cost coefficient associated with 𝑘-th solar PV plant, 𝑃𝑠𝑠,𝑘 is the scheduled power from the same plant. 2.3 Cost evaluation of uncertainties in wind power A situation may arise when actual power delivered by the wind farm is less than the estimated value. This is termed as overestimation of power from the uncertain source. The system operator needs to have spinning reserve for such scenarios to provide uninterrupted supply to the consumers. The cost of committing the reserve generating units to meet over-estimated amount is termed as reserve cost [8]. Reserve cost for the 𝑗-th wind power plant is defined as: 𝑃𝑤𝑠,𝑗
𝐶𝑅𝑤,𝑗 (𝑃𝑤𝑠,𝑗 − 𝑃𝑤𝑎𝑣,𝑗 ) = 𝐾𝑅𝑤,𝑗 (𝑃𝑤𝑠,𝑗 − 𝑃𝑤𝑎𝑣,𝑗 ) = 𝐾𝑅𝑤,𝑗 ∫
(𝑃𝑤𝑠,𝑗 − 𝑝𝑤,𝑗 )𝑓𝑤 (𝑝𝑤,𝑗 )𝑑𝑝𝑤,𝑗
(5)
0
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where, 𝐾𝑅𝑤,𝑗 is the reserve cost coefficient pertaining to 𝑗-th wind power plant, 𝑃𝑤𝑎𝑣,𝑗 is the actual available power from the same plant. 𝑓𝑤 (𝑝𝑤,𝑗 ) is the wind power probability density function for 𝑗-th wind power plant. Calculation of probabilities of output power at various wind speeds are discussed in section 3.2. On contrary to the overestimation case, there may be condition in the network when actual power delivered by the wind farm is higher than the estimated value. The renewable source output is underestimated in such case. The surplus power will be wasted if not possible to utilize by reducing power output from conventional generators. ISO needs to pay a penalty cost corresponding to the surplus amount. Penalty cost for the 𝑗-th wind power plant is defined as: 𝑃𝑤𝑟,𝑗
𝐶𝑃𝑤,𝑗 (𝑃𝑤𝑎𝑣,𝑗 − 𝑃𝑤𝑠,𝑗 ) = 𝐾𝑃𝑤,𝑗 (𝑃𝑤𝑎𝑣,𝑗 − 𝑃𝑤𝑠,𝑗 ) = 𝐾𝑃𝑤,𝑗 ∫
(𝑝𝑤,𝑗 − 𝑃𝑤𝑠,𝑗 )𝑓𝑤 (𝑝𝑤,𝑗 )𝑑𝑝𝑤,𝑗
(6)
𝑃𝑤𝑠,𝑗
34 35 36 37 38 39 40 41 42 43 44
where, 𝐾𝑃𝑤,𝑗 is the penalty cost coefficient for the 𝑗-th wind power plant, 𝑃𝑤𝑟,𝑗 is rated output power from the same windfarm. 2.4 Cost evaluation of uncertainties in solar photovoltaic power Like wind power plant, solar PV plant also have intermittent and uncertain output. In principle, approach to over and under estimation of solar power shall be same as the wind power. However, as solar radiation follows lognormal PDF [25], different from wind distribution which is well known for trailing Weibull PDF, for convenience in calculation the reserve and penalty cost models are built based on the concept presented in [9]. Further details are presented in section 3 in calculating stochastic wind and solar PV power. Reserve cost for the 𝑘-th solar PV plant is: 𝐶𝑅𝑠,𝑘 (𝑃𝑠𝑠,𝑘 − 𝑃𝑠𝑎𝑣,𝑘 ) = 𝐾𝑅𝑠,𝑘 (𝑃𝑠𝑠,𝑘 − 𝑃𝑠𝑎𝑣,𝑘 ) = 𝐾𝑅𝑠,𝑘 ∗ 𝑓𝑠 (𝑃𝑠𝑎𝑣,𝑘 < 𝑃𝑠𝑠,𝑘 ) ∗ [𝑃𝑠𝑠,𝑘 − 𝐸(𝑃𝑠𝑎𝑣,𝑘 < 𝑃𝑠𝑠,𝑘 )]
45
(7)
where, 𝐾𝑅𝑠,𝑘 is the reserve cost coefficient pertaining to 𝑘-th solar PV plant, 𝑃𝑠𝑎𝑣,𝑘 is the actual available power 4
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from the same plant. 𝑓𝑠 (𝑃𝑠𝑎𝑣,𝑘 < 𝑃𝑠𝑠,𝑘 ) is the probability of solar power shortage occurrence than the scheduled power (𝑃𝑠𝑠,𝑘 ), 𝐸(𝑃𝑠𝑎𝑣,𝑘 < 𝑃𝑠𝑠,𝑘 ) is the expectation of solar PV power below 𝑃𝑠𝑠,𝑘 . Penalty cost for the underestimation of 𝑘-th solar PV plant is: 𝐶𝑃𝑠,𝑘 (𝑃𝑠𝑎𝑣,𝑘 − 𝑃𝑠𝑠,𝑘 ) = 𝐾𝑃𝑠,𝑘 (𝑃𝑠𝑎𝑣,𝑘 − 𝑃𝑠𝑠,𝑘 ) = 𝐾𝑃𝑠,𝑘 ∗ 𝑓𝑠 (𝑃𝑠𝑎𝑣,𝑘 > 𝑃𝑠𝑠,𝑘 ) ∗ [𝐸(𝑃𝑠𝑎𝑣,𝑘 > 𝑃𝑠𝑠,𝑘 ) − 𝑃𝑠𝑠,𝑘 ]
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(8)
where, 𝐾𝑃𝑠,𝑘 is the penalty cost coefficient pertaining to 𝑘-th solar PV plant, 𝑓𝑠 (𝑃𝑠𝑎𝑣,𝑘 > 𝑃𝑠𝑠,𝑘 ) is the probability of solar power surplus than the scheduled power (𝑃𝑠𝑠,𝑘 ), 𝐸(𝑃𝑠𝑎𝑣,𝑘 > 𝑃𝑠𝑠,𝑘 ) is the expectation of solar PV power above 𝑃𝑠𝑠,𝑘 . 2.5 Emission and carbon tax It is well known that generating power from conventional sources of energy emits harmful gases into the environment. The emission of SOx, NOx increases with increase in generated power (in p.u. MW) from thermal power generators following the relationship in equation (9). Emission in tonnes per hour (t/h) is calculated by: 𝑁𝑇𝐺
Emisson, 𝐸 = ∑[(𝛼𝑖 + 𝛽𝑖 𝑃𝑇𝐺𝑖 + 𝛾𝑖 𝑃𝑇𝐺𝑖 2 ) × 0.01 + 𝜔𝑖 𝑒 (𝜇𝑖𝑃𝑇𝐺𝑖) ]
(9)
𝑖=1
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where, 𝛼𝑖 , 𝛽𝑖 , 𝛾𝑖 , 𝜔𝑖 and 𝜇𝑖 are all emission coefficients corresponding to the 𝑖-th thermal generator. Emission coefficients for the thermal generating units are provided in Table 2. The coefficients are same as in [3] with some minor adjustment made in μ value for the generator connected to bus 1. Table 2:
Cost and emission coefficients of thermal generators for the system under study [3] Generator
Bus
𝑎
𝑏
𝑐
𝑑
𝑒
α
β
γ
ω
μ
𝑇𝐺1
1
0
2
0.00375
18
0.037
4.091
-5.554
6.49
0.0002
6.667
𝑇𝐺2
2
0
1.75
0.0175
16
0.038
2.543
-6.047
5.638
0.0005
3.333
𝑇𝐺3
8
0
3.25
0.00834
12
0.045
5.326
-3.55
3.38
0.002
2
In recent years, due to global warming, many countries are putting enormous pressure on entire energy industry to reduce carbon emission [19]. To encourage investment in cleaner forms of power like wind and solar, carbon tax (𝐶𝑡𝑎𝑥 ) is imposed on per unit amount of emitted greenhouse gases. The cost of emission (in $/h) is represented as: Emisson cost, 𝐶𝐸 = 𝐶𝑡𝑎𝑥 𝐸
