Some Observations on the Dynamical Property of a Small Wind Turbine Takashi KARASUDANI, Kimihiko WATANABE and Yuji OHYA of corresponding author
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1 九州大学学術情報リポジトリ Kyushu University Institutional Repository 小型風車の動特性に関するいくつかの考察 烏谷, 隆九州大学応用力学研究所 渡辺, 公彦九州大学応用力学研究所 大屋, 裕二九州大学応用力学研究所 Karasudani, Takashi Research Institute for Applied Mechanics, Kyushu University 他 出版情報 : 九州大学応用力学研究所所報. 137, pp , Research Institute for Applied Mechanics, Kyushu University バージョン :published 権利関係 :
2 Some Observations on the Dynamical Property of a Small Wind Turbine Takashi KARASUDANI, Kimihiko WATANABE and Yuji OHYA of corresponding author: karasu@riam. kyushu-u. ac jp Abstract We theoretically and experimentally studied on the dynamical property of a small wind turbine. We assumed that the aerodynamic properties of a blade were given by a torque coefficient, which was a function of a tip speed ratio only. A numerical model of a wind turbine having an optimum regulation system was given. We derived formulas of response time to the change of wind speed and load as a function of the torque coefficient, inertial moment and wind speed. We showed that the deviation of the tip speed ratio from the optimum value by the change of wind speed was approximated well to the second order of wind variation. The deviation was expressed by a function of characteristic frequency. We also got the expression for the phase difference between wind and power variations. Those results were in good agreement with the numerical simulation for the model. We propose a framework to discuss the dynamical property of a small wind turbine. Key words: wind turbine, dynamical property, torque coefficient, response time, delay of phase, change of wind direction
3 Fig 1. Diagrammatic representation of torque and power coefficients CT and C vs. tip speed ratio A,.
4
5 Fig.2. Judging the sign of D from the slope of a CT( X ) curve at working point A, p. Fig.3. A drawing showing the relationship between stability and tip speed ratio.
6 Fig.4 Curves numerical model. of the torque and power coefficients of the Fig.5 Comparison of response times between the values calculated by the formulas (17),(24),(27) and the values given by the simulation. Fig.6 A sample of the simulation for km2], A U=21 m/s],. f=1 1114
7 Fig.7 Width of change of tip speed ratio by the sinusoidal change of wind speed. Ak vs. f. Fig.8 Displacement of tip speed ratio from the optimum value by the sinusoidal change of wind speed a vs. f. Fig.9 The difference of phase between wind and power as a function of frequency ratio. A regression line for the data is the dashed line. The expression of the line is given in the legend.