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(10)
2.6 Objective of optimization The objective of OPF is formulated incorporating all the cost functions as discussed above. In first objective, emission cost is not included. To study the change in generation scheduling when carbon tax is imposed, second objective function is constructed adding emission cost. First objective: Minimize – 𝑁𝑊𝐺
𝐹1 = 𝐶𝑇 (𝑃𝑇𝐺 ) + ∑ [𝐶𝑤,𝑗 (𝑃𝑤𝑠,𝑗 ) + 𝐶𝑅𝑤,𝑗 (𝑃𝑤𝑠,𝑗 − 𝑃𝑤𝑎𝑣,𝑗 ) + 𝐶𝑃𝑤,𝑗 (𝑃𝑤𝑎𝑣,𝑗 − 𝑃𝑤𝑠,𝑗 )] 𝑗=1 𝑁𝑆𝐺
(11)
+ ∑[𝐶𝑠,𝑘 (𝑃𝑠𝑠,𝑘 ) + 𝐶𝑅𝑠,𝑘 (𝑃𝑠𝑠,𝑘 − 𝑃𝑠𝑎𝑣,𝑘 ) + 𝐶𝑃𝑠,𝑘 (𝑃𝑠𝑎𝑣,𝑘 − 𝑃𝑠𝑠,𝑘 )] 𝑘=1
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where, 𝑁𝑊𝐺 and 𝑁𝑆𝐺 are the numbers of wind generators and solar PVs in the network respectively. All other cost components are calculated using equations (2) to (8). Second objective: Minimize – 𝐹2 = 𝐹1 + 𝐶𝑡𝑎𝑥 𝐸
33
(12)
The OPF objectives are subject to system equality and inequality constraints. 5
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2.6.1. Equality constraints Equality constraints are the power balance equations where both active and reactive power generated in the network must be equal to the demand and losses in the network. 𝑁𝐵
𝑃𝐺𝑖 − 𝑃𝐷𝑖 − 𝑉𝑖 ∑ 𝑉𝑗 [𝐺𝑖𝑗 cos(𝛿𝑖𝑗 ) + 𝐵𝑖𝑗 sin(𝛿𝑖𝑗 )] = 0 ∀ 𝑖 ∊ 𝑁𝐵
(13)
𝑗=1 𝑁𝐵
𝑄𝐺𝑖 − 𝑄𝐷𝑖 − 𝑉𝑖 ∑ 𝑉𝑗 [𝐺𝑖𝑗 sin(𝛿𝑖𝑗 ) − 𝐵𝑖𝑗 cos(𝛿𝑖𝑗 )] = 0 ∀ 𝑖 ∊ 𝑁𝐵
(14)
𝑗=1
6 7 8 9 10 11 12 13 14
where 𝛿𝑖𝑗 = 𝛿𝑖 − 𝛿𝑗 , is the difference in voltage angles between bus 𝑖 and bus 𝑗, 𝑁𝐵 is the total number of buses, 𝑃𝐷𝑖 and 𝑄𝐷𝑖 are active and reactive load demands respectively at bus 𝑖 . 𝑃𝐺𝑖 and 𝑄𝐺𝑖 are active and reactive power generations respectively at bus 𝑖 from any of the sources (conventional or renewable) as applicable. 𝐺𝑖𝑗 is the transfer conductance and 𝐵𝑖𝑗 is the susceptance between bus 𝑖 and bus 𝑗 respectively. 2.6.2. Inequality constraints The inequality constraints are the operating limits of the equipment, components in power system and security constraints on lines and load buses. a) Generator constraints:
b) Security constraints:
𝑚𝑖𝑛 𝑚𝑎𝑥 𝑃𝑇𝐺𝑖 ≤ 𝑃𝑇𝐺𝑖 ≤ 𝑃𝑇𝐺𝑖 ,
𝑖 = 1, … . . , 𝑁𝑇𝐺
(15)
𝑚𝑖𝑛 𝑃ws,j
𝑚𝑎𝑥 𝑃ws,j ,
𝑗 = 1, … . . , 𝑁𝑊𝐺
(16)
𝑚𝑖𝑛 𝑚𝑎𝑥 𝑃ss,k ≤ 𝑃𝑠𝑠,𝑘 ≤ 𝑃ss,k ,
𝑘 = 1, … . . , 𝑁𝑆𝐺
(17)
𝑚𝑖𝑛 𝑄𝑇𝐺𝑖
𝑖 = 1, … . . , 𝑁𝑇𝐺
(18)
𝑚𝑖𝑛 𝑚𝑎𝑥 𝑄ws,j ≤ 𝑄𝑤𝑠,𝑗 ≤ 𝑄ws,j ,
𝑗 = 1, … . . , 𝑁𝑊𝐺
(19)
𝑚𝑖𝑛 𝑄ss,k
𝑘 = 1, … . . , 𝑁𝑆𝐺
(20)
𝑚𝑖𝑛 𝑚𝑎𝑥 𝑉𝐺𝑖 ≤ 𝑉𝐺𝑖 ≤ 𝑉𝐺𝑖 ,
𝑖 = 1, … . . , 𝑁𝐺
(21)
𝑉𝐿𝑚𝑖𝑛 𝑝
𝑝 = 1, … . . , 𝑁𝐿
(22)
≤ 𝑃𝑤𝑠,𝑗 ≤
≤ 𝑄𝑇𝐺𝑖 ≤ ≤ 𝑄𝑠𝑠,𝑘 ≤ ≤ 𝑉𝐿𝑝 ≤
𝑚𝑎𝑥 𝑄𝑇𝐺𝑖 ,
𝑚𝑎𝑥 𝑄ss,k ,
𝑉𝐿𝑚𝑎𝑥 , 𝑝
𝑆𝑙𝑞 ≤ 𝑆𝑙𝑚𝑎𝑥 , 𝑞
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
𝑞 = 1, … . . , 𝑛𝑙
(23)
Equations (15) to (17) represent the active power generation limits of thermal, wind generator and solar PV respectively. Following same sequence equations (18) to (20) define the reactive power capabilities of the generators. 𝑁𝐺 is the total number of generators or generator buses. Equation (21) is for the constraints on voltage of generator buses, while equation (22) defines the voltage limits imposed on load buses (PQ buses) with 𝑁𝐿 being the number of load buses. Line capacity constraints are given by equation (23) for total 𝑛𝑙 numbers of lines in the network. It is worth mentioning that convergence of power flow to a solution ensures that the equality constraints of power balance equations are automatically satisfied. Among inequality constraints, generator active power (except slack or swing generator considered to be connected to bus 1) and generator bus voltages are termed as control variables which are self-limiting. For each such variable, the optimization algorithm selects a feasible value bound by the range of that variable. Inequality constraints on slack generator power, reactive power output of the remaining generators, load bus voltage limits and line capabilities need special attention. Handling of inequality constraints consisting of these variables is discussed in section 4. Generator reactive power capability is an important aspect in OPF study. For thermal generators, narrower ranges are implemented in the study than what have been provided in [23,26]. The reactive power capability of wind turbines has considerably evolved in recent years. Wind turbines (WTs) featuring full reactive power capability are already commercially available [27]. The reactive power capability curve of Enercon FACTS-WT shows that throughout active power output range, the WT can deliver reactive power from -0.4 p.u. to 0.5 p.u. Delivery of negative reactive power signifies reactive power absorbing capability of the generator. Rooftop solar PVs can be modelled as load bus (PQ bus) with 𝑄=0. However, utility scale solar PVs are equipped with converters for which full generator modelling (P-Q capability) becomes necessary owning to the dynamic behavior of the converters [28]. Reference [29] analyses reactive power capability of solar PV including converter and controller models. Authors 6
1 2 3 4 5 6 7
in [30] extended the PV converter capability study considering variation in solar radiation and ambient temperature. For our study purpose, we consider reactive power capability of solar PV is approximately between -0.4 p.u. to 0.5 p.u. Active (P) and reactive (Q) power limits of generators are listed in Table 5 under simulation results. In OPF problem, system parameters like real power loss in the network and voltage deviation are also important. The power loss in transmission system is unavoidable as the lines have inherent resistance. The network loss is calculated as: 𝑛𝑙
𝑃𝑙𝑜𝑠𝑠 = ∑ 𝐺𝑞(𝑖𝑗) [𝑉𝑖 2 + 𝑉𝑗 2 − 2𝑉𝑖 𝑉𝑗 cos(𝛿𝑖𝑗 )]
(24)
𝑞=1
8 9 10 11 12 13
where, 𝛿𝑖𝑗 = 𝛿𝑖 − 𝛿𝑗 , is the difference in voltage angles between bus 𝑖 and bus 𝑗 and 𝐺𝑞(𝑖𝑗) is the transfer conductance of branch 𝑞 connecting buses 𝑖 and 𝑗. Voltage deviation is a measure of voltage quality in the network. The voltage deviation indicator is formulated as cumulative deviation of voltages of all load buses (PQ buses) in the network from nominal value of 1 p.u. Mathematically it is expressed as: 𝑁𝐿
𝑉𝐷 = (∑ |𝑉𝐿𝑝 − 1|)
(25)
𝑝=1
14 15
3.
16 17
It is well established that wind speed distribution follows Weibull probability density function (PDF) [6-9]. The probability of wind speed 𝑣 m/s following Weibull PDF with shape factor (𝑘) and scale factor (𝑐) is given by:
Stochastic wind / solar power and uncertainty models
𝑘
𝑣 (𝑘−1)
𝑐
𝑐
𝑓𝑣 (𝑣) = ( ) ( )
18
𝑘
𝑒 −(𝑣/𝑐) for 0 < 𝑣 < ∞
Mean of Weibull distribution is defined as: 𝑀𝑤𝑏𝑙 = 𝑐 ∗ 𝛤(1 + 𝑘 −1 )
19
(26)
(27)
where gamma function 𝛤(𝑥) is described as: ∞
𝛤(𝑥) = ∫ 𝑒 −𝑡 𝑡 𝑥−1 𝑑𝑡
(28)
0
20 21 22 23 24 25 26 27 28
In our case study of IEEE-30 bus system, conventional generators in bus 5 and bus 11 are replaced with wind power generators. Values of selected Weibull shape (𝑘) and scale (𝑐) parameters are provided in Table 3. Unless mentioned otherwise for a specific case study, we follow these PDF parameters throughout. Weibull fitting and wind frequency distributions in Fig. 1 and Fig. 2 are obtained after running 8000 Monte-Carlo scenarios. Standard [31] stipulates the design requirement of wind turbines and defines highest turbulent class IA of the turbine which is certified to perform for maximum annual average wind speed of 10 m/s at hub height. Care is taken in choosing shape (𝑘) and scale (𝑐 ) parameters for the windfarms such that maximum Weibull mean value remains around 10. Besides, different PDF parameter values for two windfarms characterize the realistic geographical diversity of the sites.
7
1 2
3 4 5 6 7 8
Fig. 1: Wind speed distribution for wind farm #1 at bus 5 (𝑐 = 9, 𝑘 = 2)
Fig. 2: Wind speed distribution for wind farm #2 at bus 11 (𝑐 = 10, 𝑘 = 2)
Conventional generator at bus 13 of IEEE-30 bus system is replaced by solar PV unit. The output from the unit is dependent upon solar irradiance (𝐺) which follows lognormal PDF [25]. The probability of solar irradiance (𝐺) following lognormal PDF with mean 𝜇 and standard deviation 𝜎 is: 𝑓𝐺 (𝐺) =
9
1 𝐺𝜎√2𝜋
−(𝑙𝑛 𝐺−𝜇)2
𝑒𝑥𝑝 {
2𝜎 2
} for 𝐺 > 0
(29)
Mean of lognormal distribution is defined as: 𝜎2 (30) ) 2 Fig. 3 indicates frequency distribution and lognormal fitting of solar irradiance after running Monte Carlo simulation with a sample size of 8000. Table 3 summarizes the selected parameters for lognormal PDF. Unless mentioned otherwise for a specific case study, we follow these PDF parameters throughout. 𝑀𝑙𝑔𝑛 = 𝑒𝑥𝑝 (𝜇 +
10 11 12
8
1 2
3 4 5 6 7
Fig. 3: Solar irradiance distribution for solar PV at bus 13 (μ = 6, σ = 0.6) Table 3:
PDF parameters of wind power and solar PV plants Wind power generating plants Wind farm #
No. of turbines
Rated power,
𝑃𝑤𝑟 (MW) 1 (bus 5) 2 (bus 11)
8 9 10 11 12 13 14 15
16 17 18 19 20 21
25 20
75 60
Weibull PDF parameters 𝑐 𝑘 𝑐 𝑘
=9 =2 = 10 =2
Solar PV plant Weibull mean,
Rated power,
𝑀𝑤𝑏𝑙
𝑃𝑠𝑟 (MW)
𝑣 = 7.976 m/s 𝑣 = 8.862 m/s
50 (bus 13)
Lognormal PDF parameters
Lognormal mean, 𝑀𝑙𝑔𝑛
𝜇 =6 𝜎 = 0.6
𝐺 = 483 W/m2
3.1 Wind and solar photovoltaic power model We consider wind power connected to bus 5 is the cumulative output power of 25 turbines in the farm, and output of windfarm having 20 turbines is connected to bus 11. Each turbine has rated output power of 3 MW. Actual output power from a wind turbine depends on the wind speed it encounters. Power output of a turbine as a function of wind speed (𝑣) can be described as: 0, for 𝑣 < 𝑣𝑖𝑛 and 𝑣 > 𝑣𝑜𝑢𝑡 𝑣 − 𝑣𝑖𝑛 ) for 𝑣𝑖𝑛 ≤ 𝑣 ≤ 𝑣𝑟 𝑝𝑤 (𝑣) = {𝑝𝑤𝑟 ( 𝑣𝑟 − 𝑣𝑖𝑛 𝑝𝑤𝑟 for 𝑣𝑟 < 𝑣 ≤ 𝑣𝑜𝑢𝑡
(31)
where, 𝑣𝑖𝑛 , 𝑣𝑟 and 𝑣𝑜𝑢𝑡 are the cut-in, rated and cut-out wind speeds of the turbine respectively. 𝑝𝑤𝑟 is the rated output power of the wind turbine. For 3-MW wind turbine, Enercon E82-E4 product datasheet is consulted. The various speed values are 𝑣𝑖𝑛 = 3 m/s, 𝑣𝑟 = 16 m/s and 𝑣𝑜𝑢𝑡 = 25 m/s. The solar irradiance (𝐺) to energy conversion for solar PV is given by [16]: 𝐺2 ) for 0 < 𝐺 < 𝑅𝑐 𝐺𝑠𝑡𝑑 𝑅𝑐 (𝐺) (32) 𝑃𝑠 = 𝐺 𝑃𝑠𝑟 ( ) for 𝐺 ≥ 𝑅𝑐 𝐺𝑠𝑡𝑑 { where, 𝐺𝑠𝑡𝑑 is the solar irradiance in standard environment set as 800 W/m2 . 𝑅𝑐 is a certain irradiance point set as 120 W/m2 . 𝑃𝑠𝑟 is the rated output power of the solar PV unit. 𝑃𝑠𝑟 (