8 42位 相 遅 れ と風 速変 動 振 動 数 の 関 係 る影 響 を考 慮 しな けれ ばな らない.翼 に当た る風 を翼面 に垂 直 風速 が正弦的 に変動 する ときは出力 も同様 に正弦的 に変 動す な成分 と平行な成分 に分 けてそ の影響 を検討 す る.翼 面 に垂 直 る.遅 延効果 のため出力の変動 は風 速の変動 に対 して位相 が遅 な成分 は回転に よる相対 風 と常 に直交 してい るた め相 対風 を 下 れ る.風 速変化 の早 さと位相遅れ の関係を求 めた.解 析は以下 流側 へ傾 ける よ うに作用 す る.傾 きの大 き さは翼の 回転位置 に の よ うに した.風 速変 動 と出力 変動 をそれ ぞれ フー リエ変換 よ らず常 に一定で あるので,翼 は迎角 が最適 な値 にな るよ うに (FFT)し た.そ れぞれのパ ワー スペ ク トルが最大 となる周 波 この傾 きを考慮 して設 計 されてい る1,2). 数が一致す る ことを確認 し,最 大 とな る周波数 の複素 フー リエ 翼面 に平行 な成分 は回転 に伴 い相 対風 の大 き さと向 きを変 え 成分 のクロススペ ク トル を求 めた.ク ロス スペ ク トル よ り風 速 る よ うに作 用す る.平 行な成分 に よる相対 風の向 きの変化 は翼 に対す る出力 の位 相の遅れ θを次式 で求 めた. 面に沿 った方向で あ り,迎 角 は変化 しないので,翼 の空力特 性 に及ぼす影 響は小 さい と思われ る.相 対風 の大 き さの変化 は迎 角の変化 を引き起 こす.そ の影響 は平行 な成分 をV回 転軸 か ら 上式 にお いて,風 速の複素 フー リエ成 分 をU,出 力 の複 素フー の距離 をrとす る と,V/rtOの 程度 であ る.一 方,風 車の 出力 は リエ成分 をPで 表 してい る.ま た,添 え字R,Iは それ ぞれ実数 主に翼端領域 が担 ってお り,ま た水平軸風 車では周速 比 は通 常 項 と虚数項 を表 してい る.結 果 をFig.9に 示す.図 中の点線 は 4 7程 度で ある.そ のた め,翼 端領 域ではV/rtUの値 は小 さい 数値計算 よ り求めた位 相遅れ を回帰 分析 した 曲線 である.曲 線 ので,平 行成 分が風車 の特性 に及ぼす影 響 も小 さい と期待 され の式 は次 のよ うにな った. る.こ の推論 を検証す るた め風洞 実験 を行 った. 5-1試 験 装置 の 概 要 43変 動風下の平均 出力 風速 が変化 してい る ときの風 車の 性能 を平均 出力で評 価す る. 平均す る時間をTと す ると平均 出力PAは 風 向を変 化 させ るため,風 洞内に設置 した風車 の支柱 を揺動 運動 させ た.実 験 装置の概略図 をFig.10に示 す.風 車 の支柱 は風 洞 の床 下にあ る回転 可能 な軸 受け① に取 り付 けた.サ ー ボモー タに より回転す る軸② を配置 し,軸 ①② の間 を長 方形の連 結板 で接続 した.軸 ② の回転 によ り軸① は揺動 し,軸 ① に取 り付 け た風 車は風向 を中心 と して左右 に首振 り運動 をす る.連 結板 の と書 ける. 風 速を平均値UAと 変動 値uに 分離 して式(39)に代入す る と, 取 りっ け位 置を調整 し,首 振 り運動 の最大角度 がお よそ50 にな るよ うに した.風 車の軸 とポテ ンシ ョを取 り付 けたプー リの間 をワイヤーで結び,首 振 り運動 によ るワイヤ ーの移動 をポテ ン シ ョの抵抗変化 として捉 え るこ とに よ り,ヨ ー角 φの変化 を測 定 した とな る.式(40)に式(36)を代入 して一周期 に渡って平均 する こと に よ り,風速が正弦 振動 してい る時 の平 均出力 を推定で きる10 風 速の変動成 分 と出力係数 との相 関を無視す る近似 をす ると, 風速変 動 の1次 の項 は零 になるので,出 力は次式で推 定で き る ことになる. 風 速変動 の分散 σ2は正 であるので,変 動風 時の出力が平均風 速 が同 じ定常風時の 出力 よ り増 加す るか あるいは減少す るか は, 平均 出力係 数CwAに 依 存す るこ とになる. 5風 向 変 化 に 対 す る振 る 舞 い 垂 直軸風 車では風向変化 を考慮 しな くて 良いが,水 平軸 風車 では風向 が変化す るこ とに よ り風 車の翼面 に対 して斜 めか ら風 が当た るよ うにな る.こ のた め,水 平軸風車 では風向変化 に よ Fig10. Schematic diagram ofthe experimental setup.
9 Fig.13 The ratio of the tip speed ratio of maximum power to that at yaw angle (I) for each yaw angle. Reference data is in ref (11), (12). Dashed lineiscoscb. Fig.11 Torque and power coefficients of the wind turbine used in the wind tunnel experiment Fig.12 The ratio of power to the power at for each yaw angle. Reference data is in ref (10, (12). Dashed line is cos.
10 Fig. 14 The relation between power and yaw angle for (a) T=420[s], (b) T60[s], (c) T=30[s], (d) T 20[s], (e) T=10[s], (f) T5[s]. The characteristic fi equencyf41.27[hz]. The figures are drawn for one period.
11 Fig.15 Phase delay vs. frequency ratio. Triangle marks ar calculated by the regression equation (38).
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