22 23 24
9
1 2 3 4 5
6 7
3.2 Calculation of wind power probabilities Referring to equation (31), it may be observed that the variable wind power is discrete in a couple of regions of wind speeds. When wind speed (𝑣) is below cut-in speed (𝑣𝑖𝑛 ) and above cut-out speed (𝑣𝑜𝑢𝑡 ), the power output is zero. The turbine gives rated power output 𝑝𝑤𝑟 between rated wind speed (𝑣𝑟 ) and cut-out speed (𝑣𝑜𝑢𝑡 ). For these discrete zones, probabilities are given by [13]: 𝑣𝑖𝑛 𝑘 𝑣𝑜𝑢𝑡 𝑘 (33) 𝑓𝑤 (𝑝𝑤 ){𝑝𝑤 = 0} = 1 − exp [− ( ) ] + exp [− ( ) ] 𝑐 𝑐 𝑣𝑟 𝑘 𝑣𝑜𝑢𝑡 𝑘 (34) 𝑓𝑤 (𝑝𝑤 ){𝑝𝑤 = 𝑝𝑤𝑟 } = exp [− ( ) ] − exp [− ( ) ] 𝑐 𝑐 The wind turbine power output is continuous between cut-in speed (𝑣𝑖𝑛 ) and rated speed (𝑣𝑟 ) of wind. The probability for the continuous region is calculated as [13]: 𝑝
𝑘
𝑣𝑖𝑛 + 𝑤 (𝑣𝑟 − 𝑣𝑖𝑛 ) 𝑘−1 𝑘(𝑣𝑟 − 𝑣𝑖𝑛 ) 𝑝𝑤 𝑝𝑤𝑟 𝑓𝑤 (𝑝𝑤 ) = 𝑘 [𝑣𝑖𝑛 + (𝑣 − 𝑣𝑖𝑛 )] exp [− ( ) ] 𝑐 ∗ 𝑝𝑤𝑟 𝑝𝑤𝑟 𝑟 𝑐
(35)
8 9 10 11 12 13 14 15
Fig. 4: Real power distribution (MW) of solar PV at bus 13
3.3 Calculation of solar power over/under estimation cost Histogram in Fig. 4 represents the stochastic power output from solar PV plant. The magenta dotted line indicates the schedule power the solar PV is supposed to deliver to the grid. As mentioned before, the schedule power can be any amount of power mutually agreed between ISO and the solar PV firm owner. The overestimation cost in equation (7) can be calculated as (suffix ‘𝑘’ in eq. (7) is omitted here for single solar PV plant): 𝑁−
𝐶𝑅𝑠 (𝑃𝑠𝑠 − 𝑃𝑠𝑎𝑣 ) = 𝐾𝑅𝑠 (𝑃𝑠𝑠 − 𝑃𝑠𝑎𝑣 ) = 𝐾𝑅𝑠 ∑[𝑃𝑠𝑠 − 𝑃𝑠𝑛− ] ∗ 𝑓𝑠𝑛−
(36)
𝑛=1
16 17 18 19 20 21
where, 𝑃𝑠𝑛− is the available power less than the schedule power 𝑃𝑠𝑠 , on left-half plane of 𝑃𝑠𝑠 in the histogram. 𝑓𝑠𝑛− is the relative frequency of occurrence of 𝑃𝑠𝑛− . 𝑁 − is the number of discrete bins on left-half of 𝑃𝑠𝑠 or in other words, the number of pairs (𝑃𝑠𝑛− , 𝑓𝑠𝑛− ) generated for the PDF. Increasing number of segments does improve accuracy of results to an extent. For the problem in our study, a total (𝑁) of 30 segments give fairly accurate results. Similarly, underestimation cost in equation (8) can be calculated as: 𝑁+
𝐶𝑃𝑠 (𝑃𝑠𝑎𝑣 − 𝑃𝑠𝑠 ) = 𝐾𝑃𝑠 (𝑃𝑠𝑎𝑣 − 𝑃𝑠𝑠 ) = 𝐾𝑃𝑠 ∑[𝑃𝑠𝑛+ − 𝑃𝑠𝑠 ] ∗ 𝑓𝑠𝑛+
(37)
𝑛=1
22 23 24
where, 𝑃𝑠𝑛+ is the available power more than the schedule power 𝑃𝑠𝑠 , on right-half plane of 𝑃𝑠𝑠 in the histogram. 𝑓𝑠𝑛+ is the relative frequency of occurrence of 𝑃𝑠𝑛+ . 𝑁 + is the number of discrete bins on right-half of 𝑃𝑠𝑠 or in other words, the number of pairs (𝑃𝑠𝑛+ , 𝑓𝑠𝑛+ ) generated for the PDF. 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
4.
Optimization algorithm and application
Differential Evolution (DE), introduced by Storn and Price in 1996, is stochastic, population based optimization algorithm where the individuals in the population evolve and improve their fitness through probabilistic operators like recombination and mutation. The performance of DE is found to be highly dependent on the control settings which are: the scaling factor (F), the crossover rate (CR), the population size (Np), and the chosen mutation/crossover strategies [32]. Adaptive mechanisms for adjusting the control parameters on-line during the search process have been studied by many researchers. Subsequently JADE [33], which proposed a novel mutation strategy, and SHADE [20], a history based parameter adaptation scheme were introduced. Our research presented in this paper uses SHADE algorithm, slightly altered with some adaptation technique followed in L-SHADE [34]. Superiority of feasible solution (SF) constraint handling technique is integrated with the modified SHADE algorithm. Both the algorithms are briefly described below followed by the integration approach (SHADE-SF). 4.1 Success history based adaptive differential evolution algorithm
16 17 18
Iteration of standard DE algorithm has four basic steps – Initialization, mutation, crossover and selection. SHADE being an adaptive DE, adaptation of control parameters takes place during evolution. The selection process in SHADE-SF follows rules of SF narrated subsequently in section 4.2.
19
4.1.1 Initialization
20 21 22
The first step in the DE optimization process is to create an initial population of candidate solutions by assigning random values to each decision vector of the population. Such values must lie inside the feasible bounds (between maximum & minimum) of the decision vector. We may initialize j-th component of the i-th decision vector as: (0)
(38)
𝑥𝑖,𝑗 = 𝑥𝑚𝑖𝑛,𝑗 + 𝑟𝑎𝑛𝑑𝑖𝑗 [0,1](𝑥𝑚𝑎𝑥,𝑗 − 𝑥𝑚𝑖𝑛,𝑗 )
23 24 25 26 27 28 29 30 31 32
where 𝑟𝑎𝑛𝑑𝑖𝑗 [0,1] is a uniformly distributed random number lying between 0 and 1 and superscript ‘0’ represents initialization. If ‘Np’ is the population size and ‘d’ is the dimension of decision vector, then i = 1,2,…,Np and j = 1,2,…,d.
33 34 35
After initialization, DE creates a donor/mutant vector 𝑣𝑖 corresponding to each population member or target (𝑡) vector 𝑥𝑖 in the current generation through mutation (the superscript ‘t’ denotes parameter at t-th generation). There are quite a few strategies for mutation. The one used here is ‘current-to-pbest/1’ [20]:
The OPF problem of IEEE 30-bus system has 11 decision variables, the active power schedule of all generators (except slack generator) and voltages of the buses where these generators are connected (including slack generator). (0) Therefore, each decision vector 𝑥𝑖 for i = 1,2,….,Np is formulated as a 11 dimensional vector (i.e. d = 11) with (0)
each element (i.e. decision variable) 𝑥𝑖,𝑗 bounded by its range of defined maximum and minimum values. 4.1.2 Mutation (𝑡)
(𝑡)
𝑣𝑖
36 37 38 39 40
(𝑡)
= 𝑥𝑖
(𝑡)
(𝑡)
(𝑡)
(𝑡)
(39) (𝑡)
(𝑡)
top 𝑁𝑝 × 𝑝 (𝑝 ∊ [0,1]) best individuals of current generation. The scaling factor 𝐹𝑖 is a positive control (𝑡) parameter for scaling the difference vectors at t-th generation. During evolution through mutation if an element 𝑣𝑖,𝑗 goes outside the search range boundaries [𝑥𝑚𝑖𝑛,𝑗 , 𝑥𝑚𝑎𝑥,𝑗 ], it is corrected as: (𝑥𝑚𝑖𝑛,𝑗 + 𝑥𝑖,𝑗𝑡 )/2 if 𝑣𝑖,𝑗 < 𝑥𝑚𝑖𝑛,𝑗 ( )
44 45 46 47
(𝑡) ) 𝑅2𝑖
−𝑥
The indices 𝑅1𝑖 & 𝑅2𝑖 are mutually exclusive integers randomly chosen from the population range; 𝑥𝑝𝑏𝑒𝑠𝑡 is the
(𝑡)
41 42 43
(𝑡) 𝑅1𝑖
+ 𝐹𝑖 . (𝑥𝑝𝑏𝑒𝑠𝑡 − 𝑥𝑖 ) + 𝐹𝑖 . (𝑥
𝑣𝑖,𝑗 = {
(𝑡)
(𝑥𝑚𝑎𝑥,𝑗 + 𝑥𝑖,𝑗𝑡 )/2 if 𝑣𝑖,𝑗 > 𝑥𝑚𝑎𝑥,𝑗 ( )
(𝑡)
(40)
4.1.3 Parameter Adaptation (𝑡)
At each generation t, each individual has its own 𝐹𝑖 vectors. These two parameters are adapted as: (𝑡)
𝐹𝑖
(𝑡)
𝐶𝑅𝑖
(𝑡)
and 𝐶𝑅𝑖
parameters that are used to generate new trial
(𝑡)
= 𝑟𝑎𝑛𝑑𝑐(µ𝐹𝑟𝑖 , 0.1) (𝑡)
= 𝑟𝑎𝑛𝑑𝑛(µ𝐶𝑅𝑟𝑖 , 0.1)
(41) 11
1 2 3 4 5 6 7 8 9 10 11 12 13 14
(𝑡)
(𝑡)
where 𝑟𝑎𝑛𝑑𝑐(µ𝐹𝑟𝑖 , 0.1) generates a value following Cauchy distribution with location parameter µ𝐹𝑟𝑖 (𝑡) scale parameter 0.1; 𝑟𝑎𝑛𝑑𝑛(µ𝐶𝑅𝑟𝑖 , 0.1) is the value (𝑡) (𝑡) variance 0.1. µ𝐹𝑟𝑖 & µ𝐶𝑅𝑟𝑖 are randomly chosen
sampled from Normal distribution with mean
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
and
random index ri [1, H ] is selected where H is the memory size. Initially 𝜇𝐹 and 𝜇𝐶𝑅 are both set to 0.5. At the end of generation 𝑡, the memory is updated at certain positions defined by indices 𝑘. At indices 𝑘, 𝜇𝐹 and 𝜇𝐶𝑅 are modified using the weighted Lehmer mean, the detail of which can be referred in [34]. 4.1.4 Crossover (𝑡)
(𝑡)
Through crossover the donor vector 𝑣𝑖 mixes its components with the target vector 𝑥𝑖 to form the (𝑡) (𝑡) (𝑡) (𝑡) trial/offspring vector 𝑢𝑖 = (𝑢𝑖,1 , 𝑢𝑖,2 , … . . , 𝑢𝑖,𝑑 ). Binomial crossover, which is adopted here, operates on each variable whenever a randomly generated number between 0 and 1 is less than or equal to the adapted parameter (𝑡) 𝐶𝑅𝑖 , the crossover rate. The scheme is expressed for an element as: (𝑡)
𝑢𝑖,𝑗 = {
(𝑡)
𝑣𝑖,𝑗 if 𝑗 = 𝐾 or 𝑟𝑎𝑛𝑑𝑖,𝑗 [0,1] ≤ 𝐶𝑅𝑖 , (𝑡)
𝑥𝑖,𝑗 otherwise
(42)
where K is any randomly chosen natural number in {1,2,….,d}, d being the dimension of real-valued decision vectors. 4.2 Superiority of feasible solutions constraint handling technique Due to the stochastic nature of evolutionary algorithms (EAs), it is essential to employ constraint handling method as they can make use of the information present in the infeasible solutions to drive the search process towards global feasible optima. An efficient constraint handling technique, superiority of feasible solutions (SF) is described below. A constrained optimization problem with 𝑑 parameters to be optimized is usually written as a nonlinear programming problem of the following form [35]: Minimize: f ( x), x ( x1 , x 2 ,..., x d ) and x S subject to:
g i ( x ) 0,
i 1,..., p
h j ( x ) 0,
j p 1,..., m
(43)
where 𝑆 is the whole search space, 𝑝 and (𝑚 − 𝑝) are the number of inequality and equality constraints, respectively. The equality constraints can be transformed into inequality constraints and total constraints can be represented as: i 1,... p. max{ gi ( x),0} Ti ( x) max{| hi ( x) | ,0} i p 1,..., m
(44)
where δ is a tolerance parameter for the equality constraints. Therefore, the objective is to minimize the fitness function 𝑓(𝑥) such that the optimal solution satisfies all the inequality constraints 𝑇𝑖 (𝑥). The overall constraint violation for an infeasible individual is a weighted mean of all the constraints, which is expressed as: 𝜈(𝑥) =
37 38 39 40 41 42 43 44 45 46
and
from location parameters of scale factors and means of crossover rates respectively of the successful candidates of previous generations stored in a memory 𝑀 [34]. A
(𝑡)
15
(𝑡) µ𝐶𝑅𝑟𝑖
∑𝑚 𝑖=1 𝑤𝑖 [𝑇𝑖 (𝑥)] ∑𝑚 𝑖=1 𝑤𝑖
(45)
where 𝑤𝑖 (= 1/𝑇𝑚𝑎𝑥,𝑖 ) is a weight parameter, 𝑇𝑚𝑎𝑥,𝑖 is the maximum violation of constraint 𝑇𝑖 (𝑥) obtained so far. Here, 𝑤𝑖 is set as 1/𝑇𝑚𝑎𝑥,𝑖 , which varies during the evolution to balance the contribution of every constraint in the problem irrespective of the differing numerical ranges of all constraints. In SF [21], 𝑥𝑖 is considered superior to 𝑥𝑗 when: • 𝑥𝑖 is feasible and 𝑥𝑗 is infeasible • 𝑥𝑖 and 𝑥𝑗 are both feasible and 𝑥𝑖 has a smaller objective value (in a minimization problem) than 𝑥𝑗 • 𝑥𝑖 and 𝑥𝑗 are both infeasible, but 𝑥𝑖 has a smaller overall constraint violation as computed by equation (45) Therefore, in SF, feasible ones are always considered better than the infeasible ones. Two infeasible solutions are compared based on their overall constraint violations only, while two feasible solutions are compared based on their 12
1 2 3 4 5 6 7
objective function values only. Comparison of infeasible solutions based on the overall constraint violation aims to push the infeasible solutions to feasible region, while comparison of two feasible solutions on the objective value improves the overall solution. The combined method of SHADE-SF is presented in Table 4 and flowchart in Fig. 5. While the table describes the process in detail, the flowchart gives an overview of the steps involved. Table 4: SHADE-SF algorithm SHADE-SF INPUT
• • • • •
Dimension of the problem, 𝑑 (𝑑 = 11 here) Number of population, 𝑁𝑝 (𝑁𝑝 = 60 considered) Stopping criteria, maximum number of function evaluation, 𝑚𝑎𝑥𝑒𝑣𝑎𝑙 Maximum and minimum values of 𝑑-decision variables, in vector form 𝑥𝑚𝑎𝑥 and 𝑥𝑚in 1 d 1 d 𝑥𝑚𝑎𝑥 = [𝑥𝑚𝑎𝑥 , … , 𝑥𝑚𝑎𝑥 ] and 𝑥𝑚in = [𝑥min , … , 𝑥min ]
INITIALIZATION
• • •
Set generation counter, 𝑡 = 0 and function evaluation counter 𝑛𝑓𝑒𝑣𝑎𝑙 = 0. Set SHADE parameters: Memory (𝑀) size 𝐻 = 5, initialize 𝜇𝐹 = 0.5 and 𝜇𝐶𝑅 = 0.5 POP: Create population of 𝑁𝑝 individuals uniformly distributed between [𝑥𝑚𝑎𝑥 , 𝑥𝑚𝑖𝑛 ]
EVALUATION
• •
Evaluate objective function, constraint function and constraint violation using eq. (43) to eq. (45) for each individual 𝑥i ∀𝑖 ∊{1, . . . , 𝑁𝑝} of POP Increase function evaluation counter 𝑛𝑓𝑒𝑣𝑎𝑙 by 𝑁𝑝
• • •
Perform mutation and crossover for each individual of POP; generate offspring OFS, 𝑥′i (𝑡) Generate mutant vector 𝑣𝑖 using equation (39) (𝑡) Generate trial vector element 𝑢𝑖,𝑗 performing crossover according to equation (42)
•
Formulate trial vector 𝑢𝑖
•
Compute objective function, constraint function and constraint violation using eq. (43) to eq. (45) for each individual 𝑥′i ∀𝑖 ∊{1, . . . , 𝑁𝑝} of OFS Increase function evaluation counter 𝑛𝑓𝑒𝑣𝑎𝑙 by 𝑁𝑝
ALGORITHM LOOP: STEP 1
STEP 2
• STEP 3
STEP 4
•
(𝑡)
with all elements, this is OFS vector 𝑥′i
•
In selection step, POP members for next generation are replaced with corresponding OFS members if OFS is better according to rules of SF. An OFS is considered better if it yields lesser constraint violation or zero constraint violation alongwith smaller fitness value (minimization problem) than the respective old POP member. If OFS is not better, old POP member is retained. If OFS is better, memory of 𝜇𝐹 and 𝜇𝐶𝑅 is updated as described in section 4.1.3
• • •
Is stopping criterion, 𝑚𝑎𝑥𝑒𝑣𝑎𝑙 reached? If yes, STOP If not, increase generation counter by 1, i.e. t= 𝑡 + 1. Go to algorithm loop STEP 1.
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
Initialize all parameters and generate initial population POP of size 𝑁𝑝, set function evaluation counter, 𝑛𝑓𝑒𝑣𝑎𝑙 = 0
Compare OFS member with corresponding POP member
Evaluate objective function and constraint violation for each individual of POP, 𝑛𝑓𝑒𝑣𝑎𝑙 = 𝑁𝑝
Is OFS better?
Yes
Perform mutation and crossover. Generate OFS for each individual of POP
No
Replace the respective POP member with OFS, update memory of control parameters
Evaluate objective function and constraint violation for each individual of OFS
Retain the old POP member
𝑛𝑓𝑒𝑣𝑎𝑙 = 𝑛𝑓𝑒𝑣𝑎𝑙 + 𝑁𝑝 No
Is stopping criterion reached?
Yes
Fig. 5: Flowchart for implementation of SHADE-SF
Stop and output final solutions
The proposed algorithms are developed using MATLAB software and simulations are carried out on a computer with Intel Core i5 CPU @2.7GHz and 4GB RAM. Simulation results are discussed in section 5.
5.
Case studies and results
Several case studies are performed for adapted IEEE-30 bus system. Results of the case studies with application of SHADE-SF algorithm are tabulated and explanation are provided in this section. The first 2 study cases are to examine variation of generation costs of wind and solar power with change in respective schedule power and PDF parameters. Remaining case studies optimize the schedule generation from all sources. In each optimization case study, a maximum of 24000 function evaluations (𝑚𝑎𝑥𝑒𝑣𝑎𝑙 = 24000) are performed in a single complete run of the algorithm. Each case is run 5 times and the best value of the objective function thus found and corresponding control variable settings are recorded. 5.1. Case 1: Scheduled power vs cost (wind and solar photovoltaic) The Weibull PDF parameters in this case are same as mentioned in Table 3. Relevant wind turbine parameters are provided in section 3.1. Direct cost coefficients of wind power are 𝑔1 = 1.6, 𝑔2 = 1.75. Penalty cost coefficient for not fully utilizing wind power is assumed as 𝐾𝑃𝑤,1 = 𝐾𝑃𝑤,2 = 1.5 and reserve cost coefficient for overestimation is 𝐾𝑅𝑤,1 = 𝐾𝑅𝑤,2 = 3. Note that direct cost of renewable power is less than the average cost of thermal power while the penalty cost for not using available wind power is less than the direct cost [36]. The scheduled power is varied from 0 to wind farm rated power and variations of reserve, penalty, direct and total costs are plotted in Fig. 6 and Fig. 7 for the two windfarms. Total cost is the sum of direct, reserve and penalty cost corresponding to the scheduled power. Direct cost follows linear relationship with scheduled power. As scheduled power increases larger spinning reserve is necessary which escalates the reserve cost and consequently overall generation cost moves upwards. The penalty cost rightly decreases, however at a lower rate, with increase in scheduled power.
14
1 2 3 4 5 6 7 8 9
Fig. 6: Variation of wind power cost vs scheduled power for wind generator 𝑊𝐺1
Similar to wind power, cost variations of solar power over/under-estimation are plotted against schedule power in Fig. 8. Yearly operating and maintenance cost for solar PV power plant is almost in similar range of that of onshore wind power plant [37]. Therefore, for our study purpose the direct, penalty and reserve cost coefficients for solar PV are assumed to be ℎ = 1.6 , 𝐾𝑃𝑠 = 1.5 and 𝐾𝑅𝑠 = 3 respectively. Other related solar PV parameters are discussed in section 3.1. With the selected PDF parameters for solar irradiance, the total solar power cost is not monotonically increasing. Indeed, the minimum cost is reported somewhere around 15 MW of scheduled power.
10
11 12 13 14
Fig. 7: Variation of wind power cost vs scheduled power for wind generator 𝑊𝐺2
15
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Fig. 8: Variation of solar power cost vs schedule power for solar PV unit
5.2. Case 2: Probability density function parameter vs cost (wind and solar photovoltaic) In this case study, scale parameter (𝑐) of Weibull distribution is varied (with shape parameter, 𝑘 = 2) to observe change in costs of wind power for a fixed arbitrarily selected schedule power. The schedule power for 𝑊𝐺1 is fixed at 25 MW and that of 𝑊𝐺2 is at 20 MW, i.e. 1/3-rd of the installed capacity. The assumption of schedule power is reasonable as practical wind farm has a capacity factor somewhere between 30% to 45% [37]. Cost coefficients are same as in case 1. Fig. 9 and Fig. 10 represent curves for cost vs Weibull scale parameter for windfarm#1 and windfarm#2 respectively. The total minimum cost is at an intermediate value of scale parameter. As scale parameter increases, higher wind speeds with certain probabilities prevail. With scheduled power remaining same, the penalty cost escalates raising the total power cost. The rate of decrement in reserve cost is not significant after a certain value of scale parameter.
15 16 17
Fig. 9: Variation of wind power cost vs Weibull scale parameter (𝑐) for windfarm#1 (bus 5)
16
1 2 3 4 5 6 7 8 9 10 11 12 13
Fig. 10: Variation of wind power cost vs Weibull scale parameter (𝑐) for windfarm#2 (bus 11)
For assessing variation in solar power cost with change in lognormal PDF mean μ, value of μ is altered from 2 to 7 in steps of 0.5. Standard deviation σ is 0.6; scheduled power is fixed at 20 MW. Cost coefficients are same as in case 1. Cost plots are given in Fig. 11. Total solar power cost is found to be gradually decreasing to minimum value at μ=5.5. At about μ=5.8, penalty cost and reserve cost are same. Thereafter, the penalty cost takes a sharp upward turn, thus suddenly mounting the total generation cost of solar power. Solar irradiance is highly sensitive to mean μ of lognormal distribution, so is output power. At low values of μ, irradiance and output power is so low, almost full reserve is necessary. However, if irradiance follows frequency distribution having higher μ values, output power from solar PV can be much larger. So, care must be taken in selecting appropriate value of scheduled solar PV power. If μ is low, smaller value of scheduled power is recommended.
14 Fig. 11: Variation of solar PV power cost vs lognormal mean (μ) for SPV (bus 13)
15 16 17 18
5.3. Case 3: Minimization of generation cost
19 20 21 22 23
Case 3 performs optimization of generation schedule for all thermal and renewable source generators to minimize total generation cost given by equation (11). Cost coefficients are same as in case 1, PDF parameters are provided in Table 3. The convergence of SHADE-SF algorithm is indicated in Fig. 12. As can be seen from the diagram, the optimum cost is achieved within 10000 function evaluations. Optimum settings of all control variables, generator reactive power (Q), total generation cost and other useful calculated parameters are summarized in Table 5. Voltage 17
1 2 3 4 5 6
𝑉𝑖 in the table signifies the voltage at 𝑖 -th bus; 𝑃𝑙𝑜𝑠𝑠 and 𝑉𝐷 are calculated using equations (24) and (25) respectively. It may be note that, 𝑃𝑤𝑠,1 signifies the schedule power from wind generator 𝑊𝐺1 and likewise. With the generation schedules listed in the table, minimum generation cost that can be achieved is 782.503 $/h. Table 5: Simulation results for optimization case studies: IEEE 30-bus system Control variables
Min
Max
Case 3
Case 6
Parameters
Min
Max
Case 3
Case 6
𝑃𝑇𝐺1 (MW) 𝑃𝑇𝐺2 (MW)
50 20
140 80
134.908 28.564
123.525 33.047
𝑄𝑇𝐺1 (MVAr) 𝑄𝑇𝐺2 (MVAr)
-20 -20
150 60
-1.903 13.261
-2.678 12.319
𝑃𝑇𝐺3 (MW)
10
35
10
10
𝑄𝑇𝐺3 (MVAr)
-15
40
35.101
35.27
𝑃𝑤𝑠1 (MW)
0
75
43.774
46.021
𝑄𝑤𝑠1 (MVAr)
-30
35
23.181
22.964
𝑃𝑤𝑠2 (MW)
0
60
36.949
38.748
𝑄𝑤𝑠2 (MVAr)
-25
30
30
30
𝑃𝑠𝑠 (MW)
0
50
34.976
37.336
𝑄𝑠𝑠 (MVAr)
-20
25
17.346
17.779
V1 (p.u.)
0.95
1.10
1.072
1.071
Total cost ($/h)
782.503
810.346
V2 (p.u.)
0.95
1.10
1.057
1.057
Emission (t/h)
1.762
0.891
V5 (p.u.)
0.95
1.10
1.035
1.036
Carbon tax ($/h)
-
17.83
V8 (p.u.)
0.95
1.10
1.04
1.04
𝑃𝑙𝑜𝑠𝑠 (MW)
5.770
5.276
V11 (p.u.)
0.95
1.10
1.1
1.099
𝑉𝐷 (p.u.)
0.463
0.469
V13 (p.u.)
0.95
1.10
1.055
1.056
7 8 850 Case-3
840
Case-6
Total cost ($/h)
830 820 810 800 790 780 770 0
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
4000
8000 12000 16000 20000 Number of function evaluations
24000
Fig. 12: Convergence of optimization Case 3 and Case 6
5.4. Case 4: Optimized cost vs reserve cost This study case is similar to Case 3 with all parameters being same as in Case 3 except reserve cost coefficients. The reserve cost coefficients for both wind and solar power are varied from 𝐾𝑅𝑤,1 = 𝐾𝑅𝑤,2 = 𝐾𝑅𝑠 = 𝐾𝑅(𝑠𝑎𝑦) = 4 to 𝐾𝑅 = 6 in discrete steps of 1. Penalty cost coefficient for all the renewable sources, say 𝐾𝑃 = 1.5 as in Case 1 and Case 3. Optimized schedule of generators is indicated by bar chart in Fig. 13. Notation of cases used - Case 4a (𝐾𝑅 = 4), Case 4b (𝐾𝑅 = 5) and Case 4c (𝐾𝑅 = 6). As reserve cost coefficient increases, the optimum power scheduled from wind generator and solar PV decreases as lowering the scheduled power requires less spinning reserve. Lesser outputs from the renewable sources are compensated by thermal generators. Therefore, thermal generator cost increases as can be observed from ‘TG cost’ profile in Fig. 14. Costs of both wind (‘WG cost’) and solar PV (‘SPV cost’) power gradually decrease to an extent. ‘WG cost’ includes cost of power for both the wind generators. Overall cost (‘Total cost’) rises with increase in reserve cost coefficient.
18
Scheduled real power (MW)
160
TG1
TG2
TG3
WG1
WG2
SPV
140 120 100 80 60 40 20 0
1 2
KR = 3 (Case 3)
KR = 4 (Case 4a)
KR = 5 (Case 4b)
KR = 6 (Case 4c)
Fig. 13: Optimal scheduled real power (MW) vs reserve cost coefficient (𝐾𝑅) Total cost
TG cost
WG cost
KR = 4 (Case 4a)
KR = 5 (Case 4b)
SPV cost
1000 900 800
Cost ($/h)
700 600 500 400 300 200 100 0
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
KR = 3 (Case 3)
KR = 6 (Case 4c)
Fig. 14: Cost curves for change in reserve cost (𝐾𝑅) coefficient
5.5. Case 5: Optimized cost vs penalty cost All parameters of this case study are same as in Case 3 except penalty cost coefficients. The penalty cost coefficients for both wind and solar power are varied from 𝐾𝑃𝑤,1 = 𝐾𝑃𝑤,2 = 𝐾𝑃𝑠 = 𝐾𝑃(𝑠𝑎𝑦) = 1.5 to 𝐾𝑃 = 3 (Case 5a), 𝐾𝑃 = 4 (Case 5b) and 𝐾𝑃 = 5 (Case 5c). Reserve cost coefficient for all the renewable sources is, 𝐾𝑅 = 3 as in Case 1 and Case 3. Optimized schedule of generators is specified by bar chart in Fig. 15. As penalty cost coefficient increases scheduled outputs from renewable sources tend to increase as raising the scheduled power would help to bring down penalty cost when wind speed or solar irradiance is high. However, unlike Case 4 where both wind and solar PV power monotonically decrease, here the increment is not uniform across all renewable sources. Indeed, output from solar PV occasionally decreases with increase in value of 𝐾𝑃. This can be attributed to highly non-linear relationships between probability distribution and penalty/reserve cost associated with both wind and solar PV power. Wind power cost (‘WG cost’) is found progressively increasing in Fig. 16. Slight fluctuation in cost is observed for solar PV power, owing to fluctuating scheduled output from it. Thermal generation cost remains almost constant and overall cost rightly shows steady rise. Voltage of generator buses are combined in Fig. 17 for all scenarios of changing reserve and penalty costs performed under Case 4 and Case 5. The bus voltages are all within specified range of 0.95 to 1.10 p.u. Except in bus 8, the voltages remain quite steady for changing values of cost coefficients. Varying reactive power output from 𝑇𝐺3 connected to bus 8 causes the variation in voltage at that bus under different case studies.
19
160
TG1
TG2
TG3
WG1
WG2
SPV
Scheduled real power (MW)
140 120 100 80 60 40 20 0 KP = 1.5 (Case 3)
1 2
KP = 3 (Case 5a)
KP = 4 (Case 5b)
KP = 5 (Case 5c)
Fig. 15: Optimal scheduled real power (MW) vs penalty cost coefficient
3 Total cost
TG cost
WG cost
KP = 3 (Case 5a)
KP = 4 (Case 5b)
SPV cost
900 800 700
Cost ($/h)
600 500 400 300 200 100 0 KP = 1.5 (Case 3)
4 5 6 7
KP = 5 (Case 5c)
Fig. 16: Cost curves for change in penalty cost (𝐾𝑃) coefficient
V1
V2
V5
V8
V11
V13
1.12 1.1
Voltage (in p.u.)
1.08 1.06 1.04 1.02 1 0.98
8 9 10
Case 4a
Case 4b
Case 4c
Case 5a
Case 5b
Case 5c
Fig. 17: Generator bus voltages for Case 4 and Case 5
20
1 2 3 4 5 6 7 8
Generator reactive power is state or dependent variable in OPF problem. The constraint on reactive power must be satisfied during optimization. Fig. 18 represents schedule of all generator reactive power. Looking at the limits given in Table 5, the operation of generators 𝑇𝐺3 and 𝑊𝐺2 is at their limits of reactive power capability for many cases. So, care must be taken in dealing with the constraint on reactive power during optimization by an algorithm. The superiority of proper constraint handling method lies in the fact that it allows the network components to operate close to the limits without violating those. Therefore, such criteria justify the application of SF constraint handling method in non-linear, constrained OPF problem. 45
TG1
TG2
TG3
WG1
WG2
SPV
Generator reactive power (MVAr)
40 35 30 25 20 15 10 5 0
9 10 11 12 13 14 15 16 17 18 19 20 21
-5
Case 4a
Case 4b
Case 4c
Case 5a
Case 5b
Case 5c
Fig. 18: Generator reactive power schedule for Case 4 and Case 5
5.6. Case 6: Minimization of generation cost with carbon tax This study case minimizes total generation cost that includes carbon tax imposed on the emission from conventional thermal power generators. The cumulative cost, given by equation (12), is to be minimized. Carbon tax rate, 𝐶𝑡𝑎𝑥 is assumed to be $20/tonne [19]. As wind and solar power are clean form of energy, the penetration of these sources is expected to increase due to carbon tax component. Optimum generation schedule, generator reactive power, total generation cost (including carbon tax) and other calculated parameters are listed in Table 5. It is observed that penetration of both wind and solar energy is higher when carbon tax is levied in Case 6 than in Case 3 with no penalty on emission. As an obvious fact, the extent of increase in optimum generation schedule of the renewable sources depends on emission volume and rate of carbon tax imposed. 1.06 Upper limit
Voltage (p.u.)
1.04 1.02 1 0.98
Case 3
0.96
Case 6 Lower limit
0.94 3
22 23
6
9
12
15 18 Load bus no.
21
24
27
30
Fig. 19: Load bus voltage profiles for Case 3 and Case 6 21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
In OPF problem, constraint on load bus voltage is also critical as operating voltages of load buses are often found be close to their limits. In our study, the load bus voltage must be maintained within 0.95 p.u. to 1.05 p.u. The voltage profiles of load buses are drawn in Fig. 19 for optimization Case 3 and Case 6. For other remaining optimization cases, the voltage profiles follow similar pattern, therefore, for clarity not included in the diagram. Clearly bus voltages are all within specified limits.
6.
Conclusion
This paper proposes solution approach to optimal power flow (OPF) problem with stochastic wind and solar power in the network. Uncertainties of intermittent renewable sources are modelled with different probability density functions. Integration method of all the sources is discussed in detail. Generation cost incorporating all sources is optimized and variation of generation cost with change in cost coefficients of uncertain sources is studied. State-ofthe-art evolutionary algorithm, SHADE, is employed for the optimization. The algorithm is combined with an effective constraint handling technique, SF. Violating physical or security constraints of network components may compromise system safety, lead to excessive losses, malfunction and often failure of the component. So, operating the network within defined limits is pre-requisite for secure and correct operation. Without suitable constraint handling method, limits on network parameters may often unknowingly be violated. Therefore, the use of proper constraint handling method in constrained optimization problems is recommended. Besides, due to fast convergence of SHADE-SF algorithm to global optima, it can successfully be applied to other multimodal, highly non-linear, constrained optimization problems. For future work on OPF front, the authors propose possible integration of small hydro-generators with variable river flow, storage in the form of batteries or pumped hydro in the network with large number of buses. Accurate model of doubly fed induction generators for wind turbines, FACTS devices can also be incorporated for detailed study. Acknowledgement: This project is funded by the National Research Foundation Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) program.
